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axiom 20170501-4
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  • area: main
  • in suites: bullseye, buster, sid
  • size: 1,048,504 kB
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file content (5003 lines) | stat: -rw-r--r-- 2,529,422 bytes parent folder | download | duplicates (5)
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(2456493 . 3485789244)       
(-18 A S) 
((|constructor| (NIL "One-dimensional-array aggregates serves as models for one-dimensional arrays. Categorically,{} these aggregates are finite linear aggregates with the \\spadatt{shallowlyMutable} property,{} that is,{} any component of the array may be changed without affecting the identity of the overall array. Array data structures are typically represented by a fixed area in storage and cannot efficiently grow or shrink on demand as can list structures (see however \\spadtype{FlexibleArray} for a data structure which is a cross between a list and an array). Iteration over,{} and access to,{} elements of arrays is extremely fast (and often can be optimized to open-code). Insertion and deletion however is generally slow since an entirely new data structure must be created for the result."))) 
NIL 
NIL 
(-19 S) 
((|constructor| (NIL "One-dimensional-array aggregates serves as models for one-dimensional arrays. Categorically,{} these aggregates are finite linear aggregates with the \\spadatt{shallowlyMutable} property,{} that is,{} any component of the array may be changed without affecting the identity of the overall array. Array data structures are typically represented by a fixed area in storage and cannot efficiently grow or shrink on demand as can list structures (see however \\spadtype{FlexibleArray} for a data structure which is a cross between a list and an array). Iteration over,{} and access to,{} elements of arrays is extremely fast (and often can be optimized to open-code). Insertion and deletion however is generally slow since an entirely new data structure must be created for the result."))) 
((-4506 . T) (-4505 . T) (-3576 . T)) 
NIL 
(-20 S) 
((|constructor| (NIL "The class of abelian groups,{} \\spadignore{i.e.} additive monoids where each element has an additive inverse. \\blankline Axioms\\spad{\\br} \\tab{5}\\spad{-(-x) = x}\\spad{\\br} \\tab{5}\\spad{x+(-x) = 0}")) (* (($ (|Integer|) $) "\\spad{n*x} is the product of \\spad{x} by the integer \\spad{n}.")) (- (($ $ $) "\\spad{x-y} is the difference of \\spad{x} and \\spad{y} \\spadignore{i.e.} \\spad{x + (-y)}.") (($ $) "\\spad{-x} is the additive inverse of \\spad{x}."))) 
NIL 
NIL 
(-21) 
((|constructor| (NIL "The class of abelian groups,{} \\spadignore{i.e.} additive monoids where each element has an additive inverse. \\blankline Axioms\\spad{\\br} \\tab{5}\\spad{-(-x) = x}\\spad{\\br} \\tab{5}\\spad{x+(-x) = 0}")) (* (($ (|Integer|) $) "\\spad{n*x} is the product of \\spad{x} by the integer \\spad{n}.")) (- (($ $ $) "\\spad{x-y} is the difference of \\spad{x} and \\spad{y} \\spadignore{i.e.} \\spad{x + (-y)}.") (($ $) "\\spad{-x} is the additive inverse of \\spad{x}."))) 
NIL 
NIL 
(-22 S) 
((|constructor| (NIL "The class of multiplicative monoids,{} \\spadignore{i.e.} semigroups with an additive identity element. \\blankline Axioms\\spad{\\br} \\tab{5}\\spad{leftIdentity(\"+\":(\\%,{}\\%)->\\%,{}0)}\\tab{5}\\spad{ 0+x=x }\\spad{\\br} \\tab{5}\\spad{rightIdentity(\"+\":(\\%,{}\\%)->\\%,{}0)}\\tab{4}\\spad{ x+0=x }")) (* (($ (|NonNegativeInteger|) $) "\\spad{n * x} is left-multiplication by a non negative integer")) (|zero?| (((|Boolean|) $) "\\spad{zero?(x)} tests if \\spad{x} is equal to 0.")) (|sample| (($) "\\spad{sample yields} a value of type \\%")) ((|Zero|) (($) "0 is the additive identity element."))) 
NIL 
NIL 
(-23) 
((|constructor| (NIL "The class of multiplicative monoids,{} \\spadignore{i.e.} semigroups with an additive identity element. \\blankline Axioms\\spad{\\br} \\tab{5}\\spad{leftIdentity(\"+\":(\\%,{}\\%)->\\%,{}0)}\\tab{5}\\spad{ 0+x=x }\\spad{\\br} \\tab{5}\\spad{rightIdentity(\"+\":(\\%,{}\\%)->\\%,{}0)}\\tab{4}\\spad{ x+0=x }")) (* (($ (|NonNegativeInteger|) $) "\\spad{n * x} is left-multiplication by a non negative integer")) (|zero?| (((|Boolean|) $) "\\spad{zero?(x)} tests if \\spad{x} is equal to 0.")) (|sample| (($) "\\spad{sample yields} a value of type \\%")) ((|Zero|) (($) "0 is the additive identity element."))) 
NIL 
NIL 
(-24 S) 
((|constructor| (NIL "The class of all additive (commutative) semigroups,{} \\spadignore{i.e.} a set with a commutative and associative operation \\spadop{+}. \\blankline Axioms\\spad{\\br} \\tab{5}\\spad{associative(\"+\":(\\%,{}\\%)->\\%)}\\tab{5}\\spad{ (x+y)+z = x+(y+z) }\\spad{\\br} \\tab{6}\\spad{commutative(\"+\":(\\%,{}\\%)->\\%)}\\tab{5}\\spad{ x+y = y+x }")) (* (($ (|PositiveInteger|) $) "\\spad{n*x} computes the left-multiplication of \\spad{x} by the positive integer \\spad{n}. This is equivalent to adding \\spad{x} to itself \\spad{n} times.")) (+ (($ $ $) "\\spad{x+y} computes the sum of \\spad{x} and \\spad{y}."))) 
NIL 
NIL 
(-25) 
((|constructor| (NIL "The class of all additive (commutative) semigroups,{} \\spadignore{i.e.} a set with a commutative and associative operation \\spadop{+}. \\blankline Axioms\\spad{\\br} \\tab{5}\\spad{associative(\"+\":(\\%,{}\\%)->\\%)}\\tab{5}\\spad{ (x+y)+z = x+(y+z) }\\spad{\\br} \\tab{6}\\spad{commutative(\"+\":(\\%,{}\\%)->\\%)}\\tab{5}\\spad{ x+y = y+x }")) (* (($ (|PositiveInteger|) $) "\\spad{n*x} computes the left-multiplication of \\spad{x} by the positive integer \\spad{n}. This is equivalent to adding \\spad{x} to itself \\spad{n} times.")) (+ (($ $ $) "\\spad{x+y} computes the sum of \\spad{x} and \\spad{y}."))) 
NIL 
NIL 
(-26 S) 
((|constructor| (NIL "Model for algebraically closed fields.")) (|zerosOf| (((|List| $) (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\indented{1}{zerosOf(\\spad{p},{} \\spad{y}) returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}.} \\indented{1}{The \\spad{yi}\\spad{'s} are expressed in radicals if possible,{} and otherwise} \\indented{1}{as implicit algebraic quantities} \\indented{1}{which display as \\spad{'yi}.} \\indented{1}{The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter} \\indented{1}{to respective root values.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} zerosOf(a,{}\\spad{x})") (((|List| $) (|SparseUnivariatePolynomial| $)) "\\indented{1}{zerosOf(\\spad{p}) returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}.} \\indented{1}{The \\spad{yi}\\spad{'s} are expressed in radicals if possible,{} and otherwise} \\indented{1}{as implicit algebraic quantities.} \\indented{1}{The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter} \\indented{1}{to respective root values.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} zerosOf(a)") (((|List| $) (|Polynomial| $)) "\\indented{1}{zerosOf(\\spad{p}) returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}.} \\indented{1}{The \\spad{yi}\\spad{'s} are expressed in radicals if possible.} \\indented{1}{Otherwise they are implicit algebraic quantities.} \\indented{1}{The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter} \\indented{1}{to respective root values.} \\indented{1}{Error: if \\spad{p} has more than one variable \\spad{y}.} \\blankline \\spad{X} a:Polynomial(Integer)\\spad{:=}-3*x^2+2*x-13 \\spad{X} zerosOf(a)")) (|zeroOf| (($ (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\indented{1}{zeroOf(\\spad{p},{} \\spad{y}) returns \\spad{y} such that \\spad{p(y) = 0};} \\indented{1}{if possible,{} \\spad{y} is expressed in terms of radicals.} \\indented{1}{Otherwise it is an implicit algebraic quantity which} \\indented{1}{displays as \\spad{'y}.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} zeroOf(a,{}\\spad{x})") (($ (|SparseUnivariatePolynomial| $)) "\\indented{1}{zeroOf(\\spad{p}) returns \\spad{y} such that \\spad{p(y) = 0};} \\indented{1}{if possible,{} \\spad{y} is expressed in terms of radicals.} \\indented{1}{Otherwise it is an implicit algebraic quantity.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} zeroOf(a)") (($ (|Polynomial| $)) "\\indented{1}{zeroOf(\\spad{p}) returns \\spad{y} such that \\spad{p(y) = 0}.} \\indented{1}{If possible,{} \\spad{y} is expressed in terms of radicals.} \\indented{1}{Otherwise it is an implicit algebraic quantity.} \\indented{1}{Error: if \\spad{p} has more than one variable \\spad{y}.} \\blankline \\spad{X} a:Polynomial(Integer)\\spad{:=}-3*x^2+2*x-13 \\spad{X} zeroOf(a)")) (|rootsOf| (((|List| $) (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\indented{1}{rootsOf(\\spad{p},{} \\spad{y}) returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0};} \\indented{1}{The returned roots display as \\spad{'y1},{}...,{}\\spad{'yn}.} \\indented{1}{Note that the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter} \\indented{1}{to respective root values.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} rootsOf(a,{}\\spad{x})") (((|List| $) (|SparseUnivariatePolynomial| $)) "\\indented{1}{rootsOf(\\spad{p}) returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}.} \\indented{1}{Note that the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter} \\indented{1}{to respective root values.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} rootsOf(a)") (((|List| $) (|Polynomial| $)) "\\indented{1}{rootsOf(\\spad{p}) returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}.} \\indented{1}{Note that the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the} \\indented{1}{interpreter to respective root values.} \\indented{1}{Error: if \\spad{p} has more than one variable \\spad{y}.} \\blankline \\spad{X} a:Polynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} rootsOf(a)")) (|rootOf| (($ (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\indented{1}{rootOf(\\spad{p},{} \\spad{y}) returns \\spad{y} such that \\spad{p(y) = 0}.} \\indented{1}{The object returned displays as \\spad{'y}.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} rootOf(a,{}\\spad{x})") (($ (|SparseUnivariatePolynomial| $)) "\\indented{1}{rootOf(\\spad{p}) returns \\spad{y} such that \\spad{p(y) = 0}.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} rootOf(a)") (($ (|Polynomial| $)) "\\indented{1}{rootOf(\\spad{p}) returns \\spad{y} such that \\spad{p(y) = 0}.} \\indented{1}{Error: if \\spad{p} has more than one variable \\spad{y}.} \\blankline \\spad{X} a:Polynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} rootOf(a)"))) 
NIL 
NIL 
(-27) 
((|constructor| (NIL "Model for algebraically closed fields.")) (|zerosOf| (((|List| $) (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\indented{1}{zerosOf(\\spad{p},{} \\spad{y}) returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}.} \\indented{1}{The \\spad{yi}\\spad{'s} are expressed in radicals if possible,{} and otherwise} \\indented{1}{as implicit algebraic quantities} \\indented{1}{which display as \\spad{'yi}.} \\indented{1}{The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter} \\indented{1}{to respective root values.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} zerosOf(a,{}\\spad{x})") (((|List| $) (|SparseUnivariatePolynomial| $)) "\\indented{1}{zerosOf(\\spad{p}) returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}.} \\indented{1}{The \\spad{yi}\\spad{'s} are expressed in radicals if possible,{} and otherwise} \\indented{1}{as implicit algebraic quantities.} \\indented{1}{The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter} \\indented{1}{to respective root values.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} zerosOf(a)") (((|List| $) (|Polynomial| $)) "\\indented{1}{zerosOf(\\spad{p}) returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}.} \\indented{1}{The \\spad{yi}\\spad{'s} are expressed in radicals if possible.} \\indented{1}{Otherwise they are implicit algebraic quantities.} \\indented{1}{The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter} \\indented{1}{to respective root values.} \\indented{1}{Error: if \\spad{p} has more than one variable \\spad{y}.} \\blankline \\spad{X} a:Polynomial(Integer)\\spad{:=}-3*x^2+2*x-13 \\spad{X} zerosOf(a)")) (|zeroOf| (($ (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\indented{1}{zeroOf(\\spad{p},{} \\spad{y}) returns \\spad{y} such that \\spad{p(y) = 0};} \\indented{1}{if possible,{} \\spad{y} is expressed in terms of radicals.} \\indented{1}{Otherwise it is an implicit algebraic quantity which} \\indented{1}{displays as \\spad{'y}.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} zeroOf(a,{}\\spad{x})") (($ (|SparseUnivariatePolynomial| $)) "\\indented{1}{zeroOf(\\spad{p}) returns \\spad{y} such that \\spad{p(y) = 0};} \\indented{1}{if possible,{} \\spad{y} is expressed in terms of radicals.} \\indented{1}{Otherwise it is an implicit algebraic quantity.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} zeroOf(a)") (($ (|Polynomial| $)) "\\indented{1}{zeroOf(\\spad{p}) returns \\spad{y} such that \\spad{p(y) = 0}.} \\indented{1}{If possible,{} \\spad{y} is expressed in terms of radicals.} \\indented{1}{Otherwise it is an implicit algebraic quantity.} \\indented{1}{Error: if \\spad{p} has more than one variable \\spad{y}.} \\blankline \\spad{X} a:Polynomial(Integer)\\spad{:=}-3*x^2+2*x-13 \\spad{X} zeroOf(a)")) (|rootsOf| (((|List| $) (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\indented{1}{rootsOf(\\spad{p},{} \\spad{y}) returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0};} \\indented{1}{The returned roots display as \\spad{'y1},{}...,{}\\spad{'yn}.} \\indented{1}{Note that the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter} \\indented{1}{to respective root values.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} rootsOf(a,{}\\spad{x})") (((|List| $) (|SparseUnivariatePolynomial| $)) "\\indented{1}{rootsOf(\\spad{p}) returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}.} \\indented{1}{Note that the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter} \\indented{1}{to respective root values.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} rootsOf(a)") (((|List| $) (|Polynomial| $)) "\\indented{1}{rootsOf(\\spad{p}) returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}.} \\indented{1}{Note that the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the} \\indented{1}{interpreter to respective root values.} \\indented{1}{Error: if \\spad{p} has more than one variable \\spad{y}.} \\blankline \\spad{X} a:Polynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} rootsOf(a)")) (|rootOf| (($ (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\indented{1}{rootOf(\\spad{p},{} \\spad{y}) returns \\spad{y} such that \\spad{p(y) = 0}.} \\indented{1}{The object returned displays as \\spad{'y}.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} rootOf(a,{}\\spad{x})") (($ (|SparseUnivariatePolynomial| $)) "\\indented{1}{rootOf(\\spad{p}) returns \\spad{y} such that \\spad{p(y) = 0}.} \\blankline \\spad{X} a:SparseUnivariatePolynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} rootOf(a)") (($ (|Polynomial| $)) "\\indented{1}{rootOf(\\spad{p}) returns \\spad{y} such that \\spad{p(y) = 0}.} \\indented{1}{Error: if \\spad{p} has more than one variable \\spad{y}.} \\blankline \\spad{X} a:Polynomial(Integer)\\spad{:=}-3*x^3+2*x+13 \\spad{X} rootOf(a)"))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-28 S R) 
((|constructor| (NIL "Model for algebraically closed function spaces.")) (|zerosOf| (((|List| $) $ (|Symbol|)) "\\spad{zerosOf(p,{} y)} returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}. The \\spad{yi}\\spad{'s} are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities which display as \\spad{'yi}. The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values.") (((|List| $) $) "\\spad{zerosOf(p)} returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}. The \\spad{yi}\\spad{'s} are expressed in radicals if possible. The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable.")) (|zeroOf| (($ $ (|Symbol|)) "\\spad{zeroOf(p,{} y)} returns \\spad{y} such that \\spad{p(y) = 0}. The value \\spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity which displays as \\spad{'y}.") (($ $) "\\spad{zeroOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. The value \\spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity. Error: if \\spad{p} has more than one variable.")) (|rootsOf| (((|List| $) $ (|Symbol|)) "\\spad{rootsOf(p,{} y)} returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}; The returned roots display as \\spad{'y1},{}...,{}\\spad{'yn}. Note that the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values.") (((|List| $) $) "\\spad{rootsOf(p,{} y)} returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}; Note that the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable \\spad{y}.")) (|rootOf| (($ $ (|Symbol|)) "\\spad{rootOf(p,{}y)} returns \\spad{y} such that \\spad{p(y) = 0}. The object returned displays as \\spad{'y}.") (($ $) "\\spad{rootOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. Error: if \\spad{p} has more than one variable \\spad{y}."))) 
NIL 
NIL 
(-29 R) 
((|constructor| (NIL "Model for algebraically closed function spaces.")) (|zerosOf| (((|List| $) $ (|Symbol|)) "\\spad{zerosOf(p,{} y)} returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}. The \\spad{yi}\\spad{'s} are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities which display as \\spad{'yi}. The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values.") (((|List| $) $) "\\spad{zerosOf(p)} returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}. The \\spad{yi}\\spad{'s} are expressed in radicals if possible. The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable.")) (|zeroOf| (($ $ (|Symbol|)) "\\spad{zeroOf(p,{} y)} returns \\spad{y} such that \\spad{p(y) = 0}. The value \\spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity which displays as \\spad{'y}.") (($ $) "\\spad{zeroOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. The value \\spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity. Error: if \\spad{p} has more than one variable.")) (|rootsOf| (((|List| $) $ (|Symbol|)) "\\spad{rootsOf(p,{} y)} returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}; The returned roots display as \\spad{'y1},{}...,{}\\spad{'yn}. Note that the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values.") (((|List| $) $) "\\spad{rootsOf(p,{} y)} returns \\spad{[y1,{}...,{}yn]} such that \\spad{p(\\spad{yi}) = 0}; Note that the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable \\spad{y}.")) (|rootOf| (($ $ (|Symbol|)) "\\spad{rootOf(p,{}y)} returns \\spad{y} such that \\spad{p(y) = 0}. The object returned displays as \\spad{'y}.") (($ $) "\\spad{rootOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. Error: if \\spad{p} has more than one variable \\spad{y}."))) 
((-4502 . T) (-4500 . T) (-4499 . T) ((-4507 "*") . T) (-4498 . T) (-4503 . T) (-4497 . T) (-3576 . T)) 
NIL 
(-30) 
((|constructor| (NIL "Plot a NON-SINGULAR plane algebraic curve \\spad{p}(\\spad{x},{}\\spad{y}) = 0.")) (|refine| (($ $ (|DoubleFloat|)) "\\indented{1}{refine(\\spad{p},{}\\spad{x}) is not documented} \\blankline \\spad{X} sketch:=makeSketch(x+y,{}\\spad{x},{}\\spad{y},{}\\spad{-1/2}..1/2,{}\\spad{-1/2}..1/2)\\$ACPLOT \\spad{X} refined:=refine(sketch,{}0.1)")) (|makeSketch| (($ (|Polynomial| (|Integer|)) (|Symbol|) (|Symbol|) (|Segment| (|Fraction| (|Integer|))) (|Segment| (|Fraction| (|Integer|)))) "\\indented{1}{makeSketch(\\spad{p},{}\\spad{x},{}\\spad{y},{}a..\\spad{b},{}\\spad{c}..\\spad{d}) creates an ACPLOT of the} \\indented{1}{curve \\spad{p = 0} in the region a \\spad{<=} \\spad{x} \\spad{<=} \\spad{b},{} \\spad{c} \\spad{<=} \\spad{y} \\spad{<=} \\spad{d}.} \\indented{1}{More specifically,{} 'makeSketch' plots a non-singular algebraic curve} \\indented{1}{\\spad{p = 0} in an rectangular region xMin \\spad{<=} \\spad{x} \\spad{<=} xMax,{}} \\indented{1}{yMin \\spad{<=} \\spad{y} \\spad{<=} yMax. The user inputs} \\indented{1}{\\spad{makeSketch(p,{}x,{}y,{}xMin..xMax,{}yMin..yMax)}.} \\indented{1}{Here \\spad{p} is a polynomial in the variables \\spad{x} and \\spad{y} with} \\indented{1}{integer coefficients (\\spad{p} belongs to the domain} \\indented{1}{\\spad{Polynomial Integer}). The case} \\indented{1}{where \\spad{p} is a polynomial in only one of the variables is} \\indented{1}{allowed.\\space{2}The variables \\spad{x} and \\spad{y} are input to specify the} \\indented{1}{the coordinate axes.\\space{2}The horizontal axis is the \\spad{x}-axis and} \\indented{1}{the vertical axis is the \\spad{y}-axis.\\space{2}The rational numbers} \\indented{1}{xMin,{}...,{}yMax specify the boundaries of the region in} \\indented{1}{which the curve is to be plotted.} \\blankline \\spad{X} makeSketch(x+y,{}\\spad{x},{}\\spad{y},{}\\spad{-1/2}..1/2,{}\\spad{-1/2}..1/2)\\$ACPLOT"))) 
NIL 
NIL 
(-31 K |symb| |PolyRing| E |ProjPt|) 
((|constructor| (NIL "The following is part of the PAFF package")) (|affineRationalPoints| (((|List| |#5|) |#3| (|PositiveInteger|)) "\\axiom{rationalPoints(\\spad{f},{}\\spad{d})} returns all points on the curve \\axiom{\\spad{f}} in the extension of the ground field of degree \\axiom{\\spad{d}}. For \\axiom{\\spad{d} > 1} this only works if \\axiom{\\spad{K}} is a \\axiomType{LocallyAlgebraicallyClosedField}"))) 
NIL 
NIL 
(-32 K |symb| |PolyRing| E |ProjPt|) 
((|constructor| (NIL "The following is part of the PAFF package"))) 
NIL 
NIL 
(-33 K) 
((|constructor| (NIL "The following is all the categories and domains related to projective space and part of the PAFF package"))) 
NIL 
NIL 
(-34 K) 
((|constructor| (NIL "The following is all the categories and domains related to projective space and part of the PAFF package"))) 
NIL 
NIL 
(-35 -3780 K) 
((|constructor| (NIL "The following is all the categories and domains related to projective space and part of the PAFF package"))) 
NIL 
NIL 
(-36 R -2262) 
((|constructor| (NIL "This package provides algebraic functions over an integral domain.")) (|iroot| ((|#2| |#1| (|Integer|)) "\\spad{iroot(p,{} n)} should be a non-exported function.")) (|definingPolynomial| ((|#2| |#2|) "\\spad{definingPolynomial(f)} returns the defining polynomial of \\spad{f} as an element of \\spad{F}. Error: if \\spad{f} is not a kernel.")) (|minPoly| (((|SparseUnivariatePolynomial| |#2|) (|Kernel| |#2|)) "\\spad{minPoly(k)} returns the defining polynomial of \\spad{k}.")) (** ((|#2| |#2| (|Fraction| (|Integer|))) "\\spad{x ** q} is \\spad{x} raised to the rational power \\spad{q}.")) (|droot| (((|OutputForm|) (|List| |#2|)) "\\spad{droot(l)} should be a non-exported function.")) (|inrootof| ((|#2| (|SparseUnivariatePolynomial| |#2|) |#2|) "\\spad{inrootof(p,{} x)} should be a non-exported function.")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} is \\spad{true} if \\spad{op} is an algebraic operator,{} that is,{} an \\spad{n}th root or implicit algebraic operator.")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\spad{F}. Error: if \\spad{op} is not an algebraic operator,{} that is,{} an \\spad{n}th root or implicit algebraic operator.")) (|rootOf| ((|#2| (|SparseUnivariatePolynomial| |#2|) (|Symbol|)) "\\spad{rootOf(p,{} y)} returns \\spad{y} such that \\spad{p(y) = 0}. The object returned displays as \\spad{'y}."))) 
NIL 
((|HasCategory| |#1| (LIST (QUOTE -1029) (QUOTE (-560))))) 
(-37 K) 
((|constructor| (NIL "The following is all the categories and domains related to projective space and part of the PAFF package")) (|pointValue| (((|List| |#1|) $) "\\spad{pointValue returns} the coordinates of the point or of the point of origin that represent an infinitly close point")) (|setelt| ((|#1| $ (|Integer|) |#1|) "\\spad{setelt sets} the value of a specified coordinates")) (|elt| ((|#1| $ (|Integer|)) "\\spad{elt returns} the value of a specified coordinates")) (|list| (((|List| |#1|) $) "\\spad{list returns} the list of the coordinates")) (|rational?| (((|Boolean|) $) "\\spad{rational?(p)} test if the point is rational according to the characteristic of the ground field.") (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{rational?(p,{}n)} test if the point is rational according to \\spad{n}.")) (|removeConjugate| (((|List| $) (|List| $)) "\\spad{removeConjugate(lp)} returns removeConjugate(\\spad{lp},{}\\spad{n}) where \\spad{n} is the characteristic of the ground field.") (((|List| $) (|List| $) (|NonNegativeInteger|)) "\\spad{removeConjugate(lp,{}n)} returns a list of points such that no points in the list is the conjugate (according to \\spad{n}) of another point.")) (|conjugate| (($ $) "\\spad{conjugate(p)} returns conjugate(\\spad{p},{}\\spad{n}) where \\spad{n} is the characteristic of the ground field.") (($ $ (|NonNegativeInteger|)) "\\spad{conjugate(p,{}n)} returns p**n,{} that is all the coordinates of \\spad{p} to the power of \\spad{n}")) (|orbit| (((|List| $) $ (|NonNegativeInteger|)) "\\spad{orbit(p,{}n)} returns the orbit of the point \\spad{p} according to \\spad{n},{} that is orbit(\\spad{p},{}\\spad{n}) = \\spad{\\{} \\spad{p},{} p**n,{} \\spad{p**}(\\spad{n**2}),{} \\spad{p**}(\\spad{n**3}),{} ..... \\spad{\\}}") (((|List| $) $) "\\spad{orbit(p)} returns the orbit of the point \\spad{p} according to the characteristic of \\spad{K},{} that is,{} for \\spad{q=} char \\spad{K},{} orbit(\\spad{p}) = \\spad{\\{} \\spad{p},{} p**q,{} \\spad{p**}(\\spad{q**2}),{} \\spad{p**}(\\spad{q**3}),{} ..... \\spad{\\}}")) (|coerce| (($ (|List| |#1|)) "\\spad{coerce a} list of \\spad{K} to a affine point.")) (|affinePoint| (($ (|List| |#1|)) "\\spad{affinePoint creates} a affine point from a list"))) 
NIL 
NIL 
(-38 S) 
((|constructor| (NIL "The notion of aggregate serves to model any data structure aggregate,{} designating any collection of objects,{} with heterogenous or homogeneous members,{} with a finite or infinite number of members,{} explicitly or implicitly represented. An aggregate can in principle represent everything from a string of characters to abstract sets such as \"the set of \\spad{x} satisfying relation \\spad{r}(\\spad{x})\" An attribute \\spadatt{finiteAggregate} is used to assert that a domain has a finite number of elements.")) (|#| (((|NonNegativeInteger|) $) "\\spad{\\# u} returns the number of items in \\spad{u}.")) (|sample| (($) "\\spad{sample yields} a value of type \\%")) (|size?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{size?(u,{}n)} tests if \\spad{u} has exactly \\spad{n} elements.")) (|more?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{more?(u,{}n)} tests if \\spad{u} has greater than \\spad{n} elements.")) (|less?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{less?(u,{}n)} tests if \\spad{u} has less than \\spad{n} elements.")) (|empty?| (((|Boolean|) $) "\\spad{empty?(u)} tests if \\spad{u} has 0 elements.")) (|empty| (($) "\\spad{empty()}\\$\\spad{D} creates an aggregate of type \\spad{D} with 0 elements. Note that The \\$\\spad{D} can be dropped if understood by context,{} \\spadignore{e.g.} \\axiom{u: \\spad{D} \\spad{:=} empty()}.")) (|copy| (($ $) "\\spad{copy(u)} returns a top-level (non-recursive) copy of \\spad{u}. Note that for collections,{} \\axiom{copy(\\spad{u}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{u}]}.")) (|eq?| (((|Boolean|) $ $) "\\spad{eq?(u,{}v)} tests if \\spad{u} and \\spad{v} are same objects."))) 
NIL 
((|HasAttribute| |#1| (QUOTE -4505))) 
(-39) 
((|constructor| (NIL "The notion of aggregate serves to model any data structure aggregate,{} designating any collection of objects,{} with heterogenous or homogeneous members,{} with a finite or infinite number of members,{} explicitly or implicitly represented. An aggregate can in principle represent everything from a string of characters to abstract sets such as \"the set of \\spad{x} satisfying relation \\spad{r}(\\spad{x})\" An attribute \\spadatt{finiteAggregate} is used to assert that a domain has a finite number of elements.")) (|#| (((|NonNegativeInteger|) $) "\\spad{\\# u} returns the number of items in \\spad{u}.")) (|sample| (($) "\\spad{sample yields} a value of type \\%")) (|size?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{size?(u,{}n)} tests if \\spad{u} has exactly \\spad{n} elements.")) (|more?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{more?(u,{}n)} tests if \\spad{u} has greater than \\spad{n} elements.")) (|less?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{less?(u,{}n)} tests if \\spad{u} has less than \\spad{n} elements.")) (|empty?| (((|Boolean|) $) "\\spad{empty?(u)} tests if \\spad{u} has 0 elements.")) (|empty| (($) "\\spad{empty()}\\$\\spad{D} creates an aggregate of type \\spad{D} with 0 elements. Note that The \\$\\spad{D} can be dropped if understood by context,{} \\spadignore{e.g.} \\axiom{u: \\spad{D} \\spad{:=} empty()}.")) (|copy| (($ $) "\\spad{copy(u)} returns a top-level (non-recursive) copy of \\spad{u}. Note that for collections,{} \\axiom{copy(\\spad{u}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{u}]}.")) (|eq?| (((|Boolean|) $ $) "\\spad{eq?(u,{}v)} tests if \\spad{u} and \\spad{v} are same objects."))) 
((-3576 . T)) 
NIL 
(-40) 
((|constructor| (NIL "Category for the inverse hyperbolic trigonometric functions.")) (|atanh| (($ $) "\\spad{atanh(x)} returns the hyperbolic arc-tangent of \\spad{x}.")) (|asinh| (($ $) "\\spad{asinh(x)} returns the hyperbolic arc-sine of \\spad{x}.")) (|asech| (($ $) "\\spad{asech(x)} returns the hyperbolic arc-secant of \\spad{x}.")) (|acsch| (($ $) "\\spad{acsch(x)} returns the hyperbolic arc-cosecant of \\spad{x}.")) (|acoth| (($ $) "\\spad{acoth(x)} returns the hyperbolic arc-cotangent of \\spad{x}.")) (|acosh| (($ $) "\\spad{acosh(x)} returns the hyperbolic arc-cosine of \\spad{x}."))) 
NIL 
NIL 
(-41 |Key| |Entry|) 
((|constructor| (NIL "An association list is a list of key entry pairs which may be viewed as a table. It is a poor mans version of a table: searching for a key is a linear operation.")) (|assoc| (((|Union| (|Record| (|:| |key| |#1|) (|:| |entry| |#2|)) "failed") |#1| $) "\\spad{assoc(k,{}u)} returns the element \\spad{x} in association list \\spad{u} stored with key \\spad{k},{} or \"failed\" if \\spad{u} has no key \\spad{k}."))) 
((-4505 . T) (-4506 . T) (-3576 . T)) 
NIL 
(-42 S R) 
((|constructor| (NIL "The category of associative algebras (modules which are themselves rings). \\blankline Axioms\\spad{\\br} \\tab{5}\\spad{(b+c)::\\% = (b::\\%) + (c::\\%)}\\spad{\\br} \\tab{5}\\spad{(b*c)::\\% = (b::\\%) * (c::\\%)}\\spad{\\br} \\tab{5}\\spad{(1::R)::\\% = 1::\\%}\\spad{\\br} \\tab{5}\\spad{b*x = (b::\\%)*x}\\spad{\\br} \\tab{5}\\spad{r*(a*b) = (r*a)*b = a*(r*b)}")) (|coerce| (($ |#2|) "\\spad{coerce(r)} maps the ring element \\spad{r} to a member of the algebra."))) 
NIL 
NIL 
(-43 R) 
((|constructor| (NIL "The category of associative algebras (modules which are themselves rings). \\blankline Axioms\\spad{\\br} \\tab{5}\\spad{(b+c)::\\% = (b::\\%) + (c::\\%)}\\spad{\\br} \\tab{5}\\spad{(b*c)::\\% = (b::\\%) * (c::\\%)}\\spad{\\br} \\tab{5}\\spad{(1::R)::\\% = 1::\\%}\\spad{\\br} \\tab{5}\\spad{b*x = (b::\\%)*x}\\spad{\\br} \\tab{5}\\spad{r*(a*b) = (r*a)*b = a*(r*b)}")) (|coerce| (($ |#1|) "\\spad{coerce(r)} maps the ring element \\spad{r} to a member of the algebra."))) 
((-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-44 UP) 
((|constructor| (NIL "Factorization of univariate polynomials with coefficients in \\spadtype{AlgebraicNumber}.")) (|doublyTransitive?| (((|Boolean|) |#1|) "\\spad{doublyTransitive?(p)} is \\spad{true} if \\spad{p} is irreducible over over the field \\spad{K} generated by its coefficients,{} and if \\spad{p(X) / (X - a)} is irreducible over \\spad{K(a)} where \\spad{p(a) = 0}.")) (|split| (((|Factored| |#1|) |#1|) "\\spad{split(p)} returns a prime factorisation of \\spad{p} over its splitting field.")) (|factor| (((|Factored| |#1|) |#1|) "\\spad{factor(p)} returns a prime factorisation of \\spad{p} over the field generated by its coefficients.") (((|Factored| |#1|) |#1| (|List| (|AlgebraicNumber|))) "\\spad{factor(p,{} [a1,{}...,{}an])} returns a prime factorisation of \\spad{p} over the field generated by its coefficients and \\spad{a1},{}...,{}an."))) 
NIL 
NIL 
(-45 -2262 UP UPUP -3542) 
((|constructor| (NIL "Function field defined by \\spad{f}(\\spad{x},{} \\spad{y}) = 0.")) (|knownInfBasis| (((|Void|) (|NonNegativeInteger|)) "\\spad{knownInfBasis(n)} is not documented"))) 
((-4498 |has| (-403 |#2|) (-359)) (-4503 |has| (-403 |#2|) (-359)) (-4497 |has| (-403 |#2|) (-359)) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| (-403 |#2|) (QUOTE (-146))) (|HasCategory| (-403 |#2|) (QUOTE (-148))) (|HasCategory| (-403 |#2|) (QUOTE (-344))) (|HasCategory| (-403 |#2|) (QUOTE (-359))) (-3322 (|HasCategory| (-403 |#2|) (QUOTE (-359))) (|HasCategory| (-403 |#2|) (QUOTE (-344)))) (|HasCategory| (-403 |#2|) (QUOTE (-364))) (|HasCategory| (-403 |#2|) (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| (-403 |#2|) (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| (-403 |#2|) (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-364))) (-3322 (|HasCategory| (-403 |#2|) (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| (-403 |#2|) (QUOTE (-359)))) (-12 (|HasCategory| (-403 |#2|) (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| (-403 |#2|) (QUOTE (-359)))) (-3322 (-12 (|HasCategory| (-403 |#2|) (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| (-403 |#2|) (QUOTE (-359)))) (-12 (|HasCategory| (-403 |#2|) (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| (-403 |#2|) (QUOTE (-344))))) (-12 (|HasCategory| (-403 |#2|) (QUOTE (-221))) (|HasCategory| (-403 |#2|) (QUOTE (-359)))) (-3322 (-12 (|HasCategory| (-403 |#2|) (QUOTE (-221))) (|HasCategory| (-403 |#2|) (QUOTE (-359)))) (|HasCategory| (-403 |#2|) (QUOTE (-344))))) 
(-46 R -2262) 
((|constructor| (NIL "AlgebraicManipulations provides functions to simplify and expand expressions involving algebraic operators.")) (|rootKerSimp| ((|#2| (|BasicOperator|) |#2| (|NonNegativeInteger|)) "\\spad{rootKerSimp(op,{}f,{}n)} should be local but conditional.")) (|rootSimp| ((|#2| |#2|) "\\spad{rootSimp(f)} transforms every radical of the form \\spad{(a * b**(q*n+r))**(1/n)} appearing in \\spad{f} into \\spad{b**q * (a * b**r)**(1/n)}. This transformation is not in general valid for all complex numbers \\spad{b}.")) (|rootProduct| ((|#2| |#2|) "\\spad{rootProduct(f)} combines every product of the form \\spad{(a**(1/n))**m * (a**(1/s))**t} into a single power of a root of \\spad{a},{} and transforms every radical power of the form \\spad{(a**(1/n))**m} into a simpler form.")) (|rootPower| ((|#2| |#2|) "\\spad{rootPower(f)} transforms every radical power of the form \\spad{(a**(1/n))**m} into a simpler form if \\spad{m} and \\spad{n} have a common factor.")) (|ratPoly| (((|SparseUnivariatePolynomial| |#2|) |#2|) "\\spad{ratPoly(f)} returns a polynomial \\spad{p} such that \\spad{p} has no algebraic coefficients,{} and \\spad{p(f) = 0}.")) (|ratDenom| ((|#2| |#2| (|List| (|Kernel| |#2|))) "\\spad{ratDenom(f,{} [a1,{}...,{}an])} removes the \\spad{ai}\\spad{'s} which are algebraic from the denominators in \\spad{f}.") ((|#2| |#2| (|List| |#2|)) "\\spad{ratDenom(f,{} [a1,{}...,{}an])} removes the \\spad{ai}\\spad{'s} which are algebraic kernels from the denominators in \\spad{f}.") ((|#2| |#2| |#2|) "\\spad{ratDenom(f,{} a)} removes \\spad{a} from the denominators in \\spad{f} if \\spad{a} is an algebraic kernel.") ((|#2| |#2|) "\\spad{ratDenom(f)} rationalizes the denominators appearing in \\spad{f} by moving all the algebraic quantities into the numerators.")) (|rootSplit| ((|#2| |#2|) "\\spad{rootSplit(f)} transforms every radical of the form \\spad{(a/b)**(1/n)} appearing in \\spad{f} into \\spad{a**(1/n) / b**(1/n)}. This transformation is not in general valid for all complex numbers \\spad{a} and \\spad{b}.")) (|coerce| (($ (|SparseMultivariatePolynomial| |#1| (|Kernel| $))) "\\spad{coerce(x)} \\undocumented")) (|denom| (((|SparseMultivariatePolynomial| |#1| (|Kernel| $)) $) "\\spad{denom(x)} \\undocumented")) (|numer| (((|SparseMultivariatePolynomial| |#1| (|Kernel| $)) $) "\\spad{numer(x)} \\undocumented"))) 
NIL 
((-12 (|HasCategory| |#1| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-447))) (|HasCategory| |#1| (QUOTE (-834))) (|HasCategory| |#2| (LIST (QUOTE -426) (|devaluate| |#1|))))) 
(-47 OV E P) 
((|constructor| (NIL "This package factors multivariate polynomials over the domain of \\spadtype{AlgebraicNumber} by allowing the user to specify a list of algebraic numbers generating the particular extension to factor over.")) (|factor| (((|Factored| (|SparseUnivariatePolynomial| |#3|)) (|SparseUnivariatePolynomial| |#3|) (|List| (|AlgebraicNumber|))) "\\spad{factor(p,{}lan)} factors the polynomial \\spad{p} over the extension generated by the algebraic numbers given by the list \\spad{lan}. \\spad{p} is presented as a univariate polynomial with multivariate coefficients.") (((|Factored| |#3|) |#3| (|List| (|AlgebraicNumber|))) "\\spad{factor(p,{}lan)} factors the polynomial \\spad{p} over the extension generated by the algebraic numbers given by the list \\spad{lan}."))) 
NIL 
NIL 
(-48 R A) 
((|constructor| (NIL "AlgebraPackage assembles a variety of useful functions for general algebras.")) (|basis| (((|Vector| |#2|) (|Vector| |#2|)) "\\spad{basis(va)} selects a basis from the elements of \\spad{va}.")) (|radicalOfLeftTraceForm| (((|List| |#2|)) "\\spad{radicalOfLeftTraceForm()} returns basis for null space of \\spad{leftTraceMatrix()},{} if the algebra is associative,{} alternative or a Jordan algebra,{} then this space equals the radical (maximal nil ideal) of the algebra.")) (|basisOfCentroid| (((|List| (|Matrix| |#1|))) "\\spad{basisOfCentroid()} returns a basis of the centroid,{} \\spadignore{i.e.} the endomorphism ring of \\spad{A} considered as \\spad{(A,{}A)}-bimodule.")) (|basisOfRightNucloid| (((|List| (|Matrix| |#1|))) "\\spad{basisOfRightNucloid()} returns a basis of the space of endomorphisms of \\spad{A} as left module. Note that right nucloid coincides with right nucleus if \\spad{A} has a unit.")) (|basisOfLeftNucloid| (((|List| (|Matrix| |#1|))) "\\spad{basisOfLeftNucloid()} returns a basis of the space of endomorphisms of \\spad{A} as right module. Note that left nucloid coincides with left nucleus if \\spad{A} has a unit.")) (|basisOfCenter| (((|List| |#2|)) "\\spad{basisOfCenter()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{commutator(x,{}a) = 0} and \\spad{associator(x,{}a,{}b) = associator(a,{}x,{}b) = associator(a,{}b,{}x) = 0} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfNucleus| (((|List| |#2|)) "\\spad{basisOfNucleus()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{associator(x,{}a,{}b) = associator(a,{}x,{}b) = associator(a,{}b,{}x) = 0} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfMiddleNucleus| (((|List| |#2|)) "\\spad{basisOfMiddleNucleus()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = associator(a,{}x,{}b)} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfRightNucleus| (((|List| |#2|)) "\\spad{basisOfRightNucleus()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = associator(a,{}b,{}x)} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfLeftNucleus| (((|List| |#2|)) "\\spad{basisOfLeftNucleus()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = associator(x,{}a,{}b)} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfRightAnnihilator| (((|List| |#2|) |#2|) "\\spad{basisOfRightAnnihilator(a)} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = a*x}.")) (|basisOfLeftAnnihilator| (((|List| |#2|) |#2|) "\\spad{basisOfLeftAnnihilator(a)} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = x*a}.")) (|basisOfCommutingElements| (((|List| |#2|)) "\\spad{basisOfCommutingElements()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = commutator(x,{}a)} for all \\spad{a} in \\spad{A}.")) (|biRank| (((|NonNegativeInteger|) |#2|) "\\spad{biRank(x)} determines the number of linearly independent elements in \\spad{x},{} \\spad{x*bi},{} \\spad{bi*x},{} \\spad{bi*x*bj},{} \\spad{i,{}j=1,{}...,{}n},{} where \\spad{b=[b1,{}...,{}bn]} is a basis. Note that if \\spad{A} has a unit,{} then doubleRank,{} weakBiRank,{} and biRank coincide.")) (|weakBiRank| (((|NonNegativeInteger|) |#2|) "\\spad{weakBiRank(x)} determines the number of linearly independent elements in the \\spad{bi*x*bj},{} \\spad{i,{}j=1,{}...,{}n},{} where \\spad{b=[b1,{}...,{}bn]} is a basis.")) (|doubleRank| (((|NonNegativeInteger|) |#2|) "\\spad{doubleRank(x)} determines the number of linearly independent elements in \\spad{b1*x},{}...,{}\\spad{x*bn},{} where \\spad{b=[b1,{}...,{}bn]} is a basis.")) (|rightRank| (((|NonNegativeInteger|) |#2|) "\\spad{rightRank(x)} determines the number of linearly independent elements in \\spad{b1*x},{}...,{}\\spad{bn*x},{} where \\spad{b=[b1,{}...,{}bn]} is a basis.")) (|leftRank| (((|NonNegativeInteger|) |#2|) "\\spad{leftRank(x)} determines the number of linearly independent elements in \\spad{x*b1},{}...,{}\\spad{x*bn},{} where \\spad{b=[b1,{}...,{}bn]} is a basis."))) 
NIL 
((|HasCategory| |#1| (QUOTE (-296)))) 
(-49 R |n| |ls| |gamma|) 
((|constructor| (NIL "AlgebraGivenByStructuralConstants implements finite rank algebras over a commutative ring,{} given by the structural constants \\spad{gamma} with respect to a fixed basis \\spad{[a1,{}..,{}an]},{} where \\spad{gamma} is an \\spad{n}-vector of \\spad{n} by \\spad{n} matrices \\spad{[(gammaijk) for k in 1..rank()]} defined by \\spad{\\spad{ai} * aj = gammaij1 * a1 + ... + gammaijn * an}. The symbols for the fixed basis have to be given as a list of symbols.")) (|coerce| (($ (|Vector| |#1|)) "\\spad{coerce(v)} converts a vector to a member of the algebra by forming a linear combination with the basis element. Note: the vector is assumed to have length equal to the dimension of the algebra."))) 
((-4502 |has| |#1| (-550)) (-4500 . T) (-4499 . T)) 
((|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-550)))) 
(-50 |Key| |Entry|) 
((|constructor| (NIL "\\spadtype{AssociationList} implements association lists. These may be viewed as lists of pairs where the first part is a key and the second is the stored value. For example,{} the key might be a string with a persons employee identification number and the value might be a record with personnel data."))) 
((-4505 . T) (-4506 . T)) 
((|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (QUOTE (-834))) (|HasCategory| |#1| (QUOTE (-834))) (|HasCategory| |#2| (QUOTE (-1082))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-1082)))) (|HasCategory| (-560) (QUOTE (-834))) (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (QUOTE (-1082))) (-3322 (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (QUOTE (-1082))) (|HasCategory| |#2| (QUOTE (-1082)))) (-3322 (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (QUOTE (-834))) (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (QUOTE (-1082))) (|HasCategory| |#2| (QUOTE (-1082)))) (-12 (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (LIST (QUOTE -298) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2286) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3071) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (QUOTE (-1082)))) (-3322 (-12 (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (LIST (QUOTE -298) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2286) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3071) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (QUOTE (-834)))) (-12 (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (LIST (QUOTE -298) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2286) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3071) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (QUOTE (-1082)))))) 
(-51 S R E) 
((|constructor| (NIL "Abelian monoid ring elements (not necessarily of finite support) of this ring are of the form formal SUM (r_i * e_i) where the r_i are coefficents and the e_i,{} elements of the ordered abelian monoid,{} are thought of as exponents or monomials. The monomials commute with each other,{} and with the coefficients (which themselves may or may not be commutative). See \\spadtype{FiniteAbelianMonoidRing} for the case of finite support a useful common model for polynomials and power series. Conceptually at least,{} only the non-zero terms are ever operated on.")) (/ (($ $ |#2|) "\\spad{p/c} divides \\spad{p} by the coefficient \\spad{c}.")) (|coefficient| ((|#2| $ |#3|) "\\spad{coefficient(p,{}e)} extracts the coefficient of the monomial with exponent \\spad{e} from polynomial \\spad{p},{} or returns zero if exponent is not present.")) (|reductum| (($ $) "\\spad{reductum(u)} returns \\spad{u} minus its leading monomial returns zero if handed the zero element.")) (|monomial| (($ |#2| |#3|) "\\spad{monomial(r,{}e)} makes a term from a coefficient \\spad{r} and an exponent \\spad{e}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(p)} tests if \\spad{p} is a single monomial.")) (|map| (($ (|Mapping| |#2| |#2|) $) "\\spad{map(fn,{}u)} maps function \\spad{fn} onto the coefficients of the non-zero monomials of \\spad{u}.")) (|degree| ((|#3| $) "\\spad{degree(p)} returns the maximum of the exponents of the terms of \\spad{p}.")) (|leadingMonomial| (($ $) "\\spad{leadingMonomial(p)} returns the monomial of \\spad{p} with the highest degree.")) (|leadingCoefficient| ((|#2| $) "\\spad{leadingCoefficient(p)} returns the coefficient highest degree term of \\spad{p}."))) 
NIL 
((|HasCategory| |#2| (LIST (QUOTE -43) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (QUOTE (-550))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-170))) (|HasCategory| |#2| (QUOTE (-359)))) 
(-52 R E) 
((|constructor| (NIL "Abelian monoid ring elements (not necessarily of finite support) of this ring are of the form formal SUM (r_i * e_i) where the r_i are coefficents and the e_i,{} elements of the ordered abelian monoid,{} are thought of as exponents or monomials. The monomials commute with each other,{} and with the coefficients (which themselves may or may not be commutative). See \\spadtype{FiniteAbelianMonoidRing} for the case of finite support a useful common model for polynomials and power series. Conceptually at least,{} only the non-zero terms are ever operated on.")) (/ (($ $ |#1|) "\\spad{p/c} divides \\spad{p} by the coefficient \\spad{c}.")) (|coefficient| ((|#1| $ |#2|) "\\spad{coefficient(p,{}e)} extracts the coefficient of the monomial with exponent \\spad{e} from polynomial \\spad{p},{} or returns zero if exponent is not present.")) (|reductum| (($ $) "\\spad{reductum(u)} returns \\spad{u} minus its leading monomial returns zero if handed the zero element.")) (|monomial| (($ |#1| |#2|) "\\spad{monomial(r,{}e)} makes a term from a coefficient \\spad{r} and an exponent \\spad{e}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(p)} tests if \\spad{p} is a single monomial.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(fn,{}u)} maps function \\spad{fn} onto the coefficients of the non-zero monomials of \\spad{u}.")) (|degree| ((|#2| $) "\\spad{degree(p)} returns the maximum of the exponents of the terms of \\spad{p}.")) (|leadingMonomial| (($ $) "\\spad{leadingMonomial(p)} returns the monomial of \\spad{p} with the highest degree.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(p)} returns the coefficient highest degree term of \\spad{p}."))) 
(((-4507 "*") |has| |#1| (-170)) (-4498 |has| |#1| (-550)) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-53) 
((|constructor| (NIL "Algebraic closure of the rational numbers,{} with mathematical =")) (|norm| (($ $ (|List| (|Kernel| $))) "\\spad{norm(f,{}l)} computes the norm of the algebraic number \\spad{f} with respect to the extension generated by kernels \\spad{l}") (($ $ (|Kernel| $)) "\\spad{norm(f,{}k)} computes the norm of the algebraic number \\spad{f} with respect to the extension generated by kernel \\spad{k}") (((|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $) (|List| (|Kernel| $))) "\\spad{norm(p,{}l)} computes the norm of the polynomial \\spad{p} with respect to the extension generated by kernels \\spad{l}") (((|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $) (|Kernel| $)) "\\spad{norm(p,{}k)} computes the norm of the polynomial \\spad{p} with respect to the extension generated by kernel \\spad{k}")) (|reduce| (($ $) "\\spad{reduce(f)} simplifies all the unreduced algebraic numbers present in \\spad{f} by applying their defining relations.")) (|denom| (((|SparseMultivariatePolynomial| (|Integer|) (|Kernel| $)) $) "\\spad{denom(f)} returns the denominator of \\spad{f} viewed as a polynomial in the kernels over \\spad{Z}.")) (|numer| (((|SparseMultivariatePolynomial| (|Integer|) (|Kernel| $)) $) "\\spad{numer(f)} returns the numerator of \\spad{f} viewed as a polynomial in the kernels over \\spad{Z}.")) (|coerce| (($ (|SparseMultivariatePolynomial| (|Integer|) (|Kernel| $))) "\\spad{coerce(p)} returns \\spad{p} viewed as an algebraic number."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| $ (QUOTE (-1039))) (|HasCategory| $ (LIST (QUOTE -1029) (QUOTE (-560))))) 
(-54) 
((|constructor| (NIL "This domain implements anonymous functions"))) 
NIL 
NIL 
(-55 R |lVar|) 
((|constructor| (NIL "The domain of antisymmetric polynomials.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,{}p)} changes each coefficient of \\spad{p} by the application of \\spad{f}.")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(p)} returns the homogeneous degree of \\spad{p}.")) (|retractable?| (((|Boolean|) $) "\\spad{retractable?(p)} tests if \\spad{p} is a 0-form,{} \\spadignore{i.e.} if degree(\\spad{p}) = 0.")) (|homogeneous?| (((|Boolean|) $) "\\spad{homogeneous?(p)} tests if all of the terms of \\spad{p} have the same degree.")) (|exp| (($ (|List| (|Integer|))) "\\spad{exp([i1,{}...in])} returns \\spad{u_1\\^{i_1} ... u_n\\^{i_n}}")) (|generator| (($ (|NonNegativeInteger|)) "\\spad{generator(n)} returns the \\spad{n}th multiplicative generator,{} a basis term.")) (|coefficient| ((|#1| $ $) "\\spad{coefficient(p,{}u)} returns the coefficient of the term in \\spad{p} containing the basis term \\spad{u} if such a term exists,{} and 0 otherwise. Error: if the second argument \\spad{u} is not a basis element.")) (|reductum| (($ $) "\\spad{reductum(p)},{} where \\spad{p} is an antisymmetric polynomial,{} returns \\spad{p} minus the leading term of \\spad{p} if \\spad{p} has at least two terms,{} and 0 otherwise.")) (|leadingBasisTerm| (($ $) "\\spad{leadingBasisTerm(p)} returns the leading basis term of antisymmetric polynomial \\spad{p}.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(p)} returns the leading coefficient of antisymmetric polynomial \\spad{p}."))) 
((-4502 . T)) 
NIL 
(-56 S) 
((|constructor| (NIL "\\spadtype{AnyFunctions1} implements several utility functions for working with \\spadtype{Any}. These functions are used to go back and forth between objects of \\spadtype{Any} and objects of other types.")) (|retract| ((|#1| (|Any|)) "\\spad{retract(a)} tries to convert \\spad{a} into an object of type \\spad{S}. If possible,{} it returns the object. Error: if no such retraction is possible.")) (|retractable?| (((|Boolean|) (|Any|)) "\\spad{retractable?(a)} tests if \\spad{a} can be converted into an object of type \\spad{S}.")) (|retractIfCan| (((|Union| |#1| "failed") (|Any|)) "\\spad{retractIfCan(a)} tries change \\spad{a} into an object of type \\spad{S}. If it can,{} then such an object is returned. Otherwise,{} \"failed\" is returned.")) (|coerce| (((|Any|) |#1|) "\\spad{coerce(s)} creates an object of \\spadtype{Any} from the object \\spad{s} of type \\spad{S}."))) 
NIL 
NIL 
(-57) 
((|constructor| (NIL "\\spadtype{Any} implements a type that packages up objects and their types in objects of \\spadtype{Any}. Roughly speaking that means that if \\spad{s : S} then when converted to \\spadtype{Any},{} the new object will include both the original object and its type. This is a way of converting arbitrary objects into a single type without losing any of the original information. Any object can be converted to one of \\spadtype{Any}.")) (|showTypeInOutput| (((|String|) (|Boolean|)) "\\spad{showTypeInOutput(bool)} affects the way objects of \\spadtype{Any} are displayed. If \\spad{bool} is \\spad{true} then the type of the original object that was converted to \\spadtype{Any} will be printed. If \\spad{bool} is \\spad{false},{} it will not be printed.")) (|obj| (((|None|) $) "\\spad{obj(a)} essentially returns the original object that was converted to \\spadtype{Any} except that the type is forced to be \\spadtype{None}.")) (|dom| (((|SExpression|) $) "\\spad{dom(a)} returns a \\spadgloss{LISP} form of the type of the original object that was converted to \\spadtype{Any}.")) (|objectOf| (((|OutputForm|) $) "\\spad{objectOf(a)} returns a printable form of the original object that was converted to \\spadtype{Any}.")) (|domainOf| (((|OutputForm|) $) "\\spad{domainOf(a)} returns a printable form of the type of the original object that was converted to \\spadtype{Any}.")) (|any| (($ (|SExpression|) (|None|)) "\\spad{any(type,{}object)} is a technical function for creating an \\spad{object} of \\spadtype{Any}. Arugment \\spad{type} is a \\spadgloss{LISP} form for the \\spad{type} of \\spad{object}."))) 
NIL 
NIL 
(-58) 
((|constructor| (NIL "This package contains useful functions that expose Axiom system internals")) (|summary| (((|Void|)) "\\indented{1}{summary() prints a short list of useful console commands} \\blankline \\spad{X} summary()")) (|credits| (((|Void|)) "\\indented{1}{credits() prints a list of people who contributed to Axiom} \\blankline \\spad{X} credits()")) (|getDomains| (((|Set| (|Symbol|)) (|Symbol|)) "\\indented{1}{The getDomains(\\spad{s}) takes a category and returns the list of domains} \\indented{1}{that have that category} \\blankline \\spad{X} getDomains 'IndexedAggregate"))) 
NIL 
NIL 
(-59 R M P) 
((|constructor| (NIL "\\spad{ApplyUnivariateSkewPolynomial} (internal) allows univariate skew polynomials to be applied to appropriate modules.")) (|apply| ((|#2| |#3| (|Mapping| |#2| |#2|) |#2|) "\\spad{apply(p,{} f,{} m)} returns \\spad{p(m)} where the action is given by \\spad{x m = f(m)}. \\spad{f} must be an \\spad{R}-pseudo linear map on \\spad{M}."))) 
NIL 
NIL 
(-60 |Base| R -2262) 
((|constructor| (NIL "This package apply rewrite rules to expressions,{} calling the pattern matcher.")) (|localUnquote| ((|#3| |#3| (|List| (|Symbol|))) "\\spad{localUnquote(f,{}ls)} is a local function.")) (|applyRules| ((|#3| (|List| (|RewriteRule| |#1| |#2| |#3|)) |#3| (|PositiveInteger|)) "\\spad{applyRules([r1,{}...,{}rn],{} expr,{} n)} applies the rules \\spad{r1},{}...,{}\\spad{rn} to \\spad{f} a most \\spad{n} times.") ((|#3| (|List| (|RewriteRule| |#1| |#2| |#3|)) |#3|) "\\spad{applyRules([r1,{}...,{}rn],{} expr)} applies the rules \\spad{r1},{}...,{}\\spad{rn} to \\spad{f} an unlimited number of times,{} \\spadignore{i.e.} until none of \\spad{r1},{}...,{}\\spad{rn} is applicable to the expression."))) 
NIL 
NIL 
(-61 S R |Row| |Col|) 
((|constructor| (NIL "Two dimensional array categories and domains")) (|map!| (($ (|Mapping| |#2| |#2|) $) "\\indented{1}{map!(\\spad{f},{}a)\\space{2}assign \\spad{a(i,{}j)} to \\spad{f(a(i,{}j))}} \\indented{1}{for all \\spad{i,{} j}} \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} map!(-,{}arr)")) (|map| (($ (|Mapping| |#2| |#2| |#2|) $ $ |#2|) "\\indented{1}{map(\\spad{f},{}a,{}\\spad{b},{}\\spad{r}) returns \\spad{c},{} where \\spad{c(i,{}j) = f(a(i,{}j),{}b(i,{}j))}} \\indented{1}{when both \\spad{a(i,{}j)} and \\spad{b(i,{}j)} exist;} \\indented{1}{else \\spad{c(i,{}j) = f(r,{} b(i,{}j))} when \\spad{a(i,{}j)} does not exist;} \\indented{1}{else \\spad{c(i,{}j) = f(a(i,{}j),{}r)} when \\spad{b(i,{}j)} does not exist;} \\indented{1}{otherwise \\spad{c(i,{}j) = f(r,{}r)}.} \\blankline \\spad{X} adder(a:Integer,{}b:Integer):Integer \\spad{==} a+b \\spad{X} \\spad{arr1} : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} \\spad{arr2} : \\spad{ARRAY2} INT \\spad{:=} new(3,{}3,{}10) \\spad{X} map(adder,{}\\spad{arr1},{}\\spad{arr2},{}17)") (($ (|Mapping| |#2| |#2| |#2|) $ $) "\\indented{1}{map(\\spad{f},{}a,{}\\spad{b}) returns \\spad{c},{} where \\spad{c(i,{}j) = f(a(i,{}j),{}b(i,{}j))}} \\indented{1}{for all \\spad{i,{} j}} \\blankline \\spad{X} adder(a:Integer,{}b:Integer):Integer \\spad{==} a+b \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} map(adder,{}arr,{}arr)") (($ (|Mapping| |#2| |#2|) $) "\\indented{1}{map(\\spad{f},{}a) returns \\spad{b},{} where \\spad{b(i,{}j) = f(a(i,{}j))}} \\indented{1}{for all \\spad{i,{} j}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} map(-,{}arr) \\spad{X} map((\\spad{x} +-> \\spad{x} + \\spad{x}),{}arr)")) (|setColumn!| (($ $ (|Integer|) |#4|) "\\indented{1}{setColumn!(\\spad{m},{}\\spad{j},{}\\spad{v}) sets to \\spad{j}th column of \\spad{m} to \\spad{v}} \\blankline \\spad{X} T1:=TwoDimensionalArray Integer \\spad{X} arr:T1:= new(5,{}4,{}0) \\spad{X} T2:=OneDimensionalArray Integer \\spad{X} acol:=construct([1,{}2,{}3,{}4,{}5]::List(INT))\\$\\spad{T2} \\spad{X} setColumn!(arr,{}1,{}acol)\\$\\spad{T1}")) (|setRow!| (($ $ (|Integer|) |#3|) "\\indented{1}{setRow!(\\spad{m},{}\\spad{i},{}\\spad{v}) sets to \\spad{i}th row of \\spad{m} to \\spad{v}} \\blankline \\spad{X} T1:=TwoDimensionalArray Integer \\spad{X} arr:T1:= new(5,{}4,{}0) \\spad{X} T2:=OneDimensionalArray Integer \\spad{X} arow:=construct([1,{}2,{}3,{}4]::List(INT))\\$\\spad{T2} \\spad{X} setRow!(arr,{}1,{}arow)\\$\\spad{T1}")) (|qsetelt!| ((|#2| $ (|Integer|) (|Integer|) |#2|) "\\indented{1}{qsetelt!(\\spad{m},{}\\spad{i},{}\\spad{j},{}\\spad{r}) sets the element in the \\spad{i}th row and \\spad{j}th} \\indented{1}{column of \\spad{m} to \\spad{r}} \\indented{1}{NO error check to determine if indices are in proper ranges} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}0) \\spad{X} qsetelt!(arr,{}1,{}1,{}17)")) (|setelt| ((|#2| $ (|Integer|) (|Integer|) |#2|) "\\indented{1}{setelt(\\spad{m},{}\\spad{i},{}\\spad{j},{}\\spad{r}) sets the element in the \\spad{i}th row and \\spad{j}th} \\indented{1}{column of \\spad{m} to \\spad{r}} \\indented{1}{error check to determine if indices are in proper ranges} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}0) \\spad{X} setelt(arr,{}1,{}1,{}17)")) (|parts| (((|List| |#2|) $) "\\indented{1}{parts(\\spad{m}) returns a list of the elements of \\spad{m} in row major order} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} parts(arr)")) (|column| ((|#4| $ (|Integer|)) "\\indented{1}{column(\\spad{m},{}\\spad{j}) returns the \\spad{j}th column of \\spad{m}} \\indented{1}{error check to determine if index is in proper ranges} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} column(arr,{}1)")) (|row| ((|#3| $ (|Integer|)) "\\indented{1}{row(\\spad{m},{}\\spad{i}) returns the \\spad{i}th row of \\spad{m}} \\indented{1}{error check to determine if index is in proper ranges} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} row(arr,{}1)")) (|qelt| ((|#2| $ (|Integer|) (|Integer|)) "\\indented{1}{qelt(\\spad{m},{}\\spad{i},{}\\spad{j}) returns the element in the \\spad{i}th row and \\spad{j}th} \\indented{1}{column of the array \\spad{m}} \\indented{1}{NO error check to determine if indices are in proper ranges} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} qelt(arr,{}1,{}1)")) (|elt| ((|#2| $ (|Integer|) (|Integer|) |#2|) "\\indented{1}{elt(\\spad{m},{}\\spad{i},{}\\spad{j},{}\\spad{r}) returns the element in the \\spad{i}th row and \\spad{j}th} \\indented{1}{column of the array \\spad{m},{} if \\spad{m} has an \\spad{i}th row and a \\spad{j}th column,{}} \\indented{1}{and returns \\spad{r} otherwise} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} elt(arr,{}1,{}1,{}6) \\spad{X} elt(arr,{}1,{}10,{}6)") ((|#2| $ (|Integer|) (|Integer|)) "\\indented{1}{elt(\\spad{m},{}\\spad{i},{}\\spad{j}) returns the element in the \\spad{i}th row and \\spad{j}th} \\indented{1}{column of the array \\spad{m}} \\indented{1}{error check to determine if indices are in proper ranges} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} elt(arr,{}1,{}1)")) (|ncols| (((|NonNegativeInteger|) $) "\\indented{1}{ncols(\\spad{m}) returns the number of columns in the array \\spad{m}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} ncols(arr)")) (|nrows| (((|NonNegativeInteger|) $) "\\indented{1}{nrows(\\spad{m}) returns the number of rows in the array \\spad{m}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} nrows(arr)")) (|maxColIndex| (((|Integer|) $) "\\indented{1}{maxColIndex(\\spad{m}) returns the index of the 'last' column of the array \\spad{m}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} maxColIndex(arr)")) (|minColIndex| (((|Integer|) $) "\\indented{1}{minColIndex(\\spad{m}) returns the index of the 'first' column of the array \\spad{m}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} minColIndex(arr)")) (|maxRowIndex| (((|Integer|) $) "\\indented{1}{maxRowIndex(\\spad{m}) returns the index of the 'last' row of the array \\spad{m}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} maxRowIndex(arr)")) (|minRowIndex| (((|Integer|) $) "\\indented{1}{minRowIndex(\\spad{m}) returns the index of the 'first' row of the array \\spad{m}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} minRowIndex(arr)")) (|fill!| (($ $ |#2|) "\\indented{1}{fill!(\\spad{m},{}\\spad{r}) fills \\spad{m} with \\spad{r}\\spad{'s}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}0) \\spad{X} fill!(arr,{}10)")) (|new| (($ (|NonNegativeInteger|) (|NonNegativeInteger|) |#2|) "\\indented{1}{new(\\spad{m},{}\\spad{n},{}\\spad{r}) is an \\spad{m}-by-\\spad{n} array all of whose entries are \\spad{r}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}0)")) (|finiteAggregate| ((|attribute|) "two-dimensional arrays are finite")) (|shallowlyMutable| ((|attribute|) "one may destructively alter arrays"))) 
NIL 
NIL 
(-62 R |Row| |Col|) 
((|constructor| (NIL "Two dimensional array categories and domains")) (|map!| (($ (|Mapping| |#1| |#1|) $) "\\indented{1}{map!(\\spad{f},{}a)\\space{2}assign \\spad{a(i,{}j)} to \\spad{f(a(i,{}j))}} \\indented{1}{for all \\spad{i,{} j}} \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} map!(-,{}arr)")) (|map| (($ (|Mapping| |#1| |#1| |#1|) $ $ |#1|) "\\indented{1}{map(\\spad{f},{}a,{}\\spad{b},{}\\spad{r}) returns \\spad{c},{} where \\spad{c(i,{}j) = f(a(i,{}j),{}b(i,{}j))}} \\indented{1}{when both \\spad{a(i,{}j)} and \\spad{b(i,{}j)} exist;} \\indented{1}{else \\spad{c(i,{}j) = f(r,{} b(i,{}j))} when \\spad{a(i,{}j)} does not exist;} \\indented{1}{else \\spad{c(i,{}j) = f(a(i,{}j),{}r)} when \\spad{b(i,{}j)} does not exist;} \\indented{1}{otherwise \\spad{c(i,{}j) = f(r,{}r)}.} \\blankline \\spad{X} adder(a:Integer,{}b:Integer):Integer \\spad{==} a+b \\spad{X} \\spad{arr1} : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} \\spad{arr2} : \\spad{ARRAY2} INT \\spad{:=} new(3,{}3,{}10) \\spad{X} map(adder,{}\\spad{arr1},{}\\spad{arr2},{}17)") (($ (|Mapping| |#1| |#1| |#1|) $ $) "\\indented{1}{map(\\spad{f},{}a,{}\\spad{b}) returns \\spad{c},{} where \\spad{c(i,{}j) = f(a(i,{}j),{}b(i,{}j))}} \\indented{1}{for all \\spad{i,{} j}} \\blankline \\spad{X} adder(a:Integer,{}b:Integer):Integer \\spad{==} a+b \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} map(adder,{}arr,{}arr)") (($ (|Mapping| |#1| |#1|) $) "\\indented{1}{map(\\spad{f},{}a) returns \\spad{b},{} where \\spad{b(i,{}j) = f(a(i,{}j))}} \\indented{1}{for all \\spad{i,{} j}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} map(-,{}arr) \\spad{X} map((\\spad{x} +-> \\spad{x} + \\spad{x}),{}arr)")) (|setColumn!| (($ $ (|Integer|) |#3|) "\\indented{1}{setColumn!(\\spad{m},{}\\spad{j},{}\\spad{v}) sets to \\spad{j}th column of \\spad{m} to \\spad{v}} \\blankline \\spad{X} T1:=TwoDimensionalArray Integer \\spad{X} arr:T1:= new(5,{}4,{}0) \\spad{X} T2:=OneDimensionalArray Integer \\spad{X} acol:=construct([1,{}2,{}3,{}4,{}5]::List(INT))\\$\\spad{T2} \\spad{X} setColumn!(arr,{}1,{}acol)\\$\\spad{T1}")) (|setRow!| (($ $ (|Integer|) |#2|) "\\indented{1}{setRow!(\\spad{m},{}\\spad{i},{}\\spad{v}) sets to \\spad{i}th row of \\spad{m} to \\spad{v}} \\blankline \\spad{X} T1:=TwoDimensionalArray Integer \\spad{X} arr:T1:= new(5,{}4,{}0) \\spad{X} T2:=OneDimensionalArray Integer \\spad{X} arow:=construct([1,{}2,{}3,{}4]::List(INT))\\$\\spad{T2} \\spad{X} setRow!(arr,{}1,{}arow)\\$\\spad{T1}")) (|qsetelt!| ((|#1| $ (|Integer|) (|Integer|) |#1|) "\\indented{1}{qsetelt!(\\spad{m},{}\\spad{i},{}\\spad{j},{}\\spad{r}) sets the element in the \\spad{i}th row and \\spad{j}th} \\indented{1}{column of \\spad{m} to \\spad{r}} \\indented{1}{NO error check to determine if indices are in proper ranges} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}0) \\spad{X} qsetelt!(arr,{}1,{}1,{}17)")) (|setelt| ((|#1| $ (|Integer|) (|Integer|) |#1|) "\\indented{1}{setelt(\\spad{m},{}\\spad{i},{}\\spad{j},{}\\spad{r}) sets the element in the \\spad{i}th row and \\spad{j}th} \\indented{1}{column of \\spad{m} to \\spad{r}} \\indented{1}{error check to determine if indices are in proper ranges} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}0) \\spad{X} setelt(arr,{}1,{}1,{}17)")) (|parts| (((|List| |#1|) $) "\\indented{1}{parts(\\spad{m}) returns a list of the elements of \\spad{m} in row major order} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} parts(arr)")) (|column| ((|#3| $ (|Integer|)) "\\indented{1}{column(\\spad{m},{}\\spad{j}) returns the \\spad{j}th column of \\spad{m}} \\indented{1}{error check to determine if index is in proper ranges} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} column(arr,{}1)")) (|row| ((|#2| $ (|Integer|)) "\\indented{1}{row(\\spad{m},{}\\spad{i}) returns the \\spad{i}th row of \\spad{m}} \\indented{1}{error check to determine if index is in proper ranges} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} row(arr,{}1)")) (|qelt| ((|#1| $ (|Integer|) (|Integer|)) "\\indented{1}{qelt(\\spad{m},{}\\spad{i},{}\\spad{j}) returns the element in the \\spad{i}th row and \\spad{j}th} \\indented{1}{column of the array \\spad{m}} \\indented{1}{NO error check to determine if indices are in proper ranges} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} qelt(arr,{}1,{}1)")) (|elt| ((|#1| $ (|Integer|) (|Integer|) |#1|) "\\indented{1}{elt(\\spad{m},{}\\spad{i},{}\\spad{j},{}\\spad{r}) returns the element in the \\spad{i}th row and \\spad{j}th} \\indented{1}{column of the array \\spad{m},{} if \\spad{m} has an \\spad{i}th row and a \\spad{j}th column,{}} \\indented{1}{and returns \\spad{r} otherwise} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} elt(arr,{}1,{}1,{}6) \\spad{X} elt(arr,{}1,{}10,{}6)") ((|#1| $ (|Integer|) (|Integer|)) "\\indented{1}{elt(\\spad{m},{}\\spad{i},{}\\spad{j}) returns the element in the \\spad{i}th row and \\spad{j}th} \\indented{1}{column of the array \\spad{m}} \\indented{1}{error check to determine if indices are in proper ranges} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} elt(arr,{}1,{}1)")) (|ncols| (((|NonNegativeInteger|) $) "\\indented{1}{ncols(\\spad{m}) returns the number of columns in the array \\spad{m}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} ncols(arr)")) (|nrows| (((|NonNegativeInteger|) $) "\\indented{1}{nrows(\\spad{m}) returns the number of rows in the array \\spad{m}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} nrows(arr)")) (|maxColIndex| (((|Integer|) $) "\\indented{1}{maxColIndex(\\spad{m}) returns the index of the 'last' column of the array \\spad{m}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} maxColIndex(arr)")) (|minColIndex| (((|Integer|) $) "\\indented{1}{minColIndex(\\spad{m}) returns the index of the 'first' column of the array \\spad{m}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} minColIndex(arr)")) (|maxRowIndex| (((|Integer|) $) "\\indented{1}{maxRowIndex(\\spad{m}) returns the index of the 'last' row of the array \\spad{m}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} maxRowIndex(arr)")) (|minRowIndex| (((|Integer|) $) "\\indented{1}{minRowIndex(\\spad{m}) returns the index of the 'first' row of the array \\spad{m}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}10) \\spad{X} minRowIndex(arr)")) (|fill!| (($ $ |#1|) "\\indented{1}{fill!(\\spad{m},{}\\spad{r}) fills \\spad{m} with \\spad{r}\\spad{'s}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}0) \\spad{X} fill!(arr,{}10)")) (|new| (($ (|NonNegativeInteger|) (|NonNegativeInteger|) |#1|) "\\indented{1}{new(\\spad{m},{}\\spad{n},{}\\spad{r}) is an \\spad{m}-by-\\spad{n} array all of whose entries are \\spad{r}} \\blankline \\spad{X} arr : \\spad{ARRAY2} INT \\spad{:=} new(5,{}4,{}0)")) (|finiteAggregate| ((|attribute|) "two-dimensional arrays are finite")) (|shallowlyMutable| ((|attribute|) "one may destructively alter arrays"))) 
((-4505 . T) (-4506 . T) (-3576 . T)) 
NIL 
(-63 A B) 
((|constructor| (NIL "This package provides tools for operating on one-dimensional arrays with unary and binary functions involving different underlying types")) (|map| (((|OneDimensionalArray| |#2|) (|Mapping| |#2| |#1|) (|OneDimensionalArray| |#1|)) "\\indented{1}{map(\\spad{f},{}a) applies function \\spad{f} to each member of one-dimensional array} \\indented{1}{\\spad{a} resulting in a new one-dimensional array over a} \\indented{1}{possibly different underlying domain.} \\blankline \\spad{X} \\spad{T1:=OneDimensionalArrayFunctions2}(Integer,{}Integer) \\spad{X} map(\\spad{x+}-\\spad{>x+2},{}[\\spad{i} for \\spad{i} in 1..10])\\$\\spad{T1}")) (|reduce| ((|#2| (|Mapping| |#2| |#1| |#2|) (|OneDimensionalArray| |#1|) |#2|) "\\indented{1}{reduce(\\spad{f},{}a,{}\\spad{r}) applies function \\spad{f} to each} \\indented{1}{successive element of the} \\indented{1}{one-dimensional array \\spad{a} and an accumulant initialized to \\spad{r}.} \\indented{1}{For example,{} \\spad{reduce(_+\\$Integer,{}[1,{}2,{}3],{}0)}} \\indented{1}{does \\spad{3+(2+(1+0))}. Note that third argument \\spad{r}} \\indented{1}{may be regarded as the identity element for the function \\spad{f}.} \\blankline \\spad{X} \\spad{T1:=OneDimensionalArrayFunctions2}(Integer,{}Integer) \\spad{X} adder(a:Integer,{}b:Integer):Integer \\spad{==} a+b \\spad{X} reduce(adder,{}[\\spad{i} for \\spad{i} in 1..10],{}0)\\$\\spad{T1}")) (|scan| (((|OneDimensionalArray| |#2|) (|Mapping| |#2| |#1| |#2|) (|OneDimensionalArray| |#1|) |#2|) "\\indented{1}{scan(\\spad{f},{}a,{}\\spad{r}) successively applies} \\indented{1}{\\spad{reduce(f,{}x,{}r)} to more and more leading sub-arrays} \\indented{1}{\\spad{x} of one-dimensional array \\spad{a}.} \\indented{1}{More precisely,{} if \\spad{a} is \\spad{[a1,{}a2,{}...]},{} then} \\indented{1}{\\spad{scan(f,{}a,{}r)} returns} \\indented{1}{\\spad{[reduce(f,{}[a1],{}r),{}reduce(f,{}[a1,{}a2],{}r),{}...]}.} \\blankline \\spad{X} \\spad{T1:=OneDimensionalArrayFunctions2}(Integer,{}Integer) \\spad{X} adder(a:Integer,{}b:Integer):Integer \\spad{==} a+b \\spad{X} scan(adder,{}[\\spad{i} for \\spad{i} in 1..10],{}0)\\$\\spad{T1}"))) 
NIL 
NIL 
(-64 S) 
((|constructor| (NIL "This is the domain of 1-based one dimensional arrays")) (|oneDimensionalArray| (($ (|NonNegativeInteger|) |#1|) "\\indented{1}{oneDimensionalArray(\\spad{n},{}\\spad{s}) creates an array from \\spad{n} copies of element \\spad{s}} \\blankline \\spad{X} oneDimensionalArray(10,{}0.0)") (($ (|List| |#1|)) "\\indented{1}{oneDimensionalArray(\\spad{l}) creates an array from a list of elements \\spad{l}} \\blankline \\spad{X} oneDimensionalArray [\\spad{i**2} for \\spad{i} in 1..10]"))) 
((-4506 . T) (-4505 . T)) 
((|HasCategory| |#1| (QUOTE (-1082))) (|HasCategory| |#1| (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| |#1| (QUOTE (-834))) (-3322 (|HasCategory| |#1| (QUOTE (-834))) (|HasCategory| |#1| (QUOTE (-1082)))) (|HasCategory| (-560) (QUOTE (-834))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082)))) (-3322 (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-834)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082)))))) 
(-65 R) 
((|constructor| (NIL "A TwoDimensionalArray is a two dimensional array with 1-based indexing for both rows and columns.")) (|shallowlyMutable| ((|attribute|) "One may destructively alter TwoDimensionalArray\\spad{'s}."))) 
((-4505 . T) (-4506 . T)) 
((|HasCategory| |#1| (QUOTE (-1082))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082))))) 
(-66 -3409) 
((|constructor| (NIL "\\spadtype{ASP10} produces Fortran for Type 10 ASPs,{} needed for NAG routine d02kef. This ASP computes the values of a set of functions,{} for example: \\blankline \\tab{5}SUBROUTINE COEFFN(\\spad{P},{}\\spad{Q},{}DQDL,{}\\spad{X},{}ELAM,{}JINT)\\spad{\\br} \\tab{5}DOUBLE PRECISION ELAM,{}\\spad{P},{}\\spad{Q},{}\\spad{X},{}DQDL\\spad{\\br} \\tab{5}INTEGER JINT\\spad{\\br} \\tab{5}\\spad{P=1}.0D0\\spad{\\br} \\tab{5}\\spad{Q=}((\\spad{-1}.0D0*X**3)+ELAM*X*X-2.0D0)/(\\spad{X*X})\\spad{\\br} \\tab{5}\\spad{DQDL=1}.0D0\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct| (QUOTE JINT) (QUOTE X) (QUOTE ELAM)) (|construct|) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-67 -3409) 
((|constructor| (NIL "\\spadtype{Asp12} produces Fortran for Type 12 ASPs,{} needed for NAG routine d02kef etc.,{} for example: \\blankline \\tab{5}SUBROUTINE MONIT (MAXIT,{}IFLAG,{}ELAM,{}FINFO)\\spad{\\br} \\tab{5}DOUBLE PRECISION ELAM,{}FINFO(15)\\spad{\\br} \\tab{5}INTEGER MAXIT,{}IFLAG\\spad{\\br} \\tab{5}IF(MAXIT.EQ.\\spad{-1})THEN\\spad{\\br} \\tab{7}PRINT*,{}\"Output from Monit\"\\spad{\\br} \\tab{5}ENDIF\\spad{\\br} \\tab{5}PRINT*,{}MAXIT,{}IFLAG,{}ELAM,{}(FINFO(\\spad{I}),{}\\spad{I=1},{}4)\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END\\")) (|outputAsFortran| (((|Void|)) "\\spad{outputAsFortran()} generates the default code for \\spadtype{ASP12}."))) 
NIL 
NIL 
(-68 -3409) 
((|constructor| (NIL "\\spadtype{Asp19} produces Fortran for Type 19 ASPs,{} evaluating a set of functions and their jacobian at a given point,{} for example: \\blankline \\tab{5}SUBROUTINE \\spad{LSFUN2}(\\spad{M},{}\\spad{N},{}\\spad{XC},{}FVECC,{}FJACC,{}\\spad{LJC})\\spad{\\br} \\tab{5}DOUBLE PRECISION FVECC(\\spad{M}),{}FJACC(\\spad{LJC},{}\\spad{N}),{}\\spad{XC}(\\spad{N})\\spad{\\br} \\tab{5}INTEGER \\spad{M},{}\\spad{N},{}\\spad{LJC}\\spad{\\br} \\tab{5}INTEGER \\spad{I},{}\\spad{J}\\spad{\\br} \\tab{5}DO 25003 \\spad{I=1},{}\\spad{LJC}\\spad{\\br} \\tab{7}DO 25004 \\spad{J=1},{}\\spad{N}\\spad{\\br} \\tab{9}FJACC(\\spad{I},{}\\spad{J})\\spad{=0}.0D0\\spad{\\br} 25004 CONTINUE\\spad{\\br} 25003 CONTINUE\\spad{\\br} \\tab{5}FVECC(1)=((\\spad{XC}(1)\\spad{-0}.14D0)\\spad{*XC}(3)+(15.0D0*XC(1)\\spad{-2}.1D0)\\spad{*XC}(2)\\spad{+1}.0D0)/(\\spad{\\br} \\tab{4}\\spad{&XC}(3)\\spad{+15}.0D0*XC(2))\\spad{\\br} \\tab{5}FVECC(2)=((\\spad{XC}(1)\\spad{-0}.18D0)\\spad{*XC}(3)+(7.0D0*XC(1)\\spad{-1}.26D0)\\spad{*XC}(2)\\spad{+1}.0D0)/(\\spad{\\br} \\tab{4}\\spad{&XC}(3)\\spad{+7}.0D0*XC(2))\\spad{\\br} \\tab{5}FVECC(3)=((\\spad{XC}(1)\\spad{-0}.22D0)\\spad{*XC}(3)+(4.333333333333333D0*XC(1)\\spad{-0}.953333\\spad{\\br} \\tab{4}\\spad{&3333333333D0})\\spad{*XC}(2)\\spad{+1}.0D0)/(\\spad{XC}(3)\\spad{+4}.333333333333333D0*XC(2))\\spad{\\br} \\tab{5}FVECC(4)=((\\spad{XC}(1)\\spad{-0}.25D0)\\spad{*XC}(3)+(3.0D0*XC(1)\\spad{-0}.75D0)\\spad{*XC}(2)\\spad{+1}.0D0)/(\\spad{\\br} \\tab{4}\\spad{&XC}(3)\\spad{+3}.0D0*XC(2))\\spad{\\br} \\tab{5}FVECC(5)=((\\spad{XC}(1)\\spad{-0}.29D0)\\spad{*XC}(3)+(2.2D0*XC(1)\\spad{-0}.6379999999999999D0)*\\spad{\\br} \\tab{4}\\spad{&XC}(2)\\spad{+1}.0D0)/(\\spad{XC}(3)\\spad{+2}.2D0*XC(2))\\spad{\\br} \\tab{5}FVECC(6)=((\\spad{XC}(1)\\spad{-0}.32D0)\\spad{*XC}(3)+(1.666666666666667D0*XC(1)\\spad{-0}.533333\\spad{\\br} \\tab{4}\\spad{&3333333333D0})\\spad{*XC}(2)\\spad{+1}.0D0)/(\\spad{XC}(3)\\spad{+1}.666666666666667D0*XC(2))\\spad{\\br} \\tab{5}FVECC(7)=((\\spad{XC}(1)\\spad{-0}.35D0)\\spad{*XC}(3)+(1.285714285714286D0*XC(1)\\spad{-0}.45D0)*\\spad{\\br} \\tab{4}\\spad{&XC}(2)\\spad{+1}.0D0)/(\\spad{XC}(3)\\spad{+1}.285714285714286D0*XC(2))\\spad{\\br} \\tab{5}FVECC(8)=((\\spad{XC}(1)\\spad{-0}.39D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-0}.39D0)\\spad{*XC}(2)\\spad{+1}.0D0)/(\\spad{XC}(3)+\\spad{\\br} \\tab{4}\\spad{&XC}(2))\\spad{\\br} \\tab{5}FVECC(9)=((\\spad{XC}(1)\\spad{-0}.37D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-0}.37D0)\\spad{*XC}(2)\\spad{+1}.285714285714\\spad{\\br} \\tab{4}\\spad{&286D0})/(\\spad{XC}(3)\\spad{+XC}(2))\\spad{\\br} \\tab{5}FVECC(10)=((\\spad{XC}(1)\\spad{-0}.58D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-0}.58D0)\\spad{*XC}(2)\\spad{+1}.66666666666\\spad{\\br} \\tab{4}\\spad{&6667D0})/(\\spad{XC}(3)\\spad{+XC}(2))\\spad{\\br} \\tab{5}FVECC(11)=((\\spad{XC}(1)\\spad{-0}.73D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-0}.73D0)\\spad{*XC}(2)\\spad{+2}.2D0)/(\\spad{XC}(3)\\spad{\\br} \\tab{4}&+XC(2))\\spad{\\br} \\tab{5}FVECC(12)=((\\spad{XC}(1)\\spad{-0}.96D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-0}.96D0)\\spad{*XC}(2)\\spad{+3}.0D0)/(\\spad{XC}(3)\\spad{\\br} \\tab{4}&+XC(2))\\spad{\\br} \\tab{5}FVECC(13)=((\\spad{XC}(1)\\spad{-1}.34D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-1}.34D0)\\spad{*XC}(2)\\spad{+4}.33333333333\\spad{\\br} \\tab{4}\\spad{&3333D0})/(\\spad{XC}(3)\\spad{+XC}(2))\\spad{\\br} \\tab{5}FVECC(14)=((\\spad{XC}(1)\\spad{-2}.1D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-2}.1D0)\\spad{*XC}(2)\\spad{+7}.0D0)/(\\spad{XC}(3)\\spad{+X}\\spad{\\br} \\tab{4}\\spad{&C}(2))\\spad{\\br} \\tab{5}FVECC(15)=((\\spad{XC}(1)\\spad{-4}.39D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-4}.39D0)\\spad{*XC}(2)\\spad{+15}.0D0)/(\\spad{XC}(3\\spad{\\br} \\tab{4}&)\\spad{+XC}(2))\\spad{\\br} \\tab{5}FJACC(1,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(1,{}2)=-15.0D0/(\\spad{XC}(3)\\spad{**2+30}.0D0*XC(2)\\spad{*XC}(3)\\spad{+225}.0D0*XC(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(1,{}3)=-1.0D0/(\\spad{XC}(3)\\spad{**2+30}.0D0*XC(2)\\spad{*XC}(3)\\spad{+225}.0D0*XC(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(2,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(2,{}2)=-7.0D0/(\\spad{XC}(3)\\spad{**2+14}.0D0*XC(2)\\spad{*XC}(3)\\spad{+49}.0D0*XC(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(2,{}3)=-1.0D0/(\\spad{XC}(3)\\spad{**2+14}.0D0*XC(2)\\spad{*XC}(3)\\spad{+49}.0D0*XC(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(3,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(3,{}2)=((\\spad{-0}.1110223024625157D-15*XC(3))\\spad{-4}.333333333333333D0)/(\\spad{\\br} \\tab{4}\\spad{&XC}(3)\\spad{**2+8}.666666666666666D0*XC(2)\\spad{*XC}(3)\\spad{+18}.77777777777778D0*XC(2)\\spad{\\br} \\tab{4}\\spad{&**2})\\spad{\\br} \\tab{5}FJACC(3,{}3)=(0.1110223024625157D-15*XC(2)\\spad{-1}.0D0)/(\\spad{XC}(3)\\spad{**2+8}.666666\\spad{\\br} \\tab{4}&666666666D0*XC(2)\\spad{*XC}(3)\\spad{+18}.77777777777778D0*XC(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(4,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(4,{}2)=-3.0D0/(\\spad{XC}(3)\\spad{**2+6}.0D0*XC(2)\\spad{*XC}(3)\\spad{+9}.0D0*XC(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(4,{}3)=-1.0D0/(\\spad{XC}(3)\\spad{**2+6}.0D0*XC(2)\\spad{*XC}(3)\\spad{+9}.0D0*XC(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(5,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(5,{}2)=((\\spad{-0}.1110223024625157D-15*XC(3))\\spad{-2}.2D0)/(\\spad{XC}(3)\\spad{**2+4}.399\\spad{\\br} \\tab{4}&999999999999D0*XC(2)\\spad{*XC}(3)\\spad{+4}.839999999999998D0*XC(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(5,{}3)=(0.1110223024625157D-15*XC(2)\\spad{-1}.0D0)/(\\spad{XC}(3)\\spad{**2+4}.399999\\spad{\\br} \\tab{4}&999999999D0*XC(2)\\spad{*XC}(3)\\spad{+4}.839999999999998D0*XC(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(6,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(6,{}2)=((\\spad{-0}.2220446049250313D-15*XC(3))\\spad{-1}.666666666666667D0)/(\\spad{\\br} \\tab{4}\\spad{&XC}(3)\\spad{**2+3}.333333333333333D0*XC(2)\\spad{*XC}(3)\\spad{+2}.777777777777777D0*XC(2)\\spad{\\br} \\tab{4}\\spad{&**2})\\spad{\\br} \\tab{5}FJACC(6,{}3)=(0.2220446049250313D-15*XC(2)\\spad{-1}.0D0)/(\\spad{XC}(3)\\spad{**2+3}.333333\\spad{\\br} \\tab{4}&333333333D0*XC(2)\\spad{*XC}(3)\\spad{+2}.777777777777777D0*XC(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(7,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(7,{}2)=((\\spad{-0}.5551115123125783D-16*XC(3))\\spad{-1}.285714285714286D0)/(\\spad{\\br} \\tab{4}\\spad{&XC}(3)\\spad{**2+2}.571428571428571D0*XC(2)\\spad{*XC}(3)\\spad{+1}.653061224489796D0*XC(2)\\spad{\\br} \\tab{4}\\spad{&**2})\\spad{\\br} \\tab{5}FJACC(7,{}3)=(0.5551115123125783D-16*XC(2)\\spad{-1}.0D0)/(\\spad{XC}(3)\\spad{**2+2}.571428\\spad{\\br} \\tab{4}&571428571D0*XC(2)\\spad{*XC}(3)\\spad{+1}.653061224489796D0*XC(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(8,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(8,{}2)=-1.0D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(8,{}3)=-1.0D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(9,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(9,{}2)=-1.285714285714286D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)*\\spad{\\br} \\tab{4}\\spad{&*2})\\spad{\\br} \\tab{5}FJACC(9,{}3)=-1.285714285714286D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)*\\spad{\\br} \\tab{4}\\spad{&*2})\\spad{\\br} \\tab{5}FJACC(10,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(10,{}2)=-1.666666666666667D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)\\spad{\\br} \\tab{4}\\spad{&**2})\\spad{\\br} \\tab{5}FJACC(10,{}3)=-1.666666666666667D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)\\spad{\\br} \\tab{4}\\spad{&**2})\\spad{\\br} \\tab{5}FJACC(11,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(11,{}2)=-2.2D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(11,{}3)=-2.2D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(12,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(12,{}2)=-3.0D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(12,{}3)=-3.0D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(13,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(13,{}2)=-4.333333333333333D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)\\spad{\\br} \\tab{4}\\spad{&**2})\\spad{\\br} \\tab{5}FJACC(13,{}3)=-4.333333333333333D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)\\spad{\\br} \\tab{4}\\spad{&**2})\\spad{\\br} \\tab{5}FJACC(14,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(14,{}2)=-7.0D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(14,{}3)=-7.0D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(15,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}FJACC(15,{}2)=-15.0D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)\\spad{**2})\\spad{\\br} \\tab{5}FJACC(15,{}3)=-15.0D0/(\\spad{XC}(3)\\spad{**2+2}.0D0*XC(2)\\spad{*XC}(3)\\spad{+XC}(2)\\spad{**2})\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct|) (|construct| (QUOTE XC)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-69 -3409) 
((|constructor| (NIL "\\spadtype{Asp1} produces Fortran for Type 1 ASPs,{} needed for various NAG routines. Type 1 ASPs take a univariate expression (in the symbol \\spad{x}) and turn it into a Fortran Function like the following: \\blankline \\tab{5}DOUBLE PRECISION FUNCTION \\spad{F}(\\spad{X})\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{X}\\spad{\\br} \\tab{5}F=DSIN(\\spad{X})\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|FortranExpression| (|construct| (QUOTE X)) (|construct|) (|MachineFloat|))) "\\spad{coerce(f)} takes an object from the appropriate instantiation of \\spadtype{FortranExpression} and turns it into an ASP."))) 
NIL 
NIL 
(-70 -3409) 
((|constructor| (NIL "\\spadtype{Asp20} produces Fortran for Type 20 ASPs,{} for example: \\blankline \\tab{5}SUBROUTINE QPHESS(\\spad{N},{}NROWH,{}NCOLH,{}JTHCOL,{}HESS,{}\\spad{X},{}\\spad{HX})\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{HX}(\\spad{N}),{}\\spad{X}(\\spad{N}),{}HESS(NROWH,{}NCOLH)\\spad{\\br} \\tab{5}INTEGER JTHCOL,{}\\spad{N},{}NROWH,{}NCOLH\\spad{\\br} \\tab{5}\\spad{HX}(1)\\spad{=2}.0D0*X(1)\\spad{\\br} \\tab{5}\\spad{HX}(2)\\spad{=2}.0D0*X(2)\\spad{\\br} \\tab{5}\\spad{HX}(3)\\spad{=2}.0D0*X(4)\\spad{+2}.0D0*X(3)\\spad{\\br} \\tab{5}\\spad{HX}(4)\\spad{=2}.0D0*X(4)\\spad{+2}.0D0*X(3)\\spad{\\br} \\tab{5}\\spad{HX}(5)\\spad{=2}.0D0*X(5)\\spad{\\br} \\tab{5}\\spad{HX}(6)=(\\spad{-2}.0D0*X(7))+(\\spad{-2}.0D0*X(6))\\spad{\\br} \\tab{5}\\spad{HX}(7)=(\\spad{-2}.0D0*X(7))+(\\spad{-2}.0D0*X(6))\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|Matrix| (|FortranExpression| (|construct|) (|construct| (QUOTE X) (QUOTE HESS)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-71 -3409) 
((|constructor| (NIL "\\spadtype{Asp24} produces Fortran for Type 24 ASPs which evaluate a multivariate function at a point (needed for NAG routine e04jaf),{} for example: \\blankline \\tab{5}SUBROUTINE \\spad{FUNCT1}(\\spad{N},{}\\spad{XC},{}\\spad{FC})\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{FC},{}\\spad{XC}(\\spad{N})\\spad{\\br} \\tab{5}INTEGER \\spad{N}\\spad{\\br} \\tab{5}\\spad{FC=10}.0D0*XC(4)**4+(\\spad{-40}.0D0*XC(1)\\spad{*XC}(4)\\spad{**3})+(60.0D0*XC(1)\\spad{**2+5}\\spad{\\br} \\tab{4}&.0D0)\\spad{*XC}(4)**2+((\\spad{-10}.0D0*XC(3))+(\\spad{-40}.0D0*XC(1)\\spad{**3}))\\spad{*XC}(4)\\spad{+16}.0D0*X\\spad{\\br} \\tab{4}\\spad{&C}(3)**4+(\\spad{-32}.0D0*XC(2)\\spad{*XC}(3)\\spad{**3})+(24.0D0*XC(2)\\spad{**2+5}.0D0)\\spad{*XC}(3)**2+\\spad{\\br} \\tab{4}&(\\spad{-8}.0D0*XC(2)**3*XC(3))\\spad{+XC}(2)\\spad{**4+100}.0D0*XC(2)\\spad{**2+20}.0D0*XC(1)\\spad{*XC}(\\spad{\\br} \\tab{4}\\spad{&2})\\spad{+10}.0D0*XC(1)**4+XC(1)\\spad{**2}\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END\\spad{\\br}")) (|coerce| (($ (|FortranExpression| (|construct|) (|construct| (QUOTE XC)) (|MachineFloat|))) "\\spadtype{FortranExpression} and turns it into an ASP. coerce(\\spad{f}) takes an object from the appropriate instantiation of"))) 
NIL 
NIL 
(-72 -3409) 
((|constructor| (NIL "\\spadtype{Asp27} produces Fortran for Type 27 ASPs,{} needed for NAG routine f02fjf ,{}for example: \\blankline \\tab{5}FUNCTION DOT(IFLAG,{}\\spad{N},{}\\spad{Z},{}\\spad{W},{}RWORK,{}LRWORK,{}IWORK,{}LIWORK)\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{W}(\\spad{N}),{}\\spad{Z}(\\spad{N}),{}RWORK(LRWORK)\\spad{\\br} \\tab{5}INTEGER \\spad{N},{}LIWORK,{}IFLAG,{}LRWORK,{}IWORK(LIWORK)\\spad{\\br} \\tab{5}DOT=(\\spad{W}(16)+(\\spad{-0}.5D0*W(15)))\\spad{*Z}(16)+((\\spad{-0}.5D0*W(16))\\spad{+W}(15)+(\\spad{-0}.5D0*W(1\\spad{\\br} \\tab{4}\\spad{&4})))\\spad{*Z}(15)+((\\spad{-0}.5D0*W(15))\\spad{+W}(14)+(\\spad{-0}.5D0*W(13)))\\spad{*Z}(14)+((\\spad{-0}.5D0*W(\\spad{\\br} \\tab{4}\\spad{&14}))\\spad{+W}(13)+(\\spad{-0}.5D0*W(12)))\\spad{*Z}(13)+((\\spad{-0}.5D0*W(13))\\spad{+W}(12)+(\\spad{-0}.5D0*W(1\\spad{\\br} \\tab{4}\\spad{&1})))\\spad{*Z}(12)+((\\spad{-0}.5D0*W(12))\\spad{+W}(11)+(\\spad{-0}.5D0*W(10)))\\spad{*Z}(11)+((\\spad{-0}.5D0*W(\\spad{\\br} \\tab{4}\\spad{&11}))\\spad{+W}(10)+(\\spad{-0}.5D0*W(9)))\\spad{*Z}(10)+((\\spad{-0}.5D0*W(10))\\spad{+W}(9)+(\\spad{-0}.5D0*W(8))\\spad{\\br} \\tab{4}&)\\spad{*Z}(9)+((\\spad{-0}.5D0*W(9))\\spad{+W}(8)+(\\spad{-0}.5D0*W(7)))\\spad{*Z}(8)+((\\spad{-0}.5D0*W(8))\\spad{+W}(7)\\spad{\\br} \\tab{4}\\spad{&+}(\\spad{-0}.5D0*W(6)))\\spad{*Z}(7)+((\\spad{-0}.5D0*W(7))\\spad{+W}(6)+(\\spad{-0}.5D0*W(5)))\\spad{*Z}(6)+((\\spad{-0}.\\spad{\\br} \\tab{4}&5D0*W(6))\\spad{+W}(5)+(\\spad{-0}.5D0*W(4)))\\spad{*Z}(5)+((\\spad{-0}.5D0*W(5))\\spad{+W}(4)+(\\spad{-0}.5D0*W(3\\spad{\\br} \\tab{4}&)))\\spad{*Z}(4)+((\\spad{-0}.5D0*W(4))\\spad{+W}(3)+(\\spad{-0}.5D0*W(2)))\\spad{*Z}(3)+((\\spad{-0}.5D0*W(3))\\spad{+W}(\\spad{\\br} \\tab{4}\\spad{&2})+(\\spad{-0}.5D0*W(1)))\\spad{*Z}(2)+((\\spad{-0}.5D0*W(2))\\spad{+W}(1))\\spad{*Z}(1)\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END"))) 
NIL 
NIL 
(-73 -3409) 
((|constructor| (NIL "\\spadtype{Asp28} produces Fortran for Type 28 ASPs,{} used in NAG routine f02fjf,{} for example: \\blankline \\tab{5}SUBROUTINE IMAGE(IFLAG,{}\\spad{N},{}\\spad{Z},{}\\spad{W},{}RWORK,{}LRWORK,{}IWORK,{}LIWORK)\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{Z}(\\spad{N}),{}\\spad{W}(\\spad{N}),{}IWORK(LRWORK),{}RWORK(LRWORK)\\spad{\\br} \\tab{5}INTEGER \\spad{N},{}LIWORK,{}IFLAG,{}LRWORK\\spad{\\br} \\tab{5}\\spad{W}(1)\\spad{=0}.01707454969713436D0*Z(16)\\spad{+0}.001747395874954051D0*Z(15)\\spad{+0}.00\\spad{\\br} \\tab{4}&2106973900813502D0*Z(14)\\spad{+0}.002957434991769087D0*Z(13)+(\\spad{-0}.00700554\\spad{\\br} \\tab{4}&0882865317D0*Z(12))+(\\spad{-0}.01219194009813166D0*Z(11))\\spad{+0}.0037230647365\\spad{\\br} \\tab{4}&3087D0*Z(10)\\spad{+0}.04932374658377151D0*Z(9)+(\\spad{-0}.03586220812223305D0*Z(\\spad{\\br} \\tab{4}\\spad{&8}))+(\\spad{-0}.04723268012114625D0*Z(7))+(\\spad{-0}.02434652144032987D0*Z(6))\\spad{+0}.\\spad{\\br} \\tab{4}&2264766947290192D0*Z(5)+(\\spad{-0}.1385343580686922D0*Z(4))+(\\spad{-0}.116530050\\spad{\\br} \\tab{4}&8238904D0*Z(3))+(\\spad{-0}.2803531651057233D0*Z(2))\\spad{+1}.019463911841327D0*Z\\spad{\\br} \\tab{4}&(1)\\spad{\\br} \\tab{5}\\spad{W}(2)\\spad{=0}.0227345011107737D0*Z(16)\\spad{+0}.008812321197398072D0*Z(15)\\spad{+0}.010\\spad{\\br} \\tab{4}&94012210519586D0*Z(14)+(\\spad{-0}.01764072463999744D0*Z(13))+(\\spad{-0}.01357136\\spad{\\br} \\tab{4}&72105995D0*Z(12))\\spad{+0}.00157466157362272D0*Z(11)\\spad{+0}.05258889186338282D\\spad{\\br} \\tab{4}&0*Z(10)+(\\spad{-0}.01981532388243379D0*Z(9))+(\\spad{-0}.06095390688679697D0*Z(8)\\spad{\\br} \\tab{4}&)+(\\spad{-0}.04153119955569051D0*Z(7))\\spad{+0}.2176561076571465D0*Z(6)+(\\spad{-0}.0532\\spad{\\br} \\tab{4}&5555586632358D0*Z(5))+(\\spad{-0}.1688977368984641D0*Z(4))+(\\spad{-0}.32440166056\\spad{\\br} \\tab{4}&67343D0*Z(3))\\spad{+0}.9128222941872173D0*Z(2)+(\\spad{-0}.2419652703415429D0*Z(1\\spad{\\br} \\tab{4}&))\\spad{\\br} \\tab{5}\\spad{W}(3)\\spad{=0}.03371198197190302D0*Z(16)\\spad{+0}.02021603150122265D0*Z(15)+(\\spad{-0}.0\\spad{\\br} \\tab{4}&06607305534689702D0*Z(14))+(\\spad{-0}.03032392238968179D0*Z(13))\\spad{+0}.002033\\spad{\\br} \\tab{4}&305231024948D0*Z(12)\\spad{+0}.05375944956767728D0*Z(11)+(\\spad{-0}.0163213312502\\spad{\\br} \\tab{4}&9967D0*Z(10))+(\\spad{-0}.05483186562035512D0*Z(9))+(\\spad{-0}.04901428822579872D\\spad{\\br} \\tab{4}&0*Z(8))\\spad{+0}.2091097927887612D0*Z(7)+(\\spad{-0}.05760560341383113D0*Z(6))+(-\\spad{\\br} \\tab{4}\\spad{&0}.1236679206156403D0*Z(5))+(\\spad{-0}.3523683853026259D0*Z(4))\\spad{+0}.88929961\\spad{\\br} \\tab{4}&32269974D0*Z(3)+(\\spad{-0}.2995429545781457D0*Z(2))+(\\spad{-0}.02986582812574917\\spad{\\br} \\tab{4}&D0*Z(1))\\spad{\\br} \\tab{5}\\spad{W}(4)\\spad{=0}.05141563713660119D0*Z(16)\\spad{+0}.005239165960779299D0*Z(15)+(\\spad{-0}.\\spad{\\br} \\tab{4}&01623427735779699D0*Z(14))+(\\spad{-0}.01965809746040371D0*Z(13))\\spad{+0}.054688\\spad{\\br} \\tab{4}&97337339577D0*Z(12)+(\\spad{-0}.014224695935687D0*Z(11))+(\\spad{-0}.0505181779315\\spad{\\br} \\tab{4}&6355D0*Z(10))+(\\spad{-0}.04353074206076491D0*Z(9))\\spad{+0}.2012230497530726D0*Z\\spad{\\br} \\tab{4}&(8)+(\\spad{-0}.06630874514535952D0*Z(7))+(\\spad{-0}.1280829963720053D0*Z(6))+(\\spad{-0}\\spad{\\br} \\tab{4}&.305169742604165D0*Z(5))\\spad{+0}.8600427128450191D0*Z(4)+(\\spad{-0}.32415033802\\spad{\\br} \\tab{4}&68184D0*Z(3))+(\\spad{-0}.09033531980693314D0*Z(2))\\spad{+0}.09089205517109111D0*\\spad{\\br} \\tab{4}\\spad{&Z}(1)\\spad{\\br} \\tab{5}\\spad{W}(5)\\spad{=0}.04556369767776375D0*Z(16)+(\\spad{-0}.001822737697581869D0*Z(15))+(\\spad{\\br} \\tab{4}&-0.002512226501941856D0*Z(14))\\spad{+0}.02947046460707379D0*Z(13)+(\\spad{-0}.014\\spad{\\br} \\tab{4}&45079632086177D0*Z(12))+(\\spad{-0}.05034242196614937D0*Z(11))+(\\spad{-0}.0376966\\spad{\\br} \\tab{4}&3291725935D0*Z(10))\\spad{+0}.2171103102175198D0*Z(9)+(\\spad{-0}.0824949256021352\\spad{\\br} \\tab{4}&4D0*Z(8))+(\\spad{-0}.1473995209288945D0*Z(7))+(\\spad{-0}.315042193418466D0*Z(6))\\spad{\\br} \\tab{4}\\spad{&+0}.9591623347824002D0*Z(5)+(\\spad{-0}.3852396953763045D0*Z(4))+(\\spad{-0}.141718\\spad{\\br} \\tab{4}&5427288274D0*Z(3))+(\\spad{-0}.03423495461011043D0*Z(2))\\spad{+0}.319820917706851\\spad{\\br} \\tab{4}&6D0*Z(1)\\spad{\\br} \\tab{5}\\spad{W}(6)\\spad{=0}.04015147277405744D0*Z(16)\\spad{+0}.01328585741341559D0*Z(15)\\spad{+0}.048\\spad{\\br} \\tab{4}&26082005465965D0*Z(14)+(\\spad{-0}.04319641116207706D0*Z(13))+(\\spad{-0}.04931323\\spad{\\br} \\tab{4}&319055762D0*Z(12))+(\\spad{-0}.03526886317505474D0*Z(11))\\spad{+0}.22295383396730\\spad{\\br} \\tab{4}&01D0*Z(10)+(\\spad{-0}.07375317649315155D0*Z(9))+(\\spad{-0}.1589391311991561D0*Z(\\spad{\\br} \\tab{4}\\spad{&8}))+(\\spad{-0}.328001910890377D0*Z(7))\\spad{+0}.952576555482747D0*Z(6)+(\\spad{-0}.31583\\spad{\\br} \\tab{4}&09975786731D0*Z(5))+(\\spad{-0}.1846882042225383D0*Z(4))+(\\spad{-0}.0703762046700\\spad{\\br} \\tab{4}&4427D0*Z(3))\\spad{+0}.2311852964327382D0*Z(2)\\spad{+0}.04254083491825025D0*Z(1)\\spad{\\br} \\tab{5}\\spad{W}(7)\\spad{=0}.06069778964023718D0*Z(16)\\spad{+0}.06681263884671322D0*Z(15)+(\\spad{-0}.0\\spad{\\br} \\tab{4}&2113506688615768D0*Z(14))+(\\spad{-0}.083996867458326D0*Z(13))+(\\spad{-0}.0329843\\spad{\\br} \\tab{4}&8523869648D0*Z(12))\\spad{+0}.2276878326327734D0*Z(11)+(\\spad{-0}.067356038933017\\spad{\\br} \\tab{4}&95D0*Z(10))+(\\spad{-0}.1559813965382218D0*Z(9))+(\\spad{-0}.3363262957694705D0*Z(\\spad{\\br} \\tab{4}\\spad{&8}))\\spad{+0}.9442791158560948D0*Z(7)+(\\spad{-0}.3199955249404657D0*Z(6))+(\\spad{-0}.136\\spad{\\br} \\tab{4}&2463839920727D0*Z(5))+(\\spad{-0}.1006185171570586D0*Z(4))\\spad{+0}.2057504515015\\spad{\\br} \\tab{4}&423D0*Z(3)+(\\spad{-0}.02065879269286707D0*Z(2))\\spad{+0}.03160990266745513D0*Z(1\\spad{\\br} \\tab{4}&)\\spad{\\br} \\tab{5}\\spad{W}(8)\\spad{=0}.126386868896738D0*Z(16)\\spad{+0}.002563370039476418D0*Z(15)+(\\spad{-0}.05\\spad{\\br} \\tab{4}&581757739455641D0*Z(14))+(\\spad{-0}.07777893205900685D0*Z(13))\\spad{+0}.23117338\\spad{\\br} \\tab{4}&45834199D0*Z(12)+(\\spad{-0}.06031581134427592D0*Z(11))+(\\spad{-0}.14805474755869\\spad{\\br} \\tab{4}&52D0*Z(10))+(\\spad{-0}.3364014128402243D0*Z(9))\\spad{+0}.9364014128402244D0*Z(8)\\spad{\\br} \\tab{4}\\spad{&+}(\\spad{-0}.3269452524413048D0*Z(7))+(\\spad{-0}.1396841886557241D0*Z(6))+(\\spad{-0}.056\\spad{\\br} \\tab{4}&1733845834199D0*Z(5))\\spad{+0}.1777789320590069D0*Z(4)+(\\spad{-0}.04418242260544\\spad{\\br} \\tab{4}&359D0*Z(3))+(\\spad{-0}.02756337003947642D0*Z(2))\\spad{+0}.07361313110326199D0*Z(\\spad{\\br} \\tab{4}\\spad{&1})\\spad{\\br} \\tab{5}\\spad{W}(9)\\spad{=0}.07361313110326199D0*Z(16)+(\\spad{-0}.02756337003947642D0*Z(15))+(-\\spad{\\br} \\tab{4}\\spad{&0}.04418242260544359D0*Z(14))\\spad{+0}.1777789320590069D0*Z(13)+(\\spad{-0}.056173\\spad{\\br} \\tab{4}&3845834199D0*Z(12))+(\\spad{-0}.1396841886557241D0*Z(11))+(\\spad{-0}.326945252441\\spad{\\br} \\tab{4}&3048D0*Z(10))\\spad{+0}.9364014128402244D0*Z(9)+(\\spad{-0}.3364014128402243D0*Z(8\\spad{\\br} \\tab{4}&))+(\\spad{-0}.1480547475586952D0*Z(7))+(\\spad{-0}.06031581134427592D0*Z(6))\\spad{+0}.23\\spad{\\br} \\tab{4}&11733845834199D0*Z(5)+(\\spad{-0}.07777893205900685D0*Z(4))+(\\spad{-0}.0558175773\\spad{\\br} \\tab{4}&9455641D0*Z(3))\\spad{+0}.002563370039476418D0*Z(2)\\spad{+0}.126386868896738D0*Z(\\spad{\\br} \\tab{4}\\spad{&1})\\spad{\\br} \\tab{5}\\spad{W}(10)\\spad{=0}.03160990266745513D0*Z(16)+(\\spad{-0}.02065879269286707D0*Z(15))\\spad{+0}\\spad{\\br} \\tab{4}&.2057504515015423D0*Z(14)+(\\spad{-0}.1006185171570586D0*Z(13))+(\\spad{-0}.136246\\spad{\\br} \\tab{4}&3839920727D0*Z(12))+(\\spad{-0}.3199955249404657D0*Z(11))\\spad{+0}.94427911585609\\spad{\\br} \\tab{4}&48D0*Z(10)+(\\spad{-0}.3363262957694705D0*Z(9))+(\\spad{-0}.1559813965382218D0*Z(8\\spad{\\br} \\tab{4}&))+(\\spad{-0}.06735603893301795D0*Z(7))\\spad{+0}.2276878326327734D0*Z(6)+(\\spad{-0}.032\\spad{\\br} \\tab{4}&98438523869648D0*Z(5))+(\\spad{-0}.083996867458326D0*Z(4))+(\\spad{-0}.02113506688\\spad{\\br} \\tab{4}&615768D0*Z(3))\\spad{+0}.06681263884671322D0*Z(2)\\spad{+0}.06069778964023718D0*Z(\\spad{\\br} \\tab{4}\\spad{&1})\\spad{\\br} \\tab{5}\\spad{W}(11)\\spad{=0}.04254083491825025D0*Z(16)\\spad{+0}.2311852964327382D0*Z(15)+(\\spad{-0}.0\\spad{\\br} \\tab{4}&7037620467004427D0*Z(14))+(\\spad{-0}.1846882042225383D0*Z(13))+(\\spad{-0}.315830\\spad{\\br} \\tab{4}&9975786731D0*Z(12))\\spad{+0}.952576555482747D0*Z(11)+(\\spad{-0}.328001910890377D\\spad{\\br} \\tab{4}&0*Z(10))+(\\spad{-0}.1589391311991561D0*Z(9))+(\\spad{-0}.07375317649315155D0*Z(8)\\spad{\\br} \\tab{4}&)\\spad{+0}.2229538339673001D0*Z(7)+(\\spad{-0}.03526886317505474D0*Z(6))+(\\spad{-0}.0493\\spad{\\br} \\tab{4}&1323319055762D0*Z(5))+(\\spad{-0}.04319641116207706D0*Z(4))\\spad{+0}.048260820054\\spad{\\br} \\tab{4}&65965D0*Z(3)\\spad{+0}.01328585741341559D0*Z(2)\\spad{+0}.04015147277405744D0*Z(1)\\spad{\\br} \\tab{5}\\spad{W}(12)\\spad{=0}.3198209177068516D0*Z(16)+(\\spad{-0}.03423495461011043D0*Z(15))+(-\\spad{\\br} \\tab{4}\\spad{&0}.1417185427288274D0*Z(14))+(\\spad{-0}.3852396953763045D0*Z(13))\\spad{+0}.959162\\spad{\\br} \\tab{4}&3347824002D0*Z(12)+(\\spad{-0}.315042193418466D0*Z(11))+(\\spad{-0}.14739952092889\\spad{\\br} \\tab{4}&45D0*Z(10))+(\\spad{-0}.08249492560213524D0*Z(9))\\spad{+0}.2171103102175198D0*Z(8\\spad{\\br} \\tab{4}&)+(\\spad{-0}.03769663291725935D0*Z(7))+(\\spad{-0}.05034242196614937D0*Z(6))+(\\spad{-0}.\\spad{\\br} \\tab{4}&01445079632086177D0*Z(5))\\spad{+0}.02947046460707379D0*Z(4)+(\\spad{-0}.002512226\\spad{\\br} \\tab{4}&501941856D0*Z(3))+(\\spad{-0}.001822737697581869D0*Z(2))\\spad{+0}.045563697677763\\spad{\\br} \\tab{4}&75D0*Z(1)\\spad{\\br} \\tab{5}\\spad{W}(13)\\spad{=0}.09089205517109111D0*Z(16)+(\\spad{-0}.09033531980693314D0*Z(15))+(\\spad{\\br} \\tab{4}&-0.3241503380268184D0*Z(14))\\spad{+0}.8600427128450191D0*Z(13)+(\\spad{-0}.305169\\spad{\\br} \\tab{4}&742604165D0*Z(12))+(\\spad{-0}.1280829963720053D0*Z(11))+(\\spad{-0}.0663087451453\\spad{\\br} \\tab{4}&5952D0*Z(10))\\spad{+0}.2012230497530726D0*Z(9)+(\\spad{-0}.04353074206076491D0*Z(\\spad{\\br} \\tab{4}\\spad{&8}))+(\\spad{-0}.05051817793156355D0*Z(7))+(\\spad{-0}.014224695935687D0*Z(6))\\spad{+0}.05\\spad{\\br} \\tab{4}&468897337339577D0*Z(5)+(\\spad{-0}.01965809746040371D0*Z(4))+(\\spad{-0}.016234277\\spad{\\br} \\tab{4}&35779699D0*Z(3))\\spad{+0}.005239165960779299D0*Z(2)\\spad{+0}.05141563713660119D0\\spad{\\br} \\tab{4}\\spad{&*Z}(1)\\spad{\\br} \\tab{5}\\spad{W}(14)=(\\spad{-0}.02986582812574917D0*Z(16))+(\\spad{-0}.2995429545781457D0*Z(15))\\spad{\\br} \\tab{4}\\spad{&+0}.8892996132269974D0*Z(14)+(\\spad{-0}.3523683853026259D0*Z(13))+(\\spad{-0}.1236\\spad{\\br} \\tab{4}&679206156403D0*Z(12))+(\\spad{-0}.05760560341383113D0*Z(11))\\spad{+0}.20910979278\\spad{\\br} \\tab{4}&87612D0*Z(10)+(\\spad{-0}.04901428822579872D0*Z(9))+(\\spad{-0}.05483186562035512D\\spad{\\br} \\tab{4}&0*Z(8))+(\\spad{-0}.01632133125029967D0*Z(7))\\spad{+0}.05375944956767728D0*Z(6)\\spad{+0}\\spad{\\br} \\tab{4}&.002033305231024948D0*Z(5)+(\\spad{-0}.03032392238968179D0*Z(4))+(\\spad{-0}.00660\\spad{\\br} \\tab{4}&7305534689702D0*Z(3))\\spad{+0}.02021603150122265D0*Z(2)\\spad{+0}.033711981971903\\spad{\\br} \\tab{4}&02D0*Z(1)\\spad{\\br} \\tab{5}\\spad{W}(15)=(\\spad{-0}.2419652703415429D0*Z(16))\\spad{+0}.9128222941872173D0*Z(15)+(\\spad{-0}\\spad{\\br} \\tab{4}&.3244016605667343D0*Z(14))+(\\spad{-0}.1688977368984641D0*Z(13))+(\\spad{-0}.05325\\spad{\\br} \\tab{4}&555586632358D0*Z(12))\\spad{+0}.2176561076571465D0*Z(11)+(\\spad{-0}.0415311995556\\spad{\\br} \\tab{4}&9051D0*Z(10))+(\\spad{-0}.06095390688679697D0*Z(9))+(\\spad{-0}.01981532388243379D\\spad{\\br} \\tab{4}&0*Z(8))\\spad{+0}.05258889186338282D0*Z(7)\\spad{+0}.00157466157362272D0*Z(6)+(\\spad{-0}.\\spad{\\br} \\tab{4}&0135713672105995D0*Z(5))+(\\spad{-0}.01764072463999744D0*Z(4))\\spad{+0}.010940122\\spad{\\br} \\tab{4}&10519586D0*Z(3)\\spad{+0}.008812321197398072D0*Z(2)\\spad{+0}.0227345011107737D0*Z\\spad{\\br} \\tab{4}&(1)\\spad{\\br} \\tab{5}\\spad{W}(16)\\spad{=1}.019463911841327D0*Z(16)+(\\spad{-0}.2803531651057233D0*Z(15))+(\\spad{-0}.\\spad{\\br} \\tab{4}&1165300508238904D0*Z(14))+(\\spad{-0}.1385343580686922D0*Z(13))\\spad{+0}.22647669\\spad{\\br} \\tab{4}&47290192D0*Z(12)+(\\spad{-0}.02434652144032987D0*Z(11))+(\\spad{-0}.04723268012114\\spad{\\br} \\tab{4}&625D0*Z(10))+(\\spad{-0}.03586220812223305D0*Z(9))\\spad{+0}.04932374658377151D0*Z\\spad{\\br} \\tab{4}&(8)\\spad{+0}.00372306473653087D0*Z(7)+(\\spad{-0}.01219194009813166D0*Z(6))+(\\spad{-0}.0\\spad{\\br} \\tab{4}&07005540882865317D0*Z(5))\\spad{+0}.002957434991769087D0*Z(4)\\spad{+0}.0021069739\\spad{\\br} \\tab{4}&00813502D0*Z(3)\\spad{+0}.001747395874954051D0*Z(2)\\spad{+0}.01707454969713436D0*\\spad{\\br} \\tab{4}\\spad{&Z}(1)\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END\\spad{\\br}"))) 
NIL 
NIL 
(-74 -3409) 
((|constructor| (NIL "\\spadtype{Asp29} produces Fortran for Type 29 ASPs,{} needed for NAG routine f02fjf,{} for example: \\blankline \\tab{5}SUBROUTINE MONIT(ISTATE,{}NEXTIT,{}NEVALS,{}NEVECS,{}\\spad{K},{}\\spad{F},{}\\spad{D})\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{D}(\\spad{K}),{}\\spad{F}(\\spad{K})\\spad{\\br} \\tab{5}INTEGER \\spad{K},{}NEXTIT,{}NEVALS,{}NVECS,{}ISTATE\\spad{\\br} \\tab{5}CALL F02FJZ(ISTATE,{}NEXTIT,{}NEVALS,{}NEVECS,{}\\spad{K},{}\\spad{F},{}\\spad{D})\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END\\spad{\\br}")) (|outputAsFortran| (((|Void|)) "\\spad{outputAsFortran()} generates the default code for \\spadtype{ASP29}."))) 
NIL 
NIL 
(-75 -3409) 
((|constructor| (NIL "\\spadtype{Asp30} produces Fortran for Type 30 ASPs,{} needed for NAG routine f04qaf,{} for example: \\blankline \\tab{5}SUBROUTINE APROD(MODE,{}\\spad{M},{}\\spad{N},{}\\spad{X},{}\\spad{Y},{}RWORK,{}LRWORK,{}IWORK,{}LIWORK)\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{X}(\\spad{N}),{}\\spad{Y}(\\spad{M}),{}RWORK(LRWORK)\\spad{\\br} \\tab{5}INTEGER \\spad{M},{}\\spad{N},{}LIWORK,{}IFAIL,{}LRWORK,{}IWORK(LIWORK),{}MODE\\spad{\\br} \\tab{5}DOUBLE PRECISION A(5,{}5)\\spad{\\br} \\tab{5}EXTERNAL F06PAF\\spad{\\br} \\tab{5}A(1,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}A(1,{}2)\\spad{=0}.0D0\\spad{\\br} \\tab{5}A(1,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{5}A(1,{}4)=-1.0D0\\spad{\\br} \\tab{5}A(1,{}5)\\spad{=0}.0D0\\spad{\\br} \\tab{5}A(2,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}A(2,{}2)\\spad{=1}.0D0\\spad{\\br} \\tab{5}A(2,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{5}A(2,{}4)\\spad{=0}.0D0\\spad{\\br} \\tab{5}A(2,{}5)=-1.0D0\\spad{\\br} \\tab{5}A(3,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}A(3,{}2)\\spad{=0}.0D0\\spad{\\br} \\tab{5}A(3,{}3)\\spad{=1}.0D0\\spad{\\br} \\tab{5}A(3,{}4)=-1.0D0\\spad{\\br} \\tab{5}A(3,{}5)\\spad{=0}.0D0\\spad{\\br} \\tab{5}A(4,{}1)=-1.0D0\\spad{\\br} \\tab{5}A(4,{}2)\\spad{=0}.0D0\\spad{\\br} \\tab{5}A(4,{}3)=-1.0D0\\spad{\\br} \\tab{5}A(4,{}4)\\spad{=4}.0D0\\spad{\\br} \\tab{5}A(4,{}5)=-1.0D0\\spad{\\br} \\tab{5}A(5,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}A(5,{}2)=-1.0D0\\spad{\\br} \\tab{5}A(5,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{5}A(5,{}4)=-1.0D0\\spad{\\br} \\tab{5}A(5,{}5)\\spad{=4}.0D0\\spad{\\br} \\tab{5}IF(MODE.EQ.1)THEN\\spad{\\br} \\tab{7}CALL F06PAF(\\spad{'N'},{}\\spad{M},{}\\spad{N},{}1.0D0,{}A,{}\\spad{M},{}\\spad{X},{}1,{}1.0D0,{}\\spad{Y},{}1)\\spad{\\br} \\tab{5}ELSEIF(MODE.EQ.2)THEN\\spad{\\br} \\tab{7}CALL F06PAF(\\spad{'T'},{}\\spad{M},{}\\spad{N},{}1.0D0,{}A,{}\\spad{M},{}\\spad{Y},{}1,{}1.0D0,{}\\spad{X},{}1)\\spad{\\br} \\tab{5}ENDIF\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END"))) 
NIL 
NIL 
(-76 -3409) 
((|constructor| (NIL "\\spadtype{Asp31} produces Fortran for Type 31 ASPs,{} needed for NAG routine d02ejf,{} for example: \\blankline \\tab{5}SUBROUTINE PEDERV(\\spad{X},{}\\spad{Y},{}\\spad{PW})\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{X},{}\\spad{Y}(*)\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{PW}(3,{}3)\\spad{\\br} \\tab{5}\\spad{PW}(1,{}1)=-0.03999999999999999D0\\spad{\\br} \\tab{5}\\spad{PW}(1,{}2)\\spad{=10000}.0D0*Y(3)\\spad{\\br} \\tab{5}\\spad{PW}(1,{}3)\\spad{=10000}.0D0*Y(2)\\spad{\\br} \\tab{5}\\spad{PW}(2,{}1)\\spad{=0}.03999999999999999D0\\spad{\\br} \\tab{5}\\spad{PW}(2,{}2)=(\\spad{-10000}.0D0*Y(3))+(\\spad{-60000000}.0D0*Y(2))\\spad{\\br} \\tab{5}\\spad{PW}(2,{}3)=-10000.0D0*Y(2)\\spad{\\br} \\tab{5}\\spad{PW}(3,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{PW}(3,{}2)\\spad{=60000000}.0D0*Y(2)\\spad{\\br} \\tab{5}\\spad{PW}(3,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct| (QUOTE X)) (|construct| (QUOTE Y)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-77 -3409) 
((|constructor| (NIL "\\spadtype{Asp33} produces Fortran for Type 33 ASPs,{} needed for NAG routine d02kef. The code is a dummy ASP: \\blankline \\tab{5}SUBROUTINE REPORT(\\spad{X},{}\\spad{V},{}JINT)\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{V}(3),{}\\spad{X}\\spad{\\br} \\tab{5}INTEGER JINT\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|outputAsFortran| (((|Void|)) "\\spad{outputAsFortran()} generates the default code for \\spadtype{ASP33}."))) 
NIL 
NIL 
(-78 -3409) 
((|constructor| (NIL "\\spadtype{Asp34} produces Fortran for Type 34 ASPs,{} needed for NAG routine f04mbf,{} for example: \\blankline \\tab{5}SUBROUTINE MSOLVE(IFLAG,{}\\spad{N},{}\\spad{X},{}\\spad{Y},{}RWORK,{}LRWORK,{}IWORK,{}LIWORK)\\spad{\\br} \\tab{5}DOUBLE PRECISION RWORK(LRWORK),{}\\spad{X}(\\spad{N}),{}\\spad{Y}(\\spad{N})\\spad{\\br} \\tab{5}INTEGER \\spad{I},{}\\spad{J},{}\\spad{N},{}LIWORK,{}IFLAG,{}LRWORK,{}IWORK(LIWORK)\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{W1}(3),{}\\spad{W2}(3),{}\\spad{MS}(3,{}3)\\spad{\\br} \\tab{5}IFLAG=-1\\spad{\\br} \\tab{5}\\spad{MS}(1,{}1)\\spad{=2}.0D0\\spad{\\br} \\tab{5}\\spad{MS}(1,{}2)\\spad{=1}.0D0\\spad{\\br} \\tab{5}\\spad{MS}(1,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{MS}(2,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}\\spad{MS}(2,{}2)\\spad{=2}.0D0\\spad{\\br} \\tab{5}\\spad{MS}(2,{}3)\\spad{=1}.0D0\\spad{\\br} \\tab{5}\\spad{MS}(3,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{MS}(3,{}2)\\spad{=1}.0D0\\spad{\\br} \\tab{5}\\spad{MS}(3,{}3)\\spad{=2}.0D0\\spad{\\br} \\tab{5}CALL F04ASF(\\spad{MS},{}\\spad{N},{}\\spad{X},{}\\spad{N},{}\\spad{Y},{}\\spad{W1},{}\\spad{W2},{}IFLAG)\\spad{\\br} \\tab{5}IFLAG=-IFLAG\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END"))) 
NIL 
NIL 
(-79 -3409) 
((|constructor| (NIL "\\spadtype{Asp35} produces Fortran for Type 35 ASPs,{} needed for NAG routines c05pbf,{} c05pcf,{} for example: \\blankline \\tab{5}SUBROUTINE \\spad{FCN}(\\spad{N},{}\\spad{X},{}FVEC,{}FJAC,{}LDFJAC,{}IFLAG)\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{X}(\\spad{N}),{}FVEC(\\spad{N}),{}FJAC(LDFJAC,{}\\spad{N})\\spad{\\br} \\tab{5}INTEGER LDFJAC,{}\\spad{N},{}IFLAG\\spad{\\br} \\tab{5}IF(IFLAG.EQ.1)THEN\\spad{\\br} \\tab{7}FVEC(1)=(\\spad{-1}.0D0*X(2))\\spad{+X}(1)\\spad{\\br} \\tab{7}FVEC(2)=(\\spad{-1}.0D0*X(3))\\spad{+2}.0D0*X(2)\\spad{\\br} \\tab{7}FVEC(3)\\spad{=3}.0D0*X(3)\\spad{\\br} \\tab{5}ELSEIF(IFLAG.EQ.2)THEN\\spad{\\br} \\tab{7}FJAC(1,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{7}FJAC(1,{}2)=-1.0D0\\spad{\\br} \\tab{7}FJAC(1,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{7}FJAC(2,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{7}FJAC(2,{}2)\\spad{=2}.0D0\\spad{\\br} \\tab{7}FJAC(2,{}3)=-1.0D0\\spad{\\br} \\tab{7}FJAC(3,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{7}FJAC(3,{}2)\\spad{=0}.0D0\\spad{\\br} \\tab{7}FJAC(3,{}3)\\spad{=3}.0D0\\spad{\\br} \\tab{5}ENDIF\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct|) (|construct| (QUOTE X)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-80 |nameOne| |nameTwo| |nameThree|) 
((|constructor| (NIL "\\spadtype{Asp41} produces Fortran for Type 41 ASPs,{} needed for NAG routines d02raf and d02saf in particular. These ASPs are in fact three Fortran routines which return a vector of functions,{} and their derivatives \\spad{wrt} \\spad{Y}(\\spad{i}) and also a continuation parameter EPS,{} for example: \\blankline \\tab{5}SUBROUTINE \\spad{FCN}(\\spad{X},{}EPS,{}\\spad{Y},{}\\spad{F},{}\\spad{N})\\spad{\\br} \\tab{5}DOUBLE PRECISION EPS,{}\\spad{F}(\\spad{N}),{}\\spad{X},{}\\spad{Y}(\\spad{N})\\spad{\\br} \\tab{5}INTEGER \\spad{N}\\spad{\\br} \\tab{5}\\spad{F}(1)\\spad{=Y}(2)\\spad{\\br} \\tab{5}\\spad{F}(2)\\spad{=Y}(3)\\spad{\\br} \\tab{5}\\spad{F}(3)=(\\spad{-1}.0D0*Y(1)\\spad{*Y}(3))\\spad{+2}.0D0*EPS*Y(2)**2+(\\spad{-2}.0D0*EPS)\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END\\spad{\\br} \\tab{5}SUBROUTINE JACOBF(\\spad{X},{}EPS,{}\\spad{Y},{}\\spad{F},{}\\spad{N})\\spad{\\br} \\tab{5}DOUBLE PRECISION EPS,{}\\spad{F}(\\spad{N},{}\\spad{N}),{}\\spad{X},{}\\spad{Y}(\\spad{N})\\spad{\\br} \\tab{5}INTEGER \\spad{N}\\spad{\\br} \\tab{5}\\spad{F}(1,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{F}(1,{}2)\\spad{=1}.0D0\\spad{\\br} \\tab{5}\\spad{F}(1,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{F}(2,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{F}(2,{}2)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{F}(2,{}3)\\spad{=1}.0D0\\spad{\\br} \\tab{5}\\spad{F}(3,{}1)=-1.0D0*Y(3)\\spad{\\br} \\tab{5}\\spad{F}(3,{}2)\\spad{=4}.0D0*EPS*Y(2)\\spad{\\br} \\tab{5}\\spad{F}(3,{}3)=-1.0D0*Y(1)\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END\\spad{\\br} \\tab{5}SUBROUTINE JACEPS(\\spad{X},{}EPS,{}\\spad{Y},{}\\spad{F},{}\\spad{N})\\spad{\\br} \\tab{5}DOUBLE PRECISION EPS,{}\\spad{F}(\\spad{N}),{}\\spad{X},{}\\spad{Y}(\\spad{N})\\spad{\\br} \\tab{5}INTEGER \\spad{N}\\spad{\\br} \\tab{5}\\spad{F}(1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{F}(2)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{F}(3)\\spad{=2}.0D0*Y(2)\\spad{**2}-2.0D0\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct| (QUOTE X) (QUOTE EPS)) (|construct| (QUOTE Y)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-81 |nameOne| |nameTwo| |nameThree|) 
((|constructor| (NIL "\\spadtype{Asp42} produces Fortran for Type 42 ASPs,{} needed for NAG routines d02raf and d02saf in particular. These ASPs are in fact three Fortran routines which return a vector of functions,{} and their derivatives \\spad{wrt} \\spad{Y}(\\spad{i}) and also a continuation parameter EPS,{} for example: \\blankline \\tab{5}SUBROUTINE \\spad{G}(EPS,{}YA,{}\\spad{YB},{}\\spad{BC},{}\\spad{N})\\spad{\\br} \\tab{5}DOUBLE PRECISION EPS,{}YA(\\spad{N}),{}\\spad{YB}(\\spad{N}),{}\\spad{BC}(\\spad{N})\\spad{\\br} \\tab{5}INTEGER \\spad{N}\\spad{\\br} \\tab{5}\\spad{BC}(1)=YA(1)\\spad{\\br} \\tab{5}\\spad{BC}(2)=YA(2)\\spad{\\br} \\tab{5}\\spad{BC}(3)\\spad{=YB}(2)\\spad{-1}.0D0\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END\\spad{\\br} \\tab{5}SUBROUTINE JACOBG(EPS,{}YA,{}\\spad{YB},{}AJ,{}\\spad{BJ},{}\\spad{N})\\spad{\\br} \\tab{5}DOUBLE PRECISION EPS,{}YA(\\spad{N}),{}AJ(\\spad{N},{}\\spad{N}),{}\\spad{BJ}(\\spad{N},{}\\spad{N}),{}\\spad{YB}(\\spad{N})\\spad{\\br} \\tab{5}INTEGER \\spad{N}\\spad{\\br} \\tab{5}AJ(1,{}1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}AJ(1,{}2)\\spad{=0}.0D0\\spad{\\br} \\tab{5}AJ(1,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{5}AJ(2,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}AJ(2,{}2)\\spad{=1}.0D0\\spad{\\br} \\tab{5}AJ(2,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{5}AJ(3,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}AJ(3,{}2)\\spad{=0}.0D0\\spad{\\br} \\tab{5}AJ(3,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{BJ}(1,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{BJ}(1,{}2)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{BJ}(1,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{BJ}(2,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{BJ}(2,{}2)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{BJ}(2,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{BJ}(3,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{BJ}(3,{}2)\\spad{=1}.0D0\\spad{\\br} \\tab{5}\\spad{BJ}(3,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END\\spad{\\br} \\tab{5}SUBROUTINE JACGEP(EPS,{}YA,{}\\spad{YB},{}BCEP,{}\\spad{N})\\spad{\\br} \\tab{5}DOUBLE PRECISION EPS,{}YA(\\spad{N}),{}\\spad{YB}(\\spad{N}),{}BCEP(\\spad{N})\\spad{\\br} \\tab{5}INTEGER \\spad{N}\\spad{\\br} \\tab{5}BCEP(1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}BCEP(2)\\spad{=0}.0D0\\spad{\\br} \\tab{5}BCEP(3)\\spad{=0}.0D0\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct| (QUOTE EPS)) (|construct| (QUOTE YA) (QUOTE YB)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-82 -3409) 
((|constructor| (NIL "\\spadtype{Asp49} produces Fortran for Type 49 ASPs,{} needed for NAG routines e04dgf,{} e04ucf,{} for example: \\blankline \\tab{5}SUBROUTINE OBJFUN(MODE,{}\\spad{N},{}\\spad{X},{}OBJF,{}OBJGRD,{}NSTATE,{}IUSER,{}USER)\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{X}(\\spad{N}),{}OBJF,{}OBJGRD(\\spad{N}),{}USER(*)\\spad{\\br} \\tab{5}INTEGER \\spad{N},{}IUSER(*),{}MODE,{}NSTATE\\spad{\\br} \\tab{5}OBJF=X(4)\\spad{*X}(9)+((\\spad{-1}.0D0*X(5))\\spad{+X}(3))\\spad{*X}(8)+((\\spad{-1}.0D0*X(3))\\spad{+X}(1))\\spad{*X}(7)\\spad{\\br} \\tab{4}\\spad{&+}(\\spad{-1}.0D0*X(2)\\spad{*X}(6))\\spad{\\br} \\tab{5}OBJGRD(1)\\spad{=X}(7)\\spad{\\br} \\tab{5}OBJGRD(2)=-1.0D0*X(6)\\spad{\\br} \\tab{5}OBJGRD(3)\\spad{=X}(8)+(\\spad{-1}.0D0*X(7))\\spad{\\br} \\tab{5}OBJGRD(4)\\spad{=X}(9)\\spad{\\br} \\tab{5}OBJGRD(5)=-1.0D0*X(8)\\spad{\\br} \\tab{5}OBJGRD(6)=-1.0D0*X(2)\\spad{\\br} \\tab{5}OBJGRD(7)=(\\spad{-1}.0D0*X(3))\\spad{+X}(1)\\spad{\\br} \\tab{5}OBJGRD(8)=(\\spad{-1}.0D0*X(5))\\spad{+X}(3)\\spad{\\br} \\tab{5}OBJGRD(9)\\spad{=X}(4)\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|FortranExpression| (|construct|) (|construct| (QUOTE X)) (|MachineFloat|))) "\\spad{coerce(f)} takes an object from the appropriate instantiation of \\spadtype{FortranExpression} and turns it into an ASP."))) 
NIL 
NIL 
(-83 -3409) 
((|constructor| (NIL "\\spadtype{Asp4} produces Fortran for Type 4 ASPs,{} which take an expression in \\spad{X}(1) .. \\spad{X}(NDIM) and produce a real function of the form: \\blankline \\tab{5}DOUBLE PRECISION FUNCTION FUNCTN(NDIM,{}\\spad{X})\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{X}(NDIM)\\spad{\\br} \\tab{5}INTEGER NDIM\\spad{\\br} \\tab{5}FUNCTN=(4.0D0*X(1)\\spad{*X}(3)**2*DEXP(2.0D0*X(1)\\spad{*X}(3)))/(\\spad{X}(4)**2+(2.0D0*\\spad{\\br} \\tab{4}\\spad{&X}(2)\\spad{+2}.0D0)\\spad{*X}(4)\\spad{+X}(2)\\spad{**2+2}.0D0*X(2)\\spad{+1}.0D0)\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|FortranExpression| (|construct|) (|construct| (QUOTE X)) (|MachineFloat|))) "\\spad{coerce(f)} takes an object from the appropriate instantiation of \\spadtype{FortranExpression} and turns it into an ASP."))) 
NIL 
NIL 
(-84 -3409) 
((|constructor| (NIL "\\spadtype{Asp50} produces Fortran for Type 50 ASPs,{} needed for NAG routine e04fdf,{} for example: \\blankline \\tab{5}SUBROUTINE \\spad{LSFUN1}(\\spad{M},{}\\spad{N},{}\\spad{XC},{}FVECC)\\spad{\\br} \\tab{5}DOUBLE PRECISION FVECC(\\spad{M}),{}\\spad{XC}(\\spad{N})\\spad{\\br} \\tab{5}INTEGER \\spad{I},{}\\spad{M},{}\\spad{N}\\spad{\\br} \\tab{5}FVECC(1)=((\\spad{XC}(1)\\spad{-2}.4D0)\\spad{*XC}(3)+(15.0D0*XC(1)\\spad{-36}.0D0)\\spad{*XC}(2)\\spad{+1}.0D0)/(\\spad{\\br} \\tab{4}\\spad{&XC}(3)\\spad{+15}.0D0*XC(2))\\spad{\\br} \\tab{5}FVECC(2)=((\\spad{XC}(1)\\spad{-2}.8D0)\\spad{*XC}(3)+(7.0D0*XC(1)\\spad{-19}.6D0)\\spad{*XC}(2)\\spad{+1}.0D0)/(\\spad{X}\\spad{\\br} \\tab{4}\\spad{&C}(3)\\spad{+7}.0D0*XC(2))\\spad{\\br} \\tab{5}FVECC(3)=((\\spad{XC}(1)\\spad{-3}.2D0)\\spad{*XC}(3)+(4.333333333333333D0*XC(1)\\spad{-13}.866666\\spad{\\br} \\tab{4}\\spad{&66666667D0})\\spad{*XC}(2)\\spad{+1}.0D0)/(\\spad{XC}(3)\\spad{+4}.333333333333333D0*XC(2))\\spad{\\br} \\tab{5}FVECC(4)=((\\spad{XC}(1)\\spad{-3}.5D0)\\spad{*XC}(3)+(3.0D0*XC(1)\\spad{-10}.5D0)\\spad{*XC}(2)\\spad{+1}.0D0)/(\\spad{X}\\spad{\\br} \\tab{4}\\spad{&C}(3)\\spad{+3}.0D0*XC(2))\\spad{\\br} \\tab{5}FVECC(5)=((\\spad{XC}(1)\\spad{-3}.9D0)\\spad{*XC}(3)+(2.2D0*XC(1)\\spad{-8}.579999999999998D0)\\spad{*XC}\\spad{\\br} \\tab{4}&(2)\\spad{+1}.0D0)/(\\spad{XC}(3)\\spad{+2}.2D0*XC(2))\\spad{\\br} \\tab{5}FVECC(6)=((\\spad{XC}(1)\\spad{-4}.199999999999999D0)\\spad{*XC}(3)+(1.666666666666667D0*X\\spad{\\br} \\tab{4}\\spad{&C}(1)\\spad{-7}.0D0)\\spad{*XC}(2)\\spad{+1}.0D0)/(\\spad{XC}(3)\\spad{+1}.666666666666667D0*XC(2))\\spad{\\br} \\tab{5}FVECC(7)=((\\spad{XC}(1)\\spad{-4}.5D0)\\spad{*XC}(3)+(1.285714285714286D0*XC(1)\\spad{-5}.7857142\\spad{\\br} \\tab{4}\\spad{&85714286D0})\\spad{*XC}(2)\\spad{+1}.0D0)/(\\spad{XC}(3)\\spad{+1}.285714285714286D0*XC(2))\\spad{\\br} \\tab{5}FVECC(8)=((\\spad{XC}(1)\\spad{-4}.899999999999999D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-4}.8999999999999\\spad{\\br} \\tab{4}\\spad{&99D0})\\spad{*XC}(2)\\spad{+1}.0D0)/(\\spad{XC}(3)\\spad{+XC}(2))\\spad{\\br} \\tab{5}FVECC(9)=((\\spad{XC}(1)\\spad{-4}.699999999999999D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-4}.6999999999999\\spad{\\br} \\tab{4}\\spad{&99D0})\\spad{*XC}(2)\\spad{+1}.285714285714286D0)/(\\spad{XC}(3)\\spad{+XC}(2))\\spad{\\br} \\tab{5}FVECC(10)=((\\spad{XC}(1)\\spad{-6}.8D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-6}.8D0)\\spad{*XC}(2)\\spad{+1}.6666666666666\\spad{\\br} \\tab{4}\\spad{&67D0})/(\\spad{XC}(3)\\spad{+XC}(2))\\spad{\\br} \\tab{5}FVECC(11)=((\\spad{XC}(1)\\spad{-8}.299999999999999D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-8}.299999999999\\spad{\\br} \\tab{4}\\spad{&999D0})\\spad{*XC}(2)\\spad{+2}.2D0)/(\\spad{XC}(3)\\spad{+XC}(2))\\spad{\\br} \\tab{5}FVECC(12)=((\\spad{XC}(1)\\spad{-10}.6D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-10}.6D0)\\spad{*XC}(2)\\spad{+3}.0D0)/(\\spad{XC}(3)\\spad{\\br} \\tab{4}&+XC(2))\\spad{\\br} \\tab{5}FVECC(13)=((\\spad{XC}(1)\\spad{-1}.34D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-1}.34D0)\\spad{*XC}(2)\\spad{+4}.33333333333\\spad{\\br} \\tab{4}\\spad{&3333D0})/(\\spad{XC}(3)\\spad{+XC}(2))\\spad{\\br} \\tab{5}FVECC(14)=((\\spad{XC}(1)\\spad{-2}.1D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-2}.1D0)\\spad{*XC}(2)\\spad{+7}.0D0)/(\\spad{XC}(3)\\spad{+X}\\spad{\\br} \\tab{4}\\spad{&C}(2))\\spad{\\br} \\tab{5}FVECC(15)=((\\spad{XC}(1)\\spad{-4}.39D0)\\spad{*XC}(3)+(\\spad{XC}(1)\\spad{-4}.39D0)\\spad{*XC}(2)\\spad{+15}.0D0)/(\\spad{XC}(3\\spad{\\br} \\tab{4}&)\\spad{+XC}(2))\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct|) (|construct| (QUOTE XC)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-85 -3409) 
((|constructor| (NIL "\\spadtype{Asp55} produces Fortran for Type 55 ASPs,{} needed for NAG routines e04dgf and e04ucf,{} for example: \\blankline \\tab{5}SUBROUTINE CONFUN(MODE,{}NCNLN,{}\\spad{N},{}NROWJ,{}NEEDC,{}\\spad{X},{}\\spad{C},{}CJAC,{}NSTATE,{}IUSER\\spad{\\br} \\tab{4}&,{}USER)\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{C}(NCNLN),{}\\spad{X}(\\spad{N}),{}CJAC(NROWJ,{}\\spad{N}),{}USER(*)\\spad{\\br} \\tab{5}INTEGER \\spad{N},{}IUSER(*),{}NEEDC(NCNLN),{}NROWJ,{}MODE,{}NCNLN,{}NSTATE\\spad{\\br} \\tab{5}IF(NEEDC(1).\\spad{GT}.0)THEN\\spad{\\br} \\tab{7}\\spad{C}(1)\\spad{=X}(6)**2+X(1)\\spad{**2}\\spad{\\br} \\tab{7}CJAC(1,{}1)\\spad{=2}.0D0*X(1)\\spad{\\br} \\tab{7}CJAC(1,{}2)\\spad{=0}.0D0\\spad{\\br} \\tab{7}CJAC(1,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{7}CJAC(1,{}4)\\spad{=0}.0D0\\spad{\\br} \\tab{7}CJAC(1,{}5)\\spad{=0}.0D0\\spad{\\br} \\tab{7}CJAC(1,{}6)\\spad{=2}.0D0*X(6)\\spad{\\br} \\tab{5}ENDIF\\spad{\\br} \\tab{5}IF(NEEDC(2).\\spad{GT}.0)THEN\\spad{\\br} \\tab{7}\\spad{C}(2)\\spad{=X}(2)**2+(\\spad{-2}.0D0*X(1)\\spad{*X}(2))\\spad{+X}(1)\\spad{**2}\\spad{\\br} \\tab{7}CJAC(2,{}1)=(\\spad{-2}.0D0*X(2))\\spad{+2}.0D0*X(1)\\spad{\\br} \\tab{7}CJAC(2,{}2)\\spad{=2}.0D0*X(2)+(\\spad{-2}.0D0*X(1))\\spad{\\br} \\tab{7}CJAC(2,{}3)\\spad{=0}.0D0\\spad{\\br} \\tab{7}CJAC(2,{}4)\\spad{=0}.0D0\\spad{\\br} \\tab{7}CJAC(2,{}5)\\spad{=0}.0D0\\spad{\\br} \\tab{7}CJAC(2,{}6)\\spad{=0}.0D0\\spad{\\br} \\tab{5}ENDIF\\spad{\\br} \\tab{5}IF(NEEDC(3).\\spad{GT}.0)THEN\\spad{\\br} \\tab{7}\\spad{C}(3)\\spad{=X}(3)**2+(\\spad{-2}.0D0*X(1)\\spad{*X}(3))\\spad{+X}(2)**2+X(1)\\spad{**2}\\spad{\\br} \\tab{7}CJAC(3,{}1)=(\\spad{-2}.0D0*X(3))\\spad{+2}.0D0*X(1)\\spad{\\br} \\tab{7}CJAC(3,{}2)\\spad{=2}.0D0*X(2)\\spad{\\br} \\tab{7}CJAC(3,{}3)\\spad{=2}.0D0*X(3)+(\\spad{-2}.0D0*X(1))\\spad{\\br} \\tab{7}CJAC(3,{}4)\\spad{=0}.0D0\\spad{\\br} \\tab{7}CJAC(3,{}5)\\spad{=0}.0D0\\spad{\\br} \\tab{7}CJAC(3,{}6)\\spad{=0}.0D0\\spad{\\br} \\tab{5}ENDIF\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct|) (|construct| (QUOTE X)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-86 -3409) 
((|constructor| (NIL "\\spadtype{Asp6} produces Fortran for Type 6 ASPs,{} needed for NAG routines c05nbf,{} c05ncf. These represent vectors of functions of \\spad{X}(\\spad{i}) and look like: \\blankline \\tab{5}SUBROUTINE \\spad{FCN}(\\spad{N},{}\\spad{X},{}FVEC,{}IFLAG) \\tab{5}DOUBLE PRECISION \\spad{X}(\\spad{N}),{}FVEC(\\spad{N}) \\tab{5}INTEGER \\spad{N},{}IFLAG \\tab{5}FVEC(1)=(\\spad{-2}.0D0*X(2))+(\\spad{-2}.0D0*X(1)\\spad{**2})\\spad{+3}.0D0*X(1)\\spad{+1}.0D0 \\tab{5}FVEC(2)=(\\spad{-2}.0D0*X(3))+(\\spad{-2}.0D0*X(2)\\spad{**2})\\spad{+3}.0D0*X(2)+(\\spad{-1}.0D0*X(1))\\spad{+1}. \\tab{4}\\spad{&0D0} \\tab{5}FVEC(3)=(\\spad{-2}.0D0*X(4))+(\\spad{-2}.0D0*X(3)\\spad{**2})\\spad{+3}.0D0*X(3)+(\\spad{-1}.0D0*X(2))\\spad{+1}. \\tab{4}\\spad{&0D0} \\tab{5}FVEC(4)=(\\spad{-2}.0D0*X(5))+(\\spad{-2}.0D0*X(4)\\spad{**2})\\spad{+3}.0D0*X(4)+(\\spad{-1}.0D0*X(3))\\spad{+1}. \\tab{4}\\spad{&0D0} \\tab{5}FVEC(5)=(\\spad{-2}.0D0*X(6))+(\\spad{-2}.0D0*X(5)\\spad{**2})\\spad{+3}.0D0*X(5)+(\\spad{-1}.0D0*X(4))\\spad{+1}. \\tab{4}\\spad{&0D0} \\tab{5}FVEC(6)=(\\spad{-2}.0D0*X(7))+(\\spad{-2}.0D0*X(6)\\spad{**2})\\spad{+3}.0D0*X(6)+(\\spad{-1}.0D0*X(5))\\spad{+1}. \\tab{4}\\spad{&0D0} \\tab{5}FVEC(7)=(\\spad{-2}.0D0*X(8))+(\\spad{-2}.0D0*X(7)\\spad{**2})\\spad{+3}.0D0*X(7)+(\\spad{-1}.0D0*X(6))\\spad{+1}. \\tab{4}\\spad{&0D0} \\tab{5}FVEC(8)=(\\spad{-2}.0D0*X(9))+(\\spad{-2}.0D0*X(8)\\spad{**2})\\spad{+3}.0D0*X(8)+(\\spad{-1}.0D0*X(7))\\spad{+1}. \\tab{4}\\spad{&0D0} \\tab{5}FVEC(9)=(\\spad{-2}.0D0*X(9)\\spad{**2})\\spad{+3}.0D0*X(9)+(\\spad{-1}.0D0*X(8))\\spad{+1}.0D0 \\tab{5}RETURN \\tab{5}END")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct|) (|construct| (QUOTE X)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-87 -3409) 
((|constructor| (NIL "\\spadtype{Asp73} produces Fortran for Type 73 ASPs,{} needed for NAG routine d03eef,{} for example: \\blankline \\tab{5}SUBROUTINE PDEF(\\spad{X},{}\\spad{Y},{}ALPHA,{}BETA,{}GAMMA,{}DELTA,{}EPSOLN,{}PHI,{}PSI)\\spad{\\br} \\tab{5}DOUBLE PRECISION ALPHA,{}EPSOLN,{}PHI,{}\\spad{X},{}\\spad{Y},{}BETA,{}DELTA,{}GAMMA,{}PSI\\spad{\\br} \\tab{5}ALPHA=DSIN(\\spad{X})\\spad{\\br} \\tab{5}BETA=Y\\spad{\\br} \\tab{5}GAMMA=X*Y\\spad{\\br} \\tab{5}DELTA=DCOS(\\spad{X})*DSIN(\\spad{Y})\\spad{\\br} \\tab{5}EPSOLN=Y+X\\spad{\\br} \\tab{5}PHI=X\\spad{\\br} \\tab{5}PSI=Y\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct| (QUOTE X) (QUOTE Y)) (|construct|) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-88 -3409) 
((|constructor| (NIL "\\spadtype{Asp74} produces Fortran for Type 74 ASPs,{} needed for NAG routine d03eef,{} for example: \\blankline \\tab{5} SUBROUTINE BNDY(\\spad{X},{}\\spad{Y},{}A,{}\\spad{B},{}\\spad{C},{}IBND)\\spad{\\br} \\tab{5} DOUBLE PRECISION A,{}\\spad{B},{}\\spad{C},{}\\spad{X},{}\\spad{Y}\\spad{\\br} \\tab{5} INTEGER IBND\\spad{\\br} \\tab{5} IF(IBND.EQ.0)THEN\\spad{\\br} \\tab{7} \\spad{A=0}.0D0\\spad{\\br} \\tab{7} \\spad{B=1}.0D0\\spad{\\br} \\tab{7} \\spad{C=}-1.0D0*DSIN(\\spad{X})\\spad{\\br} \\tab{5} ELSEIF(IBND.EQ.1)THEN\\spad{\\br} \\tab{7} \\spad{A=1}.0D0\\spad{\\br} \\tab{7} \\spad{B=0}.0D0\\spad{\\br} \\tab{7} C=DSIN(\\spad{X})*DSIN(\\spad{Y})\\spad{\\br} \\tab{5} ELSEIF(IBND.EQ.2)THEN\\spad{\\br} \\tab{7} \\spad{A=1}.0D0\\spad{\\br} \\tab{7} \\spad{B=0}.0D0\\spad{\\br} \\tab{7} C=DSIN(\\spad{X})*DSIN(\\spad{Y})\\spad{\\br} \\tab{5} ELSEIF(IBND.EQ.3)THEN\\spad{\\br} \\tab{7} \\spad{A=0}.0D0\\spad{\\br} \\tab{7} \\spad{B=1}.0D0\\spad{\\br} \\tab{7} \\spad{C=}-1.0D0*DSIN(\\spad{Y})\\spad{\\br} \\tab{5} ENDIF\\spad{\\br} \\tab{5} END")) (|coerce| (($ (|Matrix| (|FortranExpression| (|construct| (QUOTE X) (QUOTE Y)) (|construct|) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-89 -3409) 
((|constructor| (NIL "\\spadtype{Asp77} produces Fortran for Type 77 ASPs,{} needed for NAG routine d02gbf,{} for example: \\blankline \\tab{5}SUBROUTINE FCNF(\\spad{X},{}\\spad{F})\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{X}\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{F}(2,{}2)\\spad{\\br} \\tab{5}\\spad{F}(1,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{F}(1,{}2)\\spad{=1}.0D0\\spad{\\br} \\tab{5}\\spad{F}(2,{}1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{F}(2,{}2)=-10.0D0\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|Matrix| (|FortranExpression| (|construct| (QUOTE X)) (|construct|) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-90 -3409) 
((|constructor| (NIL "\\spadtype{Asp78} produces Fortran for Type 78 ASPs,{} needed for NAG routine d02gbf,{} for example: \\blankline \\tab{5}SUBROUTINE FCNG(\\spad{X},{}\\spad{G})\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{G}(*),{}\\spad{X}\\spad{\\br} \\tab{5}\\spad{G}(1)\\spad{=0}.0D0\\spad{\\br} \\tab{5}\\spad{G}(2)\\spad{=0}.0D0\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct| (QUOTE X)) (|construct|) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-91 -3409) 
((|constructor| (NIL "\\spadtype{Asp7} produces Fortran for Type 7 ASPs,{} needed for NAG routines d02bbf,{} d02gaf. These represent a vector of functions of the scalar \\spad{X} and the array \\spad{Z},{} and look like: \\blankline \\tab{5}SUBROUTINE \\spad{FCN}(\\spad{X},{}\\spad{Z},{}\\spad{F})\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{F}(*),{}\\spad{X},{}\\spad{Z}(*)\\spad{\\br} \\tab{5}\\spad{F}(1)=DTAN(\\spad{Z}(3))\\spad{\\br} \\tab{5}\\spad{F}(2)=((\\spad{-0}.03199999999999999D0*DCOS(\\spad{Z}(3))*DTAN(\\spad{Z}(3)))+(\\spad{-0}.02D0*Z(2)\\spad{\\br} \\tab{4}\\spad{&**2}))/(\\spad{Z}(2)*DCOS(\\spad{Z}(3)))\\spad{\\br} \\tab{5}\\spad{F}(3)=-0.03199999999999999D0/(\\spad{X*Z}(2)\\spad{**2})\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct| (QUOTE X)) (|construct| (QUOTE Y)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-92 -3409) 
((|constructor| (NIL "\\spadtype{Asp80} produces Fortran for Type 80 ASPs,{} needed for NAG routine d02kef,{} for example: \\blankline \\tab{5}SUBROUTINE BDYVAL(\\spad{XL},{}\\spad{XR},{}ELAM,{}\\spad{YL},{}\\spad{YR})\\spad{\\br} \\tab{5}DOUBLE PRECISION ELAM,{}\\spad{XL},{}\\spad{YL}(3),{}\\spad{XR},{}\\spad{YR}(3)\\spad{\\br} \\tab{5}\\spad{YL}(1)\\spad{=XL}\\spad{\\br} \\tab{5}\\spad{YL}(2)\\spad{=2}.0D0\\spad{\\br} \\tab{5}\\spad{YR}(1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}\\spad{YR}(2)=-1.0D0*DSQRT(\\spad{XR+}(\\spad{-1}.0D0*ELAM))\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END")) (|coerce| (($ (|Matrix| (|FortranExpression| (|construct| (QUOTE XL) (QUOTE XR) (QUOTE ELAM)) (|construct|) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
NIL 
NIL 
(-93 -3409) 
((|constructor| (NIL "\\spadtype{Asp8} produces Fortran for Type 8 ASPs,{} needed for NAG routine d02bbf. This ASP prints intermediate values of the computed solution of an ODE and might look like: \\blankline \\tab{5}SUBROUTINE OUTPUT(XSOL,{}\\spad{Y},{}COUNT,{}\\spad{M},{}\\spad{N},{}RESULT,{}FORWRD)\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{Y}(\\spad{N}),{}RESULT(\\spad{M},{}\\spad{N}),{}XSOL\\spad{\\br} \\tab{5}INTEGER \\spad{M},{}\\spad{N},{}COUNT\\spad{\\br} \\tab{5}LOGICAL FORWRD\\spad{\\br} \\tab{5}DOUBLE PRECISION X02ALF,{}POINTS(8)\\spad{\\br} \\tab{5}EXTERNAL X02ALF\\spad{\\br} \\tab{5}INTEGER \\spad{I}\\spad{\\br} \\tab{5}POINTS(1)\\spad{=1}.0D0\\spad{\\br} \\tab{5}POINTS(2)\\spad{=2}.0D0\\spad{\\br} \\tab{5}POINTS(3)\\spad{=3}.0D0\\spad{\\br} \\tab{5}POINTS(4)\\spad{=4}.0D0\\spad{\\br} \\tab{5}POINTS(5)\\spad{=5}.0D0\\spad{\\br} \\tab{5}POINTS(6)\\spad{=6}.0D0\\spad{\\br} \\tab{5}POINTS(7)\\spad{=7}.0D0\\spad{\\br} \\tab{5}POINTS(8)\\spad{=8}.0D0\\spad{\\br} \\tab{5}\\spad{COUNT=COUNT+1}\\spad{\\br} \\tab{5}DO 25001 \\spad{I=1},{}\\spad{N}\\spad{\\br} \\tab{7} RESULT(COUNT,{}\\spad{I})\\spad{=Y}(\\spad{I})\\spad{\\br} 25001 CONTINUE\\spad{\\br} \\tab{5}IF(COUNT.EQ.\\spad{M})THEN\\spad{\\br} \\tab{7}IF(FORWRD)THEN\\spad{\\br} \\tab{9}XSOL=X02ALF()\\spad{\\br} \\tab{7}ELSE\\spad{\\br} \\tab{9}XSOL=-X02ALF()\\spad{\\br} \\tab{7}ENDIF\\spad{\\br} \\tab{5}ELSE\\spad{\\br} \\tab{7} XSOL=POINTS(COUNT)\\spad{\\br} \\tab{5}ENDIF\\spad{\\br} \\tab{5}END"))) 
NIL 
NIL 
(-94 -3409) 
((|constructor| (NIL "\\spadtype{Asp9} produces Fortran for Type 9 ASPs,{} needed for NAG routines d02bhf,{} d02cjf,{} d02ejf. These ASPs represent a function of a scalar \\spad{X} and a vector \\spad{Y},{} for example: \\blankline \\tab{5}DOUBLE PRECISION FUNCTION \\spad{G}(\\spad{X},{}\\spad{Y})\\spad{\\br} \\tab{5}DOUBLE PRECISION \\spad{X},{}\\spad{Y}(*)\\spad{\\br} \\tab{5}G=X+Y(1)\\spad{\\br} \\tab{5}RETURN\\spad{\\br} \\tab{5}END \\blankline If the user provides a constant value for \\spad{G},{} then extra information is added via COMMON blocks used by certain routines. This specifies that the value returned by \\spad{G} in this case is to be ignored.")) (|coerce| (($ (|FortranExpression| (|construct| (QUOTE X)) (|construct| (QUOTE Y)) (|MachineFloat|))) "\\spad{coerce(f)} takes an object from the appropriate instantiation of \\spadtype{FortranExpression} and turns it into an ASP."))) 
NIL 
NIL 
(-95 R L) 
((|constructor| (NIL "\\spadtype{AssociatedEquations} provides functions to compute the associated equations needed for factoring operators")) (|associatedEquations| (((|Record| (|:| |minor| (|List| (|PositiveInteger|))) (|:| |eq| |#2|) (|:| |minors| (|List| (|List| (|PositiveInteger|)))) (|:| |ops| (|List| |#2|))) |#2| (|PositiveInteger|)) "\\spad{associatedEquations(op,{} m)} returns \\spad{[w,{} eq,{} lw,{} lop]} such that \\spad{eq(w) = 0} where \\spad{w} is the given minor,{} and \\spad{lw_i = lop_i(w)} for all the other minors.")) (|uncouplingMatrices| (((|Vector| (|Matrix| |#1|)) (|Matrix| |#1|)) "\\spad{uncouplingMatrices(M)} returns \\spad{[A_1,{}...,{}A_n]} such that if \\spad{y = [y_1,{}...,{}y_n]} is a solution of \\spad{y' = M y},{} then \\spad{[\\$y_j',{}y_j'',{}...,{}y_j^{(n)}\\$] = \\$A_j y\\$} for all \\spad{j}\\spad{'s}.")) (|associatedSystem| (((|Record| (|:| |mat| (|Matrix| |#1|)) (|:| |vec| (|Vector| (|List| (|PositiveInteger|))))) |#2| (|PositiveInteger|)) "\\spad{associatedSystem(op,{} m)} returns \\spad{[M,{}w]} such that the \\spad{m}-th associated equation system to \\spad{L} is \\spad{w' = M w}."))) 
NIL 
((|HasCategory| |#1| (QUOTE (-359)))) 
(-96 S) 
((|constructor| (NIL "A stack represented as a flexible array.")) (|member?| (((|Boolean|) |#1| $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} member?(3,{}a)")) (|members| (((|List| |#1|) $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} members a")) (|parts| (((|List| |#1|) $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} parts a")) (|#| (((|NonNegativeInteger|) $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} \\#a")) (|count| (((|NonNegativeInteger|) |#1| $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} count(4,{}a)") (((|NonNegativeInteger|) (|Mapping| (|Boolean|) |#1|) $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} count(\\spad{x+}->(\\spad{x>2}),{}a)")) (|any?| (((|Boolean|) (|Mapping| (|Boolean|) |#1|) $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} any?(\\spad{x+}->(\\spad{x=4}),{}a)")) (|every?| (((|Boolean|) (|Mapping| (|Boolean|) |#1|) $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} every?(\\spad{x+}->(\\spad{x=4}),{}a)")) (~= (((|Boolean|) $ $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} b:=copy a \\spad{X} (a~=b)")) (= (((|Boolean|) $ $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} b:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} (a=b)@Boolean")) (|coerce| (((|OutputForm|) $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} coerce a")) (|hash| (((|SingleInteger|) $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} hash a")) (|latex| (((|String|) $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} latex a")) (|map!| (($ (|Mapping| |#1| |#1|) $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} map!(\\spad{x+}-\\spad{>x+10},{}a) \\spad{X} a")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} map(\\spad{x+}-\\spad{>x+10},{}a) \\spad{X} a")) (|eq?| (((|Boolean|) $ $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} b:=copy a \\spad{X} eq?(a,{}\\spad{b})")) (|copy| (($ $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} copy a")) (|sample| (($) "\\blankline \\spad{X} sample()\\$ArrayStack(INT)")) (|empty| (($) "\\blankline \\spad{X} b:=empty()\\$(ArrayStack INT)")) (|empty?| (((|Boolean|) $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} empty? a")) (|bag| (($ (|List| |#1|)) "\\blankline \\spad{X} bag([1,{}2,{}3,{}4,{}5])\\$ArrayStack(INT)")) (|size?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} size?(a,{}5)")) (|more?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} more?(a,{}9)")) (|less?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} less?(a,{}9)")) (|depth| (((|NonNegativeInteger|) $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} depth a")) (|top| ((|#1| $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} top a")) (|inspect| ((|#1| $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} inspect a")) (|insert!| (($ |#1| $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} insert!(8,{}a) \\spad{X} a")) (|push!| ((|#1| |#1| $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} push!(9,{}a) \\spad{X} a")) (|extract!| ((|#1| $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} extract! a \\spad{X} a")) (|pop!| ((|#1| $) "\\blankline \\spad{X} a:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5] \\spad{X} pop! a \\spad{X} a")) (|arrayStack| (($ (|List| |#1|)) "\\indented{1}{arrayStack([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) creates an array stack with first (top)} \\indented{1}{element \\spad{x},{} second element \\spad{y},{}...,{}and last element \\spad{z}.} \\blankline \\spad{E} c:ArrayStack INT:= arrayStack [1,{}2,{}3,{}4,{}5]"))) 
((-4505 . T) (-4506 . T)) 
((|HasCategory| |#1| (QUOTE (-1082))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082))))) 
(-97 S) 
((|constructor| (NIL "Category for the inverse trigonometric functions.")) (|atan| (($ $) "\\spad{atan(x)} returns the arc-tangent of \\spad{x}.")) (|asin| (($ $) "\\spad{asin(x)} returns the arc-sine of \\spad{x}.")) (|asec| (($ $) "\\spad{asec(x)} returns the arc-secant of \\spad{x}.")) (|acsc| (($ $) "\\spad{acsc(x)} returns the arc-cosecant of \\spad{x}.")) (|acot| (($ $) "\\spad{acot(x)} returns the arc-cotangent of \\spad{x}.")) (|acos| (($ $) "\\spad{acos(x)} returns the arc-cosine of \\spad{x}."))) 
NIL 
NIL 
(-98) 
((|constructor| (NIL "Category for the inverse trigonometric functions.")) (|atan| (($ $) "\\spad{atan(x)} returns the arc-tangent of \\spad{x}.")) (|asin| (($ $) "\\spad{asin(x)} returns the arc-sine of \\spad{x}.")) (|asec| (($ $) "\\spad{asec(x)} returns the arc-secant of \\spad{x}.")) (|acsc| (($ $) "\\spad{acsc(x)} returns the arc-cosecant of \\spad{x}.")) (|acot| (($ $) "\\spad{acot(x)} returns the arc-cotangent of \\spad{x}.")) (|acos| (($ $) "\\spad{acos(x)} returns the arc-cosine of \\spad{x}."))) 
NIL 
NIL 
(-99) 
((|constructor| (NIL "\\axiomType{AttributeButtons} implements a database and associated adjustment mechanisms for a set of attributes. \\blankline For ODEs these attributes are \"stiffness\",{} \"stability\" (\\spadignore{i.e.} how much affect the cosine or sine component of the solution has on the stability of the result),{} \"accuracy\" and \"expense\" (\\spadignore{i.e.} how expensive is the evaluation of the ODE). All these have bearing on the cost of calculating the solution given that reducing the step-length to achieve greater accuracy requires considerable number of evaluations and calculations. \\blankline The effect of each of these attributes can be altered by increasing or decreasing the button value. \\blankline For Integration there is a button for increasing and decreasing the preset number of function evaluations for each method. This is automatically used by ANNA when a method fails due to insufficient workspace or where the limit of function evaluations has been reached before the required accuracy is achieved.")) (|setButtonValue| (((|Float|) (|String|) (|String|) (|Float|)) "\\axiom{setButtonValue(attributeName,{}routineName,{}\\spad{n})} sets the value of the button of attribute \\spad{attributeName} to routine \\spad{routineName} to \\spad{n}. \\spad{n} must be in the range [0..1]. \\blankline \\axiom{attributeName} should be one of the values \"stiffness\",{} \"stability\",{} \"accuracy\",{} \"expense\" or \"functionEvaluations\".") (((|Float|) (|String|) (|Float|)) "\\axiom{setButtonValue(attributeName,{}\\spad{n})} sets the value of all buttons of attribute \\spad{attributeName} to \\spad{n}. \\spad{n} must be in the range [0..1]. \\blankline \\axiom{attributeName} should be one of the values \"stiffness\",{} \"stability\",{} \"accuracy\",{} \"expense\" or \"functionEvaluations\".")) (|setAttributeButtonStep| (((|Float|) (|Float|)) "\\axiom{setAttributeButtonStep(\\spad{n})} sets the value of the steps for increasing and decreasing the button values. \\axiom{\\spad{n}} must be greater than 0 and less than 1. The preset value is 0.5.")) (|resetAttributeButtons| (((|Void|)) "\\axiom{resetAttributeButtons()} resets the Attribute buttons to a neutral level.")) (|getButtonValue| (((|Float|) (|String|) (|String|)) "\\axiom{getButtonValue(routineName,{}attributeName)} returns the current value for the effect of the attribute \\axiom{attributeName} with routine \\axiom{routineName}. \\blankline \\axiom{attributeName} should be one of the values \"stiffness\",{} \"stability\",{} \"accuracy\",{} \"expense\" or \"functionEvaluations\".")) (|decrease| (((|Float|) (|String|)) "\\axiom{decrease(attributeName)} decreases the value for the effect of the attribute \\axiom{attributeName} with all routines. \\blankline \\axiom{attributeName} should be one of the values \"stiffness\",{} \"stability\",{} \"accuracy\",{} \"expense\" or \"functionEvaluations\".") (((|Float|) (|String|) (|String|)) "\\axiom{decrease(routineName,{}attributeName)} decreases the value for the effect of the attribute \\axiom{attributeName} with routine \\axiom{routineName}. \\blankline \\axiom{attributeName} should be one of the values \"stiffness\",{} \"stability\",{} \"accuracy\",{} \"expense\" or \"functionEvaluations\".")) (|increase| (((|Float|) (|String|)) "\\axiom{increase(attributeName)} increases the value for the effect of the attribute \\axiom{attributeName} with all routines. \\blankline \\axiom{attributeName} should be one of the values \"stiffness\",{} \"stability\",{} \"accuracy\",{} \"expense\" or \"functionEvaluations\".") (((|Float|) (|String|) (|String|)) "\\axiom{increase(routineName,{}attributeName)} increases the value for the effect of the attribute \\axiom{attributeName} with routine \\axiom{routineName}. \\blankline \\axiom{attributeName} should be one of the values \"stiffness\",{} \"stability\",{} \"accuracy\",{} \"expense\" or \"functionEvaluations\"."))) 
((-4505 . T)) 
NIL 
(-100) 
((|constructor| (NIL "This category exports the attributes in the AXIOM Library")) (|approximate| ((|attribute|) "\\spad{approximate} means \"is an approximation to the real numbers\".")) (|canonical| ((|attribute|) "\\spad{canonical} is \\spad{true} if and only if distinct elements have distinct data structures. For example,{} a domain of mathematical objects which has the \\spad{canonical} attribute means that two objects are mathematically equal if and only if their data structures are equal.")) (|multiplicativeValuation| ((|attribute|) "\\spad{multiplicativeValuation} implies \\spad{euclideanSize(a*b)=euclideanSize(a)*euclideanSize(b)}.")) (|additiveValuation| ((|attribute|) "\\spad{additiveValuation} implies \\spad{euclideanSize(a*b)=euclideanSize(a)+euclideanSize(b)}.")) (|noetherian| ((|attribute|) "\\spad{noetherian} is \\spad{true} if all of its ideals are finitely generated.")) (|central| ((|attribute|) "\\spad{central} is \\spad{true} if,{} given an algebra over a ring \\spad{R},{} the image of \\spad{R} is the center of the algebra,{} \\spadignore{i.e.} the set of members of the algebra which commute with all others is precisely the image of \\spad{R} in the algebra.")) (|partiallyOrderedSet| ((|attribute|) "\\spad{partiallyOrderedSet} is \\spad{true} if a set with \\spadop{<} which is transitive,{} but \\spad{not(a < b or a = b)} does not necessarily imply \\spad{b<a}.")) (|arbitraryPrecision| ((|attribute|) "\\spad{arbitraryPrecision} means the user can set the precision for subsequent calculations.")) (|canonicalsClosed| ((|attribute|) "\\spad{canonicalsClosed} is \\spad{true} if \\spad{unitCanonical(a)*unitCanonical(b) = unitCanonical(a*b)}.")) (|canonicalUnitNormal| ((|attribute|) "\\spad{canonicalUnitNormal} is \\spad{true} if we can choose a canonical representative for each class of associate elements,{} that is \\spad{associates?(a,{}b)} returns \\spad{true} if and only if \\spad{unitCanonical(a) = unitCanonical(b)}.")) (|noZeroDivisors| ((|attribute|) "\\spad{noZeroDivisors} is \\spad{true} if \\spad{x * y \\~~= 0} implies both \\spad{x} and \\spad{y} are non-zero.")) (|rightUnitary| ((|attribute|) "\\spad{rightUnitary} is \\spad{true} if \\spad{x * 1 = x} for all \\spad{x}.")) (|leftUnitary| ((|attribute|) "\\spad{leftUnitary} is \\spad{true} if \\spad{1 * x = x} for all \\spad{x}.")) (|unitsKnown| ((|attribute|) "\\spad{unitsKnown} is \\spad{true} if a monoid (a multiplicative semigroup with a 1) has \\spad{unitsKnown} means that the operation \\spadfun{recip} can only return \"failed\" if its argument is not a unit.")) (|shallowlyMutable| ((|attribute|) "\\spad{shallowlyMutable} is \\spad{true} if its values have immediate components that are updateable (mutable). Note that the properties of any component domain are irrevelant to the \\spad{shallowlyMutable} proper.")) (|commutative| ((|attribute| "*") "\\spad{commutative(\"*\")} is \\spad{true} if it has an operation \\spad{\"*\": (D,{}D) -> D} which is commutative.")) (|finiteAggregate| ((|attribute|) "\\spad{finiteAggregate} is \\spad{true} if it is an aggregate with a finite number of elements."))) 
((-4505 . T) ((-4507 "*") . T) (-4506 . T) (-4502 . T) (-4500 . T) (-4499 . T) (-4498 . T) (-4503 . T) (-4497 . T) (-4496 . T) (-4495 . T) (-4494 . T) (-4493 . T) (-4501 . T) (-4504 . T) (|NullSquare| . T) (|JacobiIdentity| . T) (-4492 . T) (-3580 . T)) 
NIL 
(-101 R) 
((|constructor| (NIL "Automorphism \\spad{R} is the multiplicative group of automorphisms of \\spad{R}.")) (|morphism| (($ (|Mapping| |#1| |#1| (|Integer|))) "\\spad{morphism(f)} returns the morphism given by \\spad{f^n(x) = f(x,{}n)}.") (($ (|Mapping| |#1| |#1|) (|Mapping| |#1| |#1|)) "\\spad{morphism(f,{} g)} returns the invertible morphism given by \\spad{f},{} where \\spad{g} is the inverse of \\spad{f}..") (($ (|Mapping| |#1| |#1|)) "\\spad{morphism(f)} returns the non-invertible morphism given by \\spad{f}."))) 
((-4502 . T)) 
NIL 
(-102) 
((|constructor| (NIL "This package provides a functions to support a web server for the new Axiom Browser functions."))) 
NIL 
NIL 
(-103 R UP) 
((|constructor| (NIL "This package provides balanced factorisations of polynomials.")) (|balancedFactorisation| (((|Factored| |#2|) |#2| (|List| |#2|)) "\\spad{balancedFactorisation(a,{} [b1,{}...,{}bn])} returns a factorisation \\spad{a = p1^e1 ... pm^em} such that each \\spad{pi} is balanced with respect to \\spad{[b1,{}...,{}bm]}.") (((|Factored| |#2|) |#2| |#2|) "\\spad{balancedFactorisation(a,{} b)} returns a factorisation \\spad{a = p1^e1 ... pm^em} such that each \\spad{\\spad{pi}} is balanced with respect to \\spad{b}."))) 
NIL 
NIL 
(-104 S) 
((|constructor| (NIL "\\spadtype{BasicType} is the basic category for describing a collection of elements with \\spadop{=} (equality).")) (~= (((|Boolean|) $ $) "\\spad{x~=y} tests if \\spad{x} and \\spad{y} are not equal.")) (= (((|Boolean|) $ $) "\\spad{x=y} tests if \\spad{x} and \\spad{y} are equal."))) 
NIL 
NIL 
(-105) 
((|constructor| (NIL "\\spadtype{BasicType} is the basic category for describing a collection of elements with \\spadop{=} (equality).")) (~= (((|Boolean|) $ $) "\\spad{x~=y} tests if \\spad{x} and \\spad{y} are not equal.")) (= (((|Boolean|) $ $) "\\spad{x=y} tests if \\spad{x} and \\spad{y} are equal."))) 
NIL 
NIL 
(-106 S) 
((|constructor| (NIL "\\spadtype{BalancedBinaryTree(S)} is the domain of balanced binary trees (bbtree). A balanced binary tree of \\spad{2**k} leaves,{} for some \\spad{k > 0},{} is symmetric,{} that is,{} the left and right subtree of each interior node have identical shape. In general,{} the left and right subtree of a given node can differ by at most leaf node.")) (|mapDown!| (($ $ |#1| (|Mapping| (|List| |#1|) |#1| |#1| |#1|)) "\\indented{1}{mapDown!(\\spad{t},{}\\spad{p},{}\\spad{f}) returns \\spad{t} after traversing \\spad{t} in \"preorder\"} \\indented{1}{(node then left then right) fashion replacing the successive} \\indented{1}{interior nodes as follows. Let \\spad{l} and \\spad{r} denote the left and} \\indented{1}{right subtrees of \\spad{t}. The root value \\spad{x} of \\spad{t} is replaced by \\spad{p}.} \\indented{1}{Then \\spad{f}(value \\spad{l},{} value \\spad{r},{} \\spad{p}),{} where \\spad{l} and \\spad{r} denote the left} \\indented{1}{and right subtrees of \\spad{t},{} is evaluated producing two values} \\indented{1}{\\spad{pl} and \\spad{pr}. Then \\spad{mapDown!(l,{}pl,{}f)} and \\spad{mapDown!(l,{}pr,{}f)}} \\indented{1}{are evaluated.} \\blankline \\spad{X} T1:=BalancedBinaryTree Integer \\spad{X} t2:=balancedBinaryTree(4,{} 0)\\$\\spad{T1} \\spad{X} setleaves!(\\spad{t2},{}[1,{}2,{}3,{}4]::List(Integer)) \\spad{X} \\spad{adder3}(i:Integer,{}j:Integer,{}k:Integer):List Integer \\spad{==} [i+j,{}\\spad{j+k}] \\spad{X} mapDown!(\\spad{t2},{}4::INT,{}\\spad{adder3}) \\spad{X} \\spad{t2}") (($ $ |#1| (|Mapping| |#1| |#1| |#1|)) "\\indented{1}{mapDown!(\\spad{t},{}\\spad{p},{}\\spad{f}) returns \\spad{t} after traversing \\spad{t} in \"preorder\"} \\indented{1}{(node then left then right) fashion replacing the successive} \\indented{1}{interior nodes as follows. The root value \\spad{x} is} \\indented{1}{replaced by \\spad{q} \\spad{:=} \\spad{f}(\\spad{p},{}\\spad{x}). The mapDown!(\\spad{l},{}\\spad{q},{}\\spad{f}) and} \\indented{1}{mapDown!(\\spad{r},{}\\spad{q},{}\\spad{f}) are evaluated for the left and right subtrees} \\indented{1}{\\spad{l} and \\spad{r} of \\spad{t}.} \\blankline \\spad{X} T1:=BalancedBinaryTree Integer \\spad{X} t2:=balancedBinaryTree(4,{} 0)\\$\\spad{T1} \\spad{X} setleaves!(\\spad{t2},{}[1,{}2,{}3,{}4]::List(Integer)) \\spad{X} adder(i:Integer,{}j:Integer):Integer \\spad{==} i+j \\spad{X} mapDown!(\\spad{t2},{}4::INT,{}adder) \\spad{X} \\spad{t2}")) (|mapUp!| (($ $ $ (|Mapping| |#1| |#1| |#1| |#1| |#1|)) "\\indented{1}{mapUp!(\\spad{t},{}\\spad{t1},{}\\spad{f}) traverses balanced binary tree \\spad{t} in an \"endorder\"} \\indented{1}{(left then right then node) fashion returning \\spad{t} with the value} \\indented{1}{at each successive interior node of \\spad{t} replaced by} \\indented{1}{\\spad{f}(\\spad{l},{}\\spad{r},{}\\spad{l1},{}\\spad{r1}) where \\spad{l} and \\spad{r} are the values at the immediate} \\indented{1}{left and right nodes. Values \\spad{l1} and \\spad{r1} are values at the} \\indented{1}{corresponding nodes of a balanced binary tree \\spad{t1},{} of identical} \\indented{1}{shape at \\spad{t}.} \\blankline \\spad{X} T1:=BalancedBinaryTree Integer \\spad{X} t2:=balancedBinaryTree(4,{} 0)\\$\\spad{T1} \\spad{X} setleaves!(\\spad{t2},{}[1,{}2,{}3,{}4]::List(Integer)) \\spad{X} \\spad{adder4}(i:INT,{}j:INT,{}k:INT,{}l:INT):INT \\spad{==} i+j+k+l \\spad{X} mapUp!(\\spad{t2},{}\\spad{t2},{}\\spad{adder4}) \\spad{X} \\spad{t2}") ((|#1| $ (|Mapping| |#1| |#1| |#1|)) "\\indented{1}{mapUp!(\\spad{t},{}\\spad{f}) traverses balanced binary tree \\spad{t} in an \"endorder\"} \\indented{1}{(left then right then node) fashion returning \\spad{t} with the value} \\indented{1}{at each successive interior node of \\spad{t} replaced by} \\indented{1}{\\spad{f}(\\spad{l},{}\\spad{r}) where \\spad{l} and \\spad{r} are the values at the immediate} \\indented{1}{left and right nodes.} \\blankline \\spad{X} T1:=BalancedBinaryTree Integer \\spad{X} t2:=balancedBinaryTree(4,{} 0)\\$\\spad{T1} \\spad{X} setleaves!(\\spad{t2},{}[1,{}2,{}3,{}4]::List(Integer)) \\spad{X} adder(a:Integer,{}b:Integer):Integer \\spad{==} a+b \\spad{X} mapUp!(\\spad{t2},{}adder) \\spad{X} \\spad{t2}")) (|setleaves!| (($ $ (|List| |#1|)) "\\indented{1}{setleaves!(\\spad{t},{} \\spad{ls}) sets the leaves of \\spad{t} in left-to-right order} \\indented{1}{to the elements of \\spad{ls}.} \\blankline \\spad{X} t1:=balancedBinaryTree(4,{} 0) \\spad{X} setleaves!(\\spad{t1},{}[1,{}2,{}3,{}4])")) (|balancedBinaryTree| (($ (|NonNegativeInteger|) |#1|) "\\indented{1}{balancedBinaryTree(\\spad{n},{} \\spad{s}) creates a balanced binary tree with} \\indented{1}{\\spad{n} nodes each with value \\spad{s}.} \\blankline \\spad{X} balancedBinaryTree(4,{} 0)"))) 
((-4505 . T) (-4506 . T)) 
((|HasCategory| |#1| (QUOTE (-1082))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082))))) 
(-107 R) 
((|constructor| (NIL "Provide linear,{} quadratic,{} and cubic spline bezier curves")) (|cubicBezier| (((|Mapping| (|List| |#1|) |#1|) (|List| |#1|) (|List| |#1|) (|List| |#1|) (|List| |#1|)) "\\indented{1}{A cubic Bezier curve is a simple interpolation between the} \\indented{1}{starting point,{} a left-middle point,{},{} a right-middle point,{}} \\indented{1}{and the ending point based on a parameter \\spad{t}.} \\indented{1}{Given a start point a=[\\spad{x1},{}\\spad{y1}],{} the left-middle point \\spad{b=}[\\spad{x2},{}\\spad{y2}],{}} \\indented{1}{the right-middle point \\spad{c=}[\\spad{x3},{}\\spad{y3}] and an endpoint \\spad{d=}[\\spad{x4},{}\\spad{y4}]} \\indented{1}{\\spad{f}(\\spad{t}) \\spad{==} [(1-\\spad{t})\\spad{^3} \\spad{x1} + 3t(1-\\spad{t})\\spad{^2} \\spad{x2} + 3t^2 (1-\\spad{t}) \\spad{x3} + \\spad{t^3} \\spad{x4},{}} \\indented{10}{(1-\\spad{t})\\spad{^3} \\spad{y1} + 3t(1-\\spad{t})\\spad{^2} \\spad{y2} + 3t^2 (1-\\spad{t}) \\spad{y3} + \\spad{t^3} \\spad{y4}]} \\blankline \\spad{X} n:=cubicBezier([2.0,{}2.0],{}[2.0,{}4.0],{}[6.0,{}4.0],{}[6.0,{}2.0]) \\spad{X} [\\spad{n}(\\spad{t/10}.0) for \\spad{t} in 0..10 by 1]")) (|quadraticBezier| (((|Mapping| (|List| |#1|) |#1|) (|List| |#1|) (|List| |#1|) (|List| |#1|)) "\\indented{1}{A quadratic Bezier curve is a simple interpolation between the} \\indented{1}{starting point,{} a middle point,{} and the ending point based on} \\indented{1}{a parameter \\spad{t}.} \\indented{1}{Given a start point a=[\\spad{x1},{}\\spad{y1}],{} a middle point \\spad{b=}[\\spad{x2},{}\\spad{y2}],{}} \\indented{1}{and an endpoint \\spad{c=}[\\spad{x3},{}\\spad{y3}]} \\indented{1}{\\spad{f}(\\spad{t}) \\spad{==} [(1-\\spad{t})\\spad{^2} \\spad{x1} + 2t(1-\\spad{t}) \\spad{x2} + \\spad{t^2} \\spad{x3},{}} \\indented{10}{(1-\\spad{t})\\spad{^2} \\spad{y1} + 2t(1-\\spad{t}) \\spad{y2} + \\spad{t^2} \\spad{y3}]} \\blankline \\spad{X} n:=quadraticBezier([2.0,{}2.0],{}[4.0,{}4.0],{}[6.0,{}2.0]) \\spad{X} [\\spad{n}(\\spad{t/10}.0) for \\spad{t} in 0..10 by 1]")) (|linearBezier| (((|Mapping| (|List| |#1|) |#1|) (|List| |#1|) (|List| |#1|)) "\\indented{1}{A linear Bezier curve is a simple interpolation between the} \\indented{1}{starting point and the ending point based on a parameter \\spad{t}.} \\indented{1}{Given a start point a=[\\spad{x1},{}\\spad{y1}] and an endpoint \\spad{b=}[\\spad{x2},{}\\spad{y2}]} \\indented{1}{\\spad{f}(\\spad{t}) \\spad{==} [(1-\\spad{t})\\spad{*x1} + \\spad{t*x2},{} (1-\\spad{t})\\spad{*y1} + \\spad{t*y2}]} \\blankline \\spad{X} n:=linearBezier([2.0,{}2.0],{}[4.0,{}4.0]) \\spad{X} [\\spad{n}(\\spad{t/10}.0) for \\spad{t} in 0..10 by 1]"))) 
NIL 
NIL 
(-108 R UP M |Row| |Col|) 
((|constructor| (NIL "\\spadtype{BezoutMatrix} contains functions for computing resultants and discriminants using Bezout matrices.")) (|bezoutDiscriminant| ((|#1| |#2|) "\\spad{bezoutDiscriminant(p)} computes the discriminant of a polynomial \\spad{p} by computing the determinant of a Bezout matrix.")) (|bezoutResultant| ((|#1| |#2| |#2|) "\\spad{bezoutResultant(p,{}q)} computes the resultant of the two polynomials \\spad{p} and \\spad{q} by computing the determinant of a Bezout matrix.")) (|bezoutMatrix| ((|#3| |#2| |#2|) "\\spad{bezoutMatrix(p,{}q)} returns the Bezout matrix for the two polynomials \\spad{p} and \\spad{q}.")) (|sylvesterMatrix| ((|#3| |#2| |#2|) "\\spad{sylvesterMatrix(p,{}q)} returns the Sylvester matrix for the two polynomials \\spad{p} and \\spad{q}."))) 
NIL 
((|HasAttribute| |#1| (QUOTE (-4507 "*")))) 
(-109) 
((|constructor| (NIL "A Domain which implements a table containing details of points at which particular functions have evaluation problems.")) (|bfEntry| (((|Record| (|:| |zeros| (|Stream| (|DoubleFloat|))) (|:| |ones| (|Stream| (|DoubleFloat|))) (|:| |singularities| (|Stream| (|DoubleFloat|)))) (|Symbol|)) "\\spad{bfEntry(k)} returns the entry in the \\axiomType{BasicFunctions} table corresponding to \\spad{k}")) (|bfKeys| (((|List| (|Symbol|))) "\\spad{bfKeys()} returns the names of each function in the \\axiomType{BasicFunctions} table"))) 
((-4505 . T)) 
NIL 
(-110 A S) 
((|constructor| (NIL "A bag aggregate is an aggregate for which one can insert and extract objects,{} and where the order in which objects are inserted determines the order of extraction. Examples of bags are stacks,{} queues,{} and dequeues.")) (|inspect| ((|#2| $) "\\spad{inspect(u)} returns an (random) element from a bag.")) (|insert!| (($ |#2| $) "\\spad{insert!(x,{}u)} inserts item \\spad{x} into bag \\spad{u}.")) (|extract!| ((|#2| $) "\\spad{extract!(u)} destructively removes a (random) item from bag \\spad{u}.")) (|bag| (($ (|List| |#2|)) "\\spad{bag([x,{}y,{}...,{}z])} creates a bag with elements \\spad{x},{}\\spad{y},{}...,{}\\spad{z}.")) (|shallowlyMutable| ((|attribute|) "shallowlyMutable means that elements of bags may be destructively changed."))) 
NIL 
NIL 
(-111 S) 
((|constructor| (NIL "A bag aggregate is an aggregate for which one can insert and extract objects,{} and where the order in which objects are inserted determines the order of extraction. Examples of bags are stacks,{} queues,{} and dequeues.")) (|inspect| ((|#1| $) "\\spad{inspect(u)} returns an (random) element from a bag.")) (|insert!| (($ |#1| $) "\\spad{insert!(x,{}u)} inserts item \\spad{x} into bag \\spad{u}.")) (|extract!| ((|#1| $) "\\spad{extract!(u)} destructively removes a (random) item from bag \\spad{u}.")) (|bag| (($ (|List| |#1|)) "\\spad{bag([x,{}y,{}...,{}z])} creates a bag with elements \\spad{x},{}\\spad{y},{}...,{}\\spad{z}.")) (|shallowlyMutable| ((|attribute|) "shallowlyMutable means that elements of bags may be destructively changed."))) 
((-4506 . T) (-3576 . T)) 
NIL 
(-112) 
((|constructor| (NIL "This domain allows rational numbers to be presented as repeating binary expansions.")) (|binary| (($ (|Fraction| (|Integer|))) "\\indented{1}{binary(\\spad{r}) converts a rational number to a binary expansion.} \\blankline \\spad{X} binary(22/7)")) (|fractionPart| (((|Fraction| (|Integer|)) $) "\\spad{fractionPart(b)} returns the fractional part of a binary expansion.")) (|coerce| (((|RadixExpansion| 2) $) "\\spad{coerce(b)} converts a binary expansion to a radix expansion with base 2.") (((|Fraction| (|Integer|)) $) "\\spad{coerce(b)} converts a binary expansion to a rational number."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| (-560) (QUOTE (-896))) (|HasCategory| (-560) (LIST (QUOTE -1029) (QUOTE (-1153)))) (|HasCategory| (-560) (QUOTE (-146))) (|HasCategory| (-560) (QUOTE (-148))) (|HasCategory| (-560) (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| (-560) (QUOTE (-1013))) (|HasCategory| (-560) (QUOTE (-807))) (|HasCategory| (-560) (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| (-560) (QUOTE (-1128))) (|HasCategory| (-560) (LIST (QUOTE -873) (QUOTE (-560)))) (|HasCategory| (-560) (LIST (QUOTE -873) (QUOTE (-375)))) (|HasCategory| (-560) (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-375))))) (|HasCategory| (-560) (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-560))))) (|HasCategory| (-560) (QUOTE (-221))) (|HasCategory| (-560) (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| (-560) (LIST (QUOTE -515) (QUOTE (-1153)) (QUOTE (-560)))) (|HasCategory| (-560) (LIST (QUOTE -298) (QUOTE (-560)))) (|HasCategory| (-560) (LIST (QUOTE -276) (QUOTE (-560)) (QUOTE (-560)))) (|HasCategory| (-560) (QUOTE (-296))) (|HasCategory| (-560) (QUOTE (-542))) (|HasCategory| (-560) (QUOTE (-834))) (-3322 (|HasCategory| (-560) (QUOTE (-807))) (|HasCategory| (-560) (QUOTE (-834)))) (|HasCategory| (-560) (LIST (QUOTE -622) (QUOTE (-560)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-560) (QUOTE (-896)))) (-3322 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-560) (QUOTE (-896)))) (|HasCategory| (-560) (QUOTE (-146))))) 
(-113) 
((|constructor| (NIL "This domain provides an implementation of binary files. Data is accessed one byte at a time as a small integer.")) (|position!| (((|SingleInteger|) $ (|SingleInteger|)) "\\spad{position!(f,{} i)} sets the current byte-position to \\spad{i}.")) (|position| (((|SingleInteger|) $) "\\spad{position(f)} returns the current byte-position in the file \\spad{f}.")) (|readIfCan!| (((|Union| (|SingleInteger|) "failed") $) "\\spad{readIfCan!(f)} returns a value from the file \\spad{f},{} if possible. If \\spad{f} is not open for reading,{} or if \\spad{f} is at the end of file then \\spad{\"failed\"} is the result."))) 
NIL 
NIL 
(-114) 
((|constructor| (NIL "\\spadtype{Bits} provides logical functions for Indexed Bits.")) (|bits| (($ (|NonNegativeInteger|) (|Boolean|)) "\\spad{bits(n,{}b)} creates bits with \\spad{n} values of \\spad{b}"))) 
((-4506 . T) (-4505 . T)) 
((|HasCategory| (-121) (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| (-121) (QUOTE (-834))) (|HasCategory| (-560) (QUOTE (-834))) (|HasCategory| (-121) (QUOTE (-1082))) (-12 (|HasCategory| (-121) (LIST (QUOTE -298) (QUOTE (-121)))) (|HasCategory| (-121) (QUOTE (-1082))))) 
(-115) 
((|constructor| (NIL "This package provides an interface to the Blas library (level 1)")) (|dcopy| (((|PrimitiveArray| (|DoubleFloat|)) (|SingleInteger|) (|PrimitiveArray| (|DoubleFloat|)) (|SingleInteger|) (|PrimitiveArray| (|DoubleFloat|)) (|SingleInteger|)) "\\indented{1}{dcopy(\\spad{n},{}\\spad{x},{}incx,{}\\spad{y},{}incy) copies \\spad{y} from \\spad{x}} \\indented{1}{for each of the chosen elements of the vectors \\spad{x} and \\spad{y}} \\indented{1}{Note that the vector \\spad{y} is modified with the results.} \\blankline \\spad{X} x:PRIMARR(DFLOAT)\\spad{:=}[[1.0,{}2.0,{}3.0,{}4.0,{}5.0,{}6.0]] \\spad{X} y:PRIMARR(DFLOAT)\\spad{:=}[[0.0,{}0.0,{}0.0,{}0.0,{}0.0,{}0.0]] \\spad{X} dcopy(6,{}\\spad{x},{}1,{}\\spad{y},{}1) \\spad{X} \\spad{y} \\spad{X} m:PRIMARR(DFLOAT)\\spad{:=}[[1.0,{}2.0,{}3.0]] \\spad{X} n:PRIMARR(DFLOAT)\\spad{:=}[[0.0,{}0.0,{}0.0,{}0.0,{}0.0,{}0.0]] \\spad{X} dcopy(3,{}\\spad{m},{}1,{}\\spad{n},{}2) \\spad{X} \\spad{n}")) (|daxpy| (((|PrimitiveArray| (|DoubleFloat|)) (|SingleInteger|) (|DoubleFloat|) (|PrimitiveArray| (|DoubleFloat|)) (|SingleInteger|) (|PrimitiveArray| (|DoubleFloat|)) (|SingleInteger|)) "\\indented{1}{daxpy(\\spad{n},{}da,{}\\spad{x},{}incx,{}\\spad{y},{}incy) computes a \\spad{y} = a*x + \\spad{y}} \\indented{1}{for each of the chosen elements of the vectors \\spad{x} and \\spad{y}} \\indented{1}{and a constant multiplier a} \\indented{1}{Note that the vector \\spad{y} is modified with the results.} \\blankline \\spad{X} x:PRIMARR(DFLOAT)\\spad{:=}[[1.0,{}2.0,{}3.0,{}4.0,{}5.0,{}6.0]] \\spad{X} y:PRIMARR(DFLOAT)\\spad{:=}[[1.0,{}2.0,{}3.0,{}4.0,{}5.0,{}6.0]] \\spad{X} daxpy(6,{}2.0,{}\\spad{x},{}1,{}\\spad{y},{}1) \\spad{X} \\spad{y} \\spad{X} m:PRIMARR(DFLOAT)\\spad{:=}[[1.0,{}2.0,{}3.0]] \\spad{X} n:PRIMARR(DFLOAT)\\spad{:=}[[1.0,{}2.0,{}3.0,{}4.0,{}5.0,{}6.0]] \\spad{X} daxpy(3,{}\\spad{-2}.0,{}\\spad{m},{}1,{}\\spad{n},{}2) \\spad{X} \\spad{n}")) (|dasum| (((|DoubleFloat|) (|SingleInteger|) (|PrimitiveArray| (|DoubleFloat|)) (|SingleInteger|)) "\\indented{1}{dasum(\\spad{n},{}array,{}incx) computes the sum of \\spad{n} elements in array} \\indented{1}{using a stride of incx} \\blankline \\spad{X} dx:PRIMARR(DFLOAT)\\spad{:=}[[1.0,{}2.0,{}3.0,{}4.0,{}5.0,{}6.0]] \\spad{X} dasum(6,{}\\spad{dx},{}1) \\spad{X} dasum(3,{}\\spad{dx},{}2)")) (|dcabs1| (((|DoubleFloat|) (|Complex| (|DoubleFloat|))) "\\indented{1}{\\spad{dcabs1}(\\spad{z}) computes (+ (abs (realpart \\spad{z})) (abs (imagpart \\spad{z})))} \\blankline \\spad{X} t1:Complex DoubleFloat \\spad{:=} complex(1.0,{}0) \\spad{X} dcabs(\\spad{t1})"))) 
NIL 
NIL 
(-116) 
((|constructor| (NIL "This domain is part of the PAFF package"))) 
((|HamburgerNoether| . T)) 
NIL 
(-117) 
NIL 
NIL 
NIL 
(-118) 
((|constructor| (NIL "This domain is part of the PAFF package"))) 
((|QuadraticTransform| . T)) 
NIL 
(-119 K |symb| |PolyRing| E BLMET) 
((|constructor| (NIL "The following is part of the PAFF package")) (|stepBlowUp| (((|Record| (|:| |mult| (|NonNegativeInteger|)) (|:| |subMult| (|NonNegativeInteger|)) (|:| |blUpRec| (|List| (|Record| (|:| |recTransStr| (|DistributedMultivariatePolynomial| (|construct| (QUOTE X) (QUOTE Y)) |#1|)) (|:| |recPoint| (|AffinePlane| |#1|)) (|:| |recChart| |#5|) (|:| |definingExtension| |#1|))))) (|DistributedMultivariatePolynomial| (|construct| (QUOTE X) (QUOTE Y)) |#1|) (|AffinePlane| |#1|) |#5| |#1|) "\\spad{stepBlowUp(pol,{}pt,{}n)} blow-up the point \\spad{pt} on the curve defined by \\spad{pol} in the affine neighbourhood specified by \\spad{n}.")) (|quadTransform| (((|DistributedMultivariatePolynomial| (|construct| (QUOTE X) (QUOTE Y)) |#1|) (|DistributedMultivariatePolynomial| (|construct| (QUOTE X) (QUOTE Y)) |#1|) (|NonNegativeInteger|) |#5|) "\\spad{quadTransform(pol,{}n,{}chart)} apply the quadratique transformation to \\spad{pol} specified by \\spad{chart} has in quadTransform(\\spad{pol},{}\\spad{chart}) and extract x**n to it,{} where \\spad{x} is the variable specified by the first integer in \\spad{chart} (blow-up exceptional coordinate).")) (|applyTransform| ((|#3| |#3| |#5|) "quadTransform(pol,{}chart) apply the quadratique transformation to pol specified by chart which consist of 3 integers. The last one indicates which varibles is set to 1,{} the first on indicates which variable remains unchange,{} and the second one indicates which variable oon which the transformation is applied. For example,{} [2,{}3,{}1] correspond to the following: \\spad{x} \\spad{->} 1,{} \\spad{y} \\spad{->} \\spad{y},{} \\spad{z} \\spad{->} \\spad{yz} (here the variable are [\\spad{x},{}\\spad{y},{}\\spad{z}] in BlUpRing)."))) 
NIL 
NIL 
(-120 R S) 
((|constructor| (NIL "A \\spadtype{BiModule} is both a left and right module with respect to potentially different rings. \\blankline Axiom\\spad{\\br} \\tab{5}\\spad{ r*(x*s) = (r*x)*s }")) (|rightUnitary| ((|attribute|) "\\spad{x * 1 = x}")) (|leftUnitary| ((|attribute|) "\\spad{1 * x = x}"))) 
((-4500 . T) (-4499 . T)) 
NIL 
(-121) 
((|constructor| (NIL "\\spadtype{Boolean} is the elementary logic with 2 values: \\spad{true} and \\spad{false}")) (|test| (((|Boolean|) $) "\\spad{test(b)} returns \\spad{b} and is provided for compatibility with the new compiler.")) (|implies| (($ $ $) "\\spad{implies(a,{}b)} returns the logical implication of Boolean \\spad{a} and \\spad{b}.")) (|nor| (($ $ $) "\\spad{nor(a,{}b)} returns the logical negation of \\spad{a} or \\spad{b}.")) (|nand| (($ $ $) "\\spad{nand(a,{}b)} returns the logical negation of \\spad{a} and \\spad{b}.")) (|xor| (($ $ $) "\\spad{xor(a,{}b)} returns the logical exclusive or of Boolean \\spad{a} and \\spad{b}.")) (|or| (($ $ $) "\\spad{a or b} returns the logical inclusive or of Boolean \\spad{a} and \\spad{b}.")) (|and| (($ $ $) "\\spad{a and b} returns the logical and of Boolean \\spad{a} and \\spad{b}.")) (|not| (($ $) "\\spad{not n} returns the negation of \\spad{n}.")) (^ (($ $) "\\spad{^ n} returns the negation of \\spad{n}.")) (|false| (($) "\\spad{false} is a logical constant.")) (|true| (($) "\\spad{true} is a logical constant."))) 
NIL 
NIL 
(-122 A) 
((|constructor| (NIL "This package exports functions to set some commonly used properties of operators,{} including properties which contain functions.")) (|constantOpIfCan| (((|Union| |#1| "failed") (|BasicOperator|)) "\\spad{constantOpIfCan(op)} returns \\spad{a} if \\spad{op} is the constant nullary operator always returning \\spad{a},{} \"failed\" otherwise.")) (|constantOperator| (((|BasicOperator|) |#1|) "\\spad{constantOperator(a)} returns a nullary operator op such that \\spad{op()} always evaluate to \\spad{a}.")) (|derivative| (((|Union| (|List| (|Mapping| |#1| (|List| |#1|))) "failed") (|BasicOperator|)) "\\spad{derivative(op)} returns the value of the \"\\%diff\" property of \\spad{op} if it has one,{} and \"failed\" otherwise.") (((|BasicOperator|) (|BasicOperator|) (|Mapping| |#1| |#1|)) "\\spad{derivative(op,{} foo)} attaches foo as the \"\\%diff\" property of \\spad{op}. If \\spad{op} has an \"\\%diff\" property \\spad{f},{} then applying a derivation \\spad{D} to \\spad{op}(a) returns \\spad{f(a) * D(a)}. Argument \\spad{op} must be unary.") (((|BasicOperator|) (|BasicOperator|) (|List| (|Mapping| |#1| (|List| |#1|)))) "\\spad{derivative(op,{} [foo1,{}...,{}foon])} attaches [\\spad{foo1},{}...,{}foon] as the \"\\%diff\" property of \\spad{op}. If \\spad{op} has an \"\\%diff\" property \\spad{[f1,{}...,{}fn]} then applying a derivation \\spad{D} to \\spad{op(a1,{}...,{}an)} returns \\spad{f1(a1,{}...,{}an) * D(a1) + ... + fn(a1,{}...,{}an) * D(an)}.")) (|evaluate| (((|Union| (|Mapping| |#1| (|List| |#1|)) "failed") (|BasicOperator|)) "\\spad{evaluate(op)} returns the value of the \"\\%eval\" property of \\spad{op} if it has one,{} and \"failed\" otherwise.") (((|BasicOperator|) (|BasicOperator|) (|Mapping| |#1| |#1|)) "\\spad{evaluate(op,{} foo)} attaches foo as the \"\\%eval\" property of \\spad{op}. If \\spad{op} has an \"\\%eval\" property \\spad{f},{} then applying \\spad{op} to a returns the result of \\spad{f(a)}. Argument \\spad{op} must be unary.") (((|BasicOperator|) (|BasicOperator|) (|Mapping| |#1| (|List| |#1|))) "\\spad{evaluate(op,{} foo)} attaches foo as the \"\\%eval\" property of \\spad{op}. If \\spad{op} has an \"\\%eval\" property \\spad{f},{} then applying \\spad{op} to \\spad{(a1,{}...,{}an)} returns the result of \\spad{f(a1,{}...,{}an)}.") (((|Union| |#1| "failed") (|BasicOperator|) (|List| |#1|)) "\\spad{evaluate(op,{} [a1,{}...,{}an])} checks if \\spad{op} has an \"\\%eval\" property \\spad{f}. If it has,{} then \\spad{f(a1,{}...,{}an)} is returned,{} and \"failed\" otherwise."))) 
NIL 
((|HasCategory| |#1| (QUOTE (-834)))) 
(-123) 
((|constructor| (NIL "Basic system operators. A basic operator is an object that can be applied to a list of arguments from a set,{} the result being a kernel over that set.")) (|setProperties| (($ $ (|AssociationList| (|String|) (|None|))) "\\spad{setProperties(op,{} l)} sets the property list of \\spad{op} to \\spad{l}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|setProperty| (($ $ (|String|) (|None|)) "\\spad{setProperty(op,{} s,{} v)} attaches property \\spad{s} to \\spad{op},{} and sets its value to \\spad{v}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|property| (((|Union| (|None|) "failed") $ (|String|)) "\\spad{property(op,{} s)} returns the value of property \\spad{s} if it is attached to \\spad{op},{} and \"failed\" otherwise.")) (|deleteProperty!| (($ $ (|String|)) "\\spad{deleteProperty!(op,{} s)} unattaches property \\spad{s} from \\spad{op}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|assert| (($ $ (|String|)) "\\spad{assert(op,{} s)} attaches property \\spad{s} to \\spad{op}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|has?| (((|Boolean|) $ (|String|)) "\\spad{has?(op,{} s)} tests if property \\spad{s} is attached to \\spad{op}.")) (|is?| (((|Boolean|) $ (|Symbol|)) "\\spad{is?(op,{} s)} tests if the name of \\spad{op} is \\spad{s}.")) (|input| (((|Union| (|Mapping| (|InputForm|) (|List| (|InputForm|))) "failed") $) "\\spad{input(op)} returns the \"\\%input\" property of \\spad{op} if it has one attached,{} \"failed\" otherwise.") (($ $ (|Mapping| (|InputForm|) (|List| (|InputForm|)))) "\\spad{input(op,{} foo)} attaches foo as the \"\\%input\" property of \\spad{op}. If \\spad{op} has a \"\\%input\" property \\spad{f},{} then \\spad{op(a1,{}...,{}an)} gets converted to InputForm as \\spad{f(a1,{}...,{}an)}.")) (|display| (($ $ (|Mapping| (|OutputForm|) (|OutputForm|))) "\\spad{display(op,{} foo)} attaches foo as the \"\\%display\" property of \\spad{op}. If \\spad{op} has a \"\\%display\" property \\spad{f},{} then \\spad{op(a)} gets converted to OutputForm as \\spad{f(a)}. Argument \\spad{op} must be unary.") (($ $ (|Mapping| (|OutputForm|) (|List| (|OutputForm|)))) "\\spad{display(op,{} foo)} attaches foo as the \"\\%display\" property of \\spad{op}. If \\spad{op} has a \"\\%display\" property \\spad{f},{} then \\spad{op(a1,{}...,{}an)} gets converted to OutputForm as \\spad{f(a1,{}...,{}an)}.") (((|Union| (|Mapping| (|OutputForm|) (|List| (|OutputForm|))) "failed") $) "\\spad{display(op)} returns the \"\\%display\" property of \\spad{op} if it has one attached,{} and \"failed\" otherwise.")) (|comparison| (($ $ (|Mapping| (|Boolean|) $ $)) "\\spad{comparison(op,{} foo?)} attaches foo? as the \"\\%less?\" property to \\spad{op}. If \\spad{op1} and \\spad{op2} have the same name,{} and one of them has a \"\\%less?\" property \\spad{f},{} then \\spad{f(op1,{} op2)} is called to decide whether \\spad{op1 < op2}.")) (|equality| (($ $ (|Mapping| (|Boolean|) $ $)) "\\spad{equality(op,{} foo?)} attaches foo? as the \"\\%equal?\" property to \\spad{op}. If \\spad{op1} and \\spad{op2} have the same name,{} and one of them has an \"\\%equal?\" property \\spad{f},{} then \\spad{f(op1,{} op2)} is called to decide whether \\spad{op1} and \\spad{op2} should be considered equal.")) (|weight| (($ $ (|NonNegativeInteger|)) "\\spad{weight(op,{} n)} attaches the weight \\spad{n} to \\spad{op}.") (((|NonNegativeInteger|) $) "\\spad{weight(op)} returns the weight attached to \\spad{op}.")) (|nary?| (((|Boolean|) $) "\\spad{nary?(op)} tests if \\spad{op} has arbitrary arity.")) (|unary?| (((|Boolean|) $) "\\spad{unary?(op)} tests if \\spad{op} is unary.")) (|nullary?| (((|Boolean|) $) "\\spad{nullary?(op)} tests if \\spad{op} is nullary.")) (|arity| (((|Union| (|NonNegativeInteger|) "failed") $) "\\spad{arity(op)} returns \\spad{n} if \\spad{op} is \\spad{n}-ary,{} and \"failed\" if \\spad{op} has arbitrary arity.")) (|operator| (($ (|Symbol|) (|NonNegativeInteger|)) "\\spad{operator(f,{} n)} makes \\spad{f} into an \\spad{n}-ary operator.") (($ (|Symbol|)) "\\spad{operator(f)} makes \\spad{f} into an operator with arbitrary arity.")) (|copy| (($ $) "\\spad{copy(op)} returns a copy of \\spad{op}.")) (|properties| (((|AssociationList| (|String|) (|None|)) $) "\\spad{properties(op)} returns the list of all the properties currently attached to \\spad{op}.")) (|name| (((|Symbol|) $) "\\spad{name(op)} returns the name of \\spad{op}."))) 
NIL 
NIL 
(-124 -2262 UP) 
((|constructor| (NIL "\\spadtype{BoundIntegerRoots} provides functions to find lower bounds on the integer roots of a polynomial.")) (|integerBound| (((|Integer|) |#2|) "\\spad{integerBound(p)} returns a lower bound on the negative integer roots of \\spad{p},{} and 0 if \\spad{p} has no negative integer roots."))) 
NIL 
NIL 
(-125 |p|) 
((|constructor| (NIL "Stream-based implementation of \\spad{Zp:} \\spad{p}-adic numbers are represented as sum(\\spad{i} = 0..,{} a[\\spad{i}] * p^i),{} where the a[\\spad{i}] lie in -(\\spad{p} - 1)\\spad{/2},{}...,{}(\\spad{p} - 1)\\spad{/2}."))) 
((-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-126 |p|) 
((|constructor| (NIL "Stream-based implementation of \\spad{Qp:} numbers are represented as sum(\\spad{i} = \\spad{k}..,{} a[\\spad{i}] * p^i),{} where the a[\\spad{i}] lie in -(\\spad{p} - 1)\\spad{/2},{}...,{}(\\spad{p} - 1)\\spad{/2}."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| (-125 |#1|) (QUOTE (-896))) (|HasCategory| (-125 |#1|) (LIST (QUOTE -1029) (QUOTE (-1153)))) (|HasCategory| (-125 |#1|) (QUOTE (-146))) (|HasCategory| (-125 |#1|) (QUOTE (-148))) (|HasCategory| (-125 |#1|) (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| (-125 |#1|) (QUOTE (-1013))) (|HasCategory| (-125 |#1|) (QUOTE (-807))) (|HasCategory| (-125 |#1|) (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| (-125 |#1|) (QUOTE (-1128))) (|HasCategory| (-125 |#1|) (LIST (QUOTE -873) (QUOTE (-560)))) (|HasCategory| (-125 |#1|) (LIST (QUOTE -873) (QUOTE (-375)))) (|HasCategory| (-125 |#1|) (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-375))))) (|HasCategory| (-125 |#1|) (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-560))))) (|HasCategory| (-125 |#1|) (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| (-125 |#1|) (QUOTE (-221))) (|HasCategory| (-125 |#1|) (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| (-125 |#1|) (LIST (QUOTE -515) (QUOTE (-1153)) (LIST (QUOTE -125) (|devaluate| |#1|)))) (|HasCategory| (-125 |#1|) (LIST (QUOTE -298) (LIST (QUOTE -125) (|devaluate| |#1|)))) (|HasCategory| (-125 |#1|) (LIST (QUOTE -276) (LIST (QUOTE -125) (|devaluate| |#1|)) (LIST (QUOTE -125) (|devaluate| |#1|)))) (|HasCategory| (-125 |#1|) (QUOTE (-296))) (|HasCategory| (-125 |#1|) (QUOTE (-542))) (|HasCategory| (-125 |#1|) (QUOTE (-834))) (-3322 (|HasCategory| (-125 |#1|) (QUOTE (-807))) (|HasCategory| (-125 |#1|) (QUOTE (-834)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-125 |#1|) (QUOTE (-896)))) (-3322 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-125 |#1|) (QUOTE (-896)))) (|HasCategory| (-125 |#1|) (QUOTE (-146))))) 
(-127 A S) 
((|constructor| (NIL "A binary-recursive aggregate has 0,{} 1 or 2 children and serves as a model for a binary tree or a doubly-linked aggregate structure")) (|setright!| (($ $ $) "\\spad{setright!(a,{}x)} sets the right child of \\spad{t} to be \\spad{x}.")) (|setleft!| (($ $ $) "\\spad{setleft!(a,{}b)} sets the left child of \\axiom{a} to be \\spad{b}.")) (|setelt| (($ $ "right" $) "\\spad{setelt(a,{}\"right\",{}b)} (also written \\axiom{\\spad{b} . right \\spad{:=} \\spad{b}}) is equivalent to \\axiom{setright!(a,{}\\spad{b})}.") (($ $ "left" $) "\\spad{setelt(a,{}\"left\",{}b)} (also written \\axiom{a . left \\spad{:=} \\spad{b}}) is equivalent to \\axiom{setleft!(a,{}\\spad{b})}.")) (|right| (($ $) "\\spad{right(a)} returns the right child.")) (|elt| (($ $ "right") "\\spad{elt(a,{}\"right\")} (also written: \\axiom{a . right}) is equivalent to \\axiom{right(a)}.") (($ $ "left") "\\spad{elt(u,{}\"left\")} (also written: \\axiom{a . left}) is equivalent to \\axiom{left(a)}.")) (|left| (($ $) "\\spad{left(u)} returns the left child."))) 
NIL 
((|HasAttribute| |#1| (QUOTE -4506))) 
(-128 S) 
((|constructor| (NIL "A binary-recursive aggregate has 0,{} 1 or 2 children and serves as a model for a binary tree or a doubly-linked aggregate structure")) (|setright!| (($ $ $) "\\spad{setright!(a,{}x)} sets the right child of \\spad{t} to be \\spad{x}.")) (|setleft!| (($ $ $) "\\spad{setleft!(a,{}b)} sets the left child of \\axiom{a} to be \\spad{b}.")) (|setelt| (($ $ "right" $) "\\spad{setelt(a,{}\"right\",{}b)} (also written \\axiom{\\spad{b} . right \\spad{:=} \\spad{b}}) is equivalent to \\axiom{setright!(a,{}\\spad{b})}.") (($ $ "left" $) "\\spad{setelt(a,{}\"left\",{}b)} (also written \\axiom{a . left \\spad{:=} \\spad{b}}) is equivalent to \\axiom{setleft!(a,{}\\spad{b})}.")) (|right| (($ $) "\\spad{right(a)} returns the right child.")) (|elt| (($ $ "right") "\\spad{elt(a,{}\"right\")} (also written: \\axiom{a . right}) is equivalent to \\axiom{right(a)}.") (($ $ "left") "\\spad{elt(u,{}\"left\")} (also written: \\axiom{a . left}) is equivalent to \\axiom{left(a)}.")) (|left| (($ $) "\\spad{left(u)} returns the left child."))) 
((-3576 . T)) 
NIL 
(-129 UP) 
((|constructor| (NIL "This package has no description")) (|noLinearFactor?| (((|Boolean|) |#1|) "\\spad{noLinearFactor?(p)} returns \\spad{true} if \\spad{p} can be shown to have no linear factor by a theorem of Lehmer,{} \\spad{false} else. \\spad{I} insist on the fact that \\spad{false} does not mean that \\spad{p} has a linear factor.")) (|brillhartTrials| (((|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{brillhartTrials(n)} sets to \\spad{n} the number of tests in \\spadfun{brillhartIrreducible?} and returns the previous value.") (((|NonNegativeInteger|)) "\\spad{brillhartTrials()} returns the number of tests in \\spadfun{brillhartIrreducible?}.")) (|brillhartIrreducible?| (((|Boolean|) |#1| (|Boolean|)) "\\spad{brillhartIrreducible?(p,{}noLinears)} returns \\spad{true} if \\spad{p} can be shown to be irreducible by a remark of Brillhart,{} \\spad{false} else. If \\spad{noLinears} is \\spad{true},{} we are being told \\spad{p} has no linear factors \\spad{false} does not mean that \\spad{p} is reducible.") (((|Boolean|) |#1|) "\\spad{brillhartIrreducible?(p)} returns \\spad{true} if \\spad{p} can be shown to be irreducible by a remark of Brillhart,{} \\spad{false} is inconclusive."))) 
NIL 
NIL 
(-130 S) 
((|constructor| (NIL "BinarySearchTree(\\spad{S}) is the domain of a binary trees where elements are ordered across the tree. A binary search tree is either empty or has a value which is an \\spad{S},{} and a right and left which are both BinaryTree(\\spad{S}) Elements are ordered across the tree.")) (|split| (((|Record| (|:| |less| $) (|:| |greater| $)) |#1| $) "\\indented{1}{split(\\spad{x},{}\\spad{b}) splits binary tree \\spad{b} into two trees,{} one with elements} \\indented{1}{greater than \\spad{x},{} the other with elements less than \\spad{x}.} \\blankline \\spad{X} t1:=binarySearchTree [1,{}2,{}3,{}4] \\spad{X} split(3,{}\\spad{t1})")) (|insertRoot!| (($ |#1| $) "\\indented{1}{insertRoot!(\\spad{x},{}\\spad{b}) inserts element \\spad{x} as a root of binary search tree \\spad{b}.} \\blankline \\spad{X} t1:=binarySearchTree [1,{}2,{}3,{}4] \\spad{X} insertRoot!(5,{}\\spad{t1})")) (|insert!| (($ |#1| $) "\\indented{1}{insert!(\\spad{x},{}\\spad{b}) inserts element \\spad{x} as leaves into binary search tree \\spad{b}.} \\blankline \\spad{X} t1:=binarySearchTree [1,{}2,{}3,{}4] \\spad{X} insert!(5,{}\\spad{t1})")) (|binarySearchTree| (($ (|List| |#1|)) "\\indented{1}{binarySearchTree(\\spad{l}) is not documented} \\blankline \\spad{X} binarySearchTree [1,{}2,{}3,{}4]"))) 
((-4505 . T) (-4506 . T)) 
((|HasCategory| |#1| (QUOTE (-1082))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082))))) 
(-131 S) 
((|constructor| (NIL "The bit aggregate category models aggregates representing large quantities of Boolean data.")) (|xor| (($ $ $) "\\spad{xor(a,{}b)} returns the logical exclusive-or of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|or| (($ $ $) "\\spad{a or b} returns the logical or of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|and| (($ $ $) "\\spad{a and b} returns the logical and of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nor| (($ $ $) "\\spad{nor(a,{}b)} returns the logical nor of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nand| (($ $ $) "\\spad{nand(a,{}b)} returns the logical nand of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (^ (($ $) "\\spad{^ b} returns the logical not of bit aggregate \\axiom{\\spad{b}}.")) (|not| (($ $) "\\spad{not(b)} returns the logical not of bit aggregate \\axiom{\\spad{b}}."))) 
NIL 
NIL 
(-132) 
((|constructor| (NIL "The bit aggregate category models aggregates representing large quantities of Boolean data.")) (|xor| (($ $ $) "\\spad{xor(a,{}b)} returns the logical exclusive-or of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|or| (($ $ $) "\\spad{a or b} returns the logical or of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|and| (($ $ $) "\\spad{a and b} returns the logical and of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nor| (($ $ $) "\\spad{nor(a,{}b)} returns the logical nor of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nand| (($ $ $) "\\spad{nand(a,{}b)} returns the logical nand of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (^ (($ $) "\\spad{^ b} returns the logical not of bit aggregate \\axiom{\\spad{b}}.")) (|not| (($ $) "\\spad{not(b)} returns the logical not of bit aggregate \\axiom{\\spad{b}}."))) 
((-4506 . T) (-4505 . T) (-3576 . T)) 
NIL 
(-133 A S) 
((|constructor| (NIL "\\spadtype{BinaryTreeCategory(S)} is the category of binary trees: a tree which is either empty or else is a \\spadfun{node} consisting of a value and a \\spadfun{left} and \\spadfun{right},{} both binary trees.")) (|node| (($ $ |#2| $) "\\spad{node(left,{}v,{}right)} creates a binary tree with value \\spad{v},{} a binary tree \\spad{left},{} and a binary tree \\spad{right}. \\blankline")) (|finiteAggregate| ((|attribute|) "Binary trees have a finite number of components")) (|shallowlyMutable| ((|attribute|) "Binary trees have updateable components"))) 
NIL 
NIL 
(-134 S) 
((|constructor| (NIL "\\spadtype{BinaryTreeCategory(S)} is the category of binary trees: a tree which is either empty or else is a \\spadfun{node} consisting of a value and a \\spadfun{left} and \\spadfun{right},{} both binary trees.")) (|node| (($ $ |#1| $) "\\spad{node(left,{}v,{}right)} creates a binary tree with value \\spad{v},{} a binary tree \\spad{left},{} and a binary tree \\spad{right}. \\blankline")) (|finiteAggregate| ((|attribute|) "Binary trees have a finite number of components")) (|shallowlyMutable| ((|attribute|) "Binary trees have updateable components"))) 
((-4505 . T) (-4506 . T) (-3576 . T)) 
NIL 
(-135 S) 
((|constructor| (NIL "BinaryTournament creates a binary tournament with the elements of \\spad{ls} as values at the nodes.")) (|insert!| (($ |#1| $) "\\indented{1}{insert!(\\spad{x},{}\\spad{b}) inserts element \\spad{x} as leaves into binary tournament \\spad{b}.} \\blankline \\spad{X} t1:=binaryTournament [1,{}2,{}3,{}4] \\spad{X} insert!(5,{}\\spad{t1}) \\spad{X} \\spad{t1}")) (|binaryTournament| (($ (|List| |#1|)) "\\indented{1}{binaryTournament(\\spad{ls}) creates a binary tournament with the} \\indented{1}{elements of \\spad{ls} as values at the nodes.} \\blankline \\spad{X} binaryTournament [1,{}2,{}3,{}4]"))) 
((-4505 . T) (-4506 . T)) 
((|HasCategory| |#1| (QUOTE (-1082))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082))))) 
(-136 S) 
((|constructor| (NIL "\\spadtype{BinaryTree(S)} is the domain of all binary trees. A binary tree over \\spad{S} is either empty or has a \\spadfun{value} which is an \\spad{S} and a \\spadfun{right} and \\spadfun{left} which are both binary trees.")) (|binaryTree| (($ $ |#1| $) "\\indented{1}{binaryTree(\\spad{l},{}\\spad{v},{}\\spad{r}) creates a binary tree with} \\indented{1}{value \\spad{v} with left subtree \\spad{l} and right subtree \\spad{r}.} \\blankline \\spad{X} t1:=binaryTree([1,{}2,{}3]) \\spad{X} t2:=binaryTree([4,{}5,{}6]) \\spad{X} binaryTree(\\spad{t1},{}[7,{}8,{}9],{}\\spad{t2})") (($ |#1|) "\\indented{1}{binaryTree(\\spad{v}) is an non-empty binary tree} \\indented{1}{with value \\spad{v},{} and left and right empty.} \\blankline \\spad{X} t1:=binaryTree([1,{}2,{}3])"))) 
((-4505 . T) (-4506 . T)) 
((|HasCategory| |#1| (QUOTE (-1082))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082))))) 
(-137) 
((|constructor| (NIL "This is an \\spadtype{AbelianMonoid} with the cancellation property,{} \\spadignore{i.e.} \\tab{5}\\spad{ a+b = a+c => b=c }.\\spad{\\br} This is formalised by the partial subtraction operator,{} which satisfies the Axioms\\spad{\\br} \\tab{5}\\spad{c = a+b <=> c-b = a}")) (|subtractIfCan| (((|Union| $ "failed") $ $) "\\spad{subtractIfCan(x,{} y)} returns an element \\spad{z} such that \\spad{z+y=x} or \"failed\" if no such element exists."))) 
NIL 
NIL 
(-138) 
((|constructor| (NIL "A cachable set is a set whose elements keep an integer as part of their structure.")) (|setPosition| (((|Void|) $ (|NonNegativeInteger|)) "\\spad{setPosition(x,{} n)} associates the integer \\spad{n} to \\spad{x}.")) (|position| (((|NonNegativeInteger|) $) "\\spad{position(x)} returns the integer \\spad{n} associated to \\spad{x}."))) 
NIL 
NIL 
(-139) 
((|constructor| (NIL "Members of the domain CardinalNumber are values indicating the cardinality of sets,{} both finite and infinite. Arithmetic operations are defined on cardinal numbers as follows. \\blankline If \\spad{x = \\#X} and \\spad{y = \\#Y} then\\spad{\\br} \\tab{5}\\spad{x+y = \\#(X+Y)} \\tab{5}disjoint union\\spad{\\br} \\tab{5}\\spad{x-y = \\#(X-Y)} \\tab{5}relative complement\\spad{\\br} \\tab{5}\\spad{x*y = \\#(X*Y)} \\tab{5}cartesian product\\spad{\\br} \\tab{5}\\spad{x**y = \\#(X**Y)} \\tab{4}\\spad{X**Y = g| g:Y->X} \\blankline The non-negative integers have a natural construction as cardinals\\spad{\\br} \\spad{0 = \\#\\{\\}},{} \\spad{1 = \\{0\\}},{} \\spad{2 = \\{0,{} 1\\}},{} ...,{} \\spad{n = \\{i| 0 <= i < n\\}}. \\blankline That \\spad{0} acts as a zero for the multiplication of cardinals is equivalent to the axiom of choice. \\blankline The generalized continuum hypothesis asserts \\spad{\\br} \\spad{2**Aleph i = Aleph(i+1)} and is independent of the axioms of set theory [Goedel 1940]. \\blankline Three commonly encountered cardinal numbers are\\spad{\\br} \\tab{5}\\spad{a = \\#Z} \\tab{5}countable infinity\\spad{\\br} \\tab{5}\\spad{c = \\#R} \\tab{5}the continuum\\spad{\\br} \\tab{5}\\spad{f = \\# g | g:[0,{}1]->R\\} \\blankline In this domain,{} these values are obtained using\\br \\tab{5}\\spad{a := Aleph 0},{} \\spad{c := 2**a},{} \\spad{f := 2**c}.")) (|generalizedContinuumHypothesisAssumed| (((|Boolean|) (|Boolean|)) "\\indented{1}{generalizedContinuumHypothesisAssumed(bool)} \\indented{1}{is used to dictate whether the hypothesis is to be assumed.} \\blankline \\spad{X} generalizedContinuumHypothesisAssumed \\spad{true} \\spad{X} a:=Aleph 0 \\spad{X} c:=2**a \\spad{X} f:=2**c")) (|generalizedContinuumHypothesisAssumed?| (((|Boolean|)) "\\indented{1}{generalizedContinuumHypothesisAssumed?()} \\indented{1}{tests if the hypothesis is currently assumed.} \\blankline \\spad{X} generalizedContinuumHypothesisAssumed?")) (|countable?| (((|Boolean|) $) "\\indented{1}{countable?(\\spad{a}) determines} \\indented{1}{whether \\spad{a} is a countable cardinal,{}} \\indented{1}{\\spadignore{i.e.} an integer or \\spad{Aleph 0}.} \\blankline \\spad{X} c2:=2::CardinalNumber \\spad{X} countable? \\spad{c2} \\spad{X} A0:=Aleph 0 \\spad{X} countable? \\spad{A0} \\spad{X} A1:=Aleph 1 \\spad{X} countable? \\spad{A1}")) (|finite?| (((|Boolean|) $) "\\indented{1}{finite?(\\spad{a}) determines whether} \\indented{1}{\\spad{a} is a finite cardinal,{} \\spadignore{i.e.} an integer.} \\blankline \\spad{X} c2:=2::CardinalNumber \\spad{X} finite? \\spad{c2} \\spad{X} A0:=Aleph 0 \\spad{X} finite? \\spad{A0}")) (|Aleph| (($ (|NonNegativeInteger|)) "\\indented{1}{Aleph(\\spad{n}) provides the named (infinite) cardinal number.} \\blankline \\spad{X} A0:=Aleph 0")) (** (($ $ $) "\\indented{1}{\\spad{x**y} returns \\spad{\\#(X**Y)} where \\spad{X**Y} is defined} \\indented{2}{as \\spad{\\{g| g:Y->X\\}}.} \\blankline \\spad{X} c2:=2::CardinalNumber \\spad{X} \\spad{c2**c2} \\spad{X} A1:=Aleph 1 \\spad{X} \\spad{A1**c2} \\spad{X} generalizedContinuumHypothesisAssumed \\spad{true} \\spad{X} \\spad{A1**A1}")) (- (((|Union| $ "failed") $ $) "\\indented{1}{\\spad{x - y} returns an element \\spad{z} such that} \\indented{1}{\\spad{z+y=x} or \"failed\" if no such element exists.} \\blankline \\spad{X} c2:=2::CardinalNumber \\spad{X} \\spad{c2}-\\spad{c2} \\spad{X} A1:=Aleph 1 \\spad{X} \\spad{A1}-\\spad{c2}")) (|commutative| ((|attribute| "*") "a domain \\spad{D} has \\spad{commutative(\"*\")} if it has an operation \\spad{\"*\": (D,{}D) -> D} which is commutative."))) 
(((-4507 "*") . T)) 
NIL 
(-140 |minix| -3780 S T$) 
((|constructor| (NIL "This package provides functions to enable conversion of tensors given conversion of the components.")) (|map| (((|CartesianTensor| |#1| |#2| |#4|) (|Mapping| |#4| |#3|) (|CartesianTensor| |#1| |#2| |#3|)) "\\spad{map(f,{}ts)} does a componentwise conversion of the tensor \\spad{ts} to a tensor with components of type \\spad{T}.")) (|reshape| (((|CartesianTensor| |#1| |#2| |#4|) (|List| |#4|) (|CartesianTensor| |#1| |#2| |#3|)) "\\spad{reshape(lt,{}ts)} organizes the list of components \\spad{lt} into a tensor with the same shape as \\spad{ts}."))) 
NIL 
NIL 
(-141 |minix| -3780 R) 
((|constructor| (NIL "CartesianTensor(minix,{}dim,{}\\spad{R}) provides Cartesian tensors with components belonging to a commutative ring \\spad{R}. These tensors can have any number of indices. Each index takes values from \\spad{minix} to \\spad{minix + dim - 1}.")) (|sample| (($) "\\spad{sample()} returns an object of type \\%.")) (|unravel| (($ (|List| |#3|)) "\\spad{unravel(t)} produces a tensor from a list of components such that \\indented{2}{\\spad{unravel(ravel(t)) = t}.}")) (|ravel| (((|List| |#3|) $) "\\indented{1}{ravel(\\spad{t}) produces a list of components from a tensor such that} \\indented{3}{\\spad{unravel(ravel(t)) = t}.} \\blankline \\spad{X} n:SquareMatrix(2,{}Integer):=matrix [[2,{}3],{}[0,{}1]] \\spad{X} tn:CartesianTensor(1,{}2,{}Integer)\\spad{:=n} \\spad{X} ravel \\spad{tn}")) (|leviCivitaSymbol| (($) "\\indented{1}{leviCivitaSymbol() is the rank \\spad{dim} tensor defined by} \\indented{1}{\\spad{leviCivitaSymbol()(i1,{}...idim) = +1/0/-1}} \\indented{1}{if \\spad{i1,{}...,{}idim} is an even/is nota /is an odd permutation} \\indented{1}{of \\spad{minix,{}...,{}minix+dim-1}.} \\blankline \\spad{X} lcs:CartesianTensor(1,{}2,{}Integer):=leviCivitaSymbol()")) (|kroneckerDelta| (($) "\\indented{1}{kroneckerDelta() is the rank 2 tensor defined by} \\indented{4}{\\spad{kroneckerDelta()(i,{}j)}} \\indented{7}{\\spad{= 1\\space{2}if i = j}} \\indented{7}{\\spad{= 0 if\\space{2}i \\^= j}} \\blankline \\spad{X} delta:CartesianTensor(1,{}2,{}Integer):=kroneckerDelta()")) (|reindex| (($ $ (|List| (|Integer|))) "\\indented{1}{reindex(\\spad{t},{}[\\spad{i1},{}...,{}idim]) permutes the indices of \\spad{t}.} \\indented{1}{For example,{} if \\spad{r = reindex(t,{} [4,{}1,{}2,{}3])}} \\indented{1}{for a rank 4 tensor \\spad{t},{}} \\indented{1}{then \\spad{r} is the rank for tensor given by} \\indented{5}{\\spad{r(i,{}j,{}k,{}l) = t(l,{}i,{}j,{}k)}.} \\blankline \\spad{X} n:SquareMatrix(2,{}Integer):=matrix [[2,{}3],{}[0,{}1]] \\spad{X} tn:CartesianTensor(1,{}2,{}Integer)\\spad{:=n} \\spad{X} p:=product(\\spad{tn},{}\\spad{tn}) \\spad{X} reindex(\\spad{p},{}[4,{}3,{}2,{}1])")) (|transpose| (($ $ (|Integer|) (|Integer|)) "\\indented{1}{transpose(\\spad{t},{}\\spad{i},{}\\spad{j}) exchanges the \\spad{i}\\spad{-}th and \\spad{j}\\spad{-}th} \\indented{1}{indices of \\spad{t}. For example,{} if \\spad{r = transpose(t,{}2,{}3)}} \\indented{1}{for a rank 4 tensor \\spad{t},{} then \\spad{r} is the rank 4 tensor} \\indented{1}{given by} \\indented{5}{\\spad{r(i,{}j,{}k,{}l) = t(i,{}k,{}j,{}l)}.} \\blankline \\spad{X} m:SquareMatrix(2,{}Integer):=matrix [[1,{}2],{}[4,{}5]] \\spad{X} tm:CartesianTensor(1,{}2,{}Integer)\\spad{:=m} \\spad{X} tn:CartesianTensor(1,{}2,{}Integer)\\spad{:=}[\\spad{tm},{}\\spad{tm}] \\spad{X} transpose(\\spad{tn},{}1,{}2)") (($ $) "\\indented{1}{transpose(\\spad{t}) exchanges the first and last indices of \\spad{t}.} \\indented{1}{For example,{} if \\spad{r = transpose(t)} for a rank 4} \\indented{1}{tensor \\spad{t},{} then \\spad{r} is the rank 4 tensor given by} \\indented{5}{\\spad{r(i,{}j,{}k,{}l) = t(l,{}j,{}k,{}i)}.} \\blankline \\spad{X} m:SquareMatrix(2,{}Integer):=matrix [[1,{}2],{}[4,{}5]] \\spad{X} Tm:CartesianTensor(1,{}2,{}Integer)\\spad{:=m} \\spad{X} transpose(\\spad{Tm})")) (|contract| (($ $ (|Integer|) (|Integer|)) "\\indented{1}{contract(\\spad{t},{}\\spad{i},{}\\spad{j}) is the contraction of tensor \\spad{t} which} \\indented{1}{sums along the \\spad{i}\\spad{-}th and \\spad{j}\\spad{-}th indices.} \\indented{1}{For example,{}\\space{2}if} \\indented{1}{\\spad{r = contract(t,{}1,{}3)} for a rank 4 tensor \\spad{t},{} then} \\indented{1}{\\spad{r} is the rank 2 \\spad{(= 4 - 2)} tensor given by} \\indented{5}{\\spad{r(i,{}j) = sum(h=1..dim,{}t(h,{}i,{}h,{}j))}.} \\blankline \\spad{X} m:SquareMatrix(2,{}Integer):=matrix [[1,{}2],{}[4,{}5]] \\spad{X} Tm:CartesianTensor(1,{}2,{}Integer)\\spad{:=m} \\spad{X} v:DirectProduct(2,{}Integer):=directProduct [3,{}4] \\spad{X} Tv:CartesianTensor(1,{}2,{}Integer)\\spad{:=v} \\spad{X} Tmv:=contract(\\spad{Tm},{}2,{}1)") (($ $ (|Integer|) $ (|Integer|)) "\\indented{1}{contract(\\spad{t},{}\\spad{i},{}\\spad{s},{}\\spad{j}) is the inner product of tenors \\spad{s} and \\spad{t}} \\indented{1}{which sums along the \\spad{k1}\\spad{-}th index of} \\indented{1}{\\spad{t} and the \\spad{k2}\\spad{-}th index of \\spad{s}.} \\indented{1}{For example,{} if \\spad{r = contract(s,{}2,{}t,{}1)} for rank 3 tensors} \\indented{1}{rank 3 tensors \\spad{s} and \\spad{t},{} then \\spad{r} is} \\indented{1}{the rank 4 \\spad{(= 3 + 3 - 2)} tensor\\space{2}given by} \\indented{5}{\\spad{r(i,{}j,{}k,{}l) = sum(h=1..dim,{}s(i,{}h,{}j)*t(h,{}k,{}l))}.} \\blankline \\spad{X} m:SquareMatrix(2,{}Integer):=matrix [[1,{}2],{}[4,{}5]] \\spad{X} Tm:CartesianTensor(1,{}2,{}Integer)\\spad{:=m} \\spad{X} v:DirectProduct(2,{}Integer):=directProduct [3,{}4] \\spad{X} Tv:CartesianTensor(1,{}2,{}Integer)\\spad{:=v} \\spad{X} Tmv:=contract(\\spad{Tm},{}2,{}\\spad{Tv},{}1)")) (* (($ $ $) "\\indented{1}{\\spad{s*t} is the inner product of the tensors \\spad{s} and \\spad{t} which contracts} \\indented{1}{the last index of \\spad{s} with the first index of \\spad{t},{} that is,{}} \\indented{5}{\\spad{t*s = contract(t,{}rank t,{} s,{} 1)}} \\indented{5}{\\spad{t*s = sum(k=1..N,{} t[i1,{}..,{}iN,{}k]*s[k,{}j1,{}..,{}jM])}} \\indented{1}{This is compatible with the use of \\spad{M*v} to denote} \\indented{1}{the matrix-vector inner product.} \\blankline \\spad{X} m:SquareMatrix(2,{}Integer):=matrix [[1,{}2],{}[4,{}5]] \\spad{X} Tm:CartesianTensor(1,{}2,{}Integer)\\spad{:=m} \\spad{X} v:DirectProduct(2,{}Integer):=directProduct [3,{}4] \\spad{X} Tv:CartesianTensor(1,{}2,{}Integer)\\spad{:=v} \\spad{X} Tm*Tv")) (|product| (($ $ $) "\\indented{1}{product(\\spad{s},{}\\spad{t}) is the outer product of the tensors \\spad{s} and \\spad{t}.} \\indented{1}{For example,{} if \\spad{r = product(s,{}t)} for rank 2 tensors} \\indented{1}{\\spad{s} and \\spad{t},{} then \\spad{r} is a rank 4 tensor given by} \\indented{5}{\\spad{r(i,{}j,{}k,{}l) = s(i,{}j)*t(k,{}l)}.} \\blankline \\spad{X} m:SquareMatrix(2,{}Integer):=matrix [[1,{}2],{}[4,{}5]] \\spad{X} Tm:CartesianTensor(1,{}2,{}Integer)\\spad{:=m} \\spad{X} n:SquareMatrix(2,{}Integer):=matrix [[2,{}3],{}[0,{}1]] \\spad{X} Tn:CartesianTensor(1,{}2,{}Integer)\\spad{:=n} \\spad{X} Tmn:=product(\\spad{Tm},{}\\spad{Tn})")) (|elt| ((|#3| $ (|List| (|Integer|))) "\\indented{1}{elt(\\spad{t},{}[\\spad{i1},{}...,{}iN]) gives a component of a rank \\spad{N} tensor.} \\blankline \\spad{X} \\spad{v:=}[2,{}3] \\spad{X} tv:CartesianTensor(1,{}2,{}Integer)\\spad{:=v} \\spad{X} tm:CartesianTensor(1,{}2,{}Integer)\\spad{:=}[\\spad{tv},{}\\spad{tv}] \\spad{X} tn:CartesianTensor(1,{}2,{}Integer)\\spad{:=}[\\spad{tm},{}\\spad{tm}] \\spad{X} tp:CartesianTensor(1,{}2,{}Integer)\\spad{:=}[\\spad{tn},{}\\spad{tn}] \\spad{X} tq:CartesianTensor(1,{}2,{}Integer)\\spad{:=}[\\spad{tp},{}\\spad{tp}] \\spad{X} elt(\\spad{tq},{}[2,{}2,{}2,{}2,{}2])") ((|#3| $ (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\indented{1}{elt(\\spad{t},{}\\spad{i},{}\\spad{j},{}\\spad{k},{}\\spad{l}) gives a component of a rank 4 tensor.} \\blankline \\spad{X} \\spad{v:=}[2,{}3] \\spad{X} tv:CartesianTensor(1,{}2,{}Integer)\\spad{:=v} \\spad{X} tm:CartesianTensor(1,{}2,{}Integer)\\spad{:=}[\\spad{tv},{}\\spad{tv}] \\spad{X} tn:CartesianTensor(1,{}2,{}Integer)\\spad{:=}[\\spad{tm},{}\\spad{tm}] \\spad{X} tp:CartesianTensor(1,{}2,{}Integer)\\spad{:=}[\\spad{tn},{}\\spad{tn}] \\spad{X} elt(\\spad{tp},{}2,{}2,{}2,{}2) \\spad{X} \\spad{tp}[2,{}2,{}2,{}2]") ((|#3| $ (|Integer|) (|Integer|) (|Integer|)) "\\indented{1}{elt(\\spad{t},{}\\spad{i},{}\\spad{j},{}\\spad{k}) gives a component of a rank 3 tensor.} \\blankline \\spad{X} \\spad{v:=}[2,{}3] \\spad{X} tv:CartesianTensor(1,{}2,{}Integer)\\spad{:=v} \\spad{X} tm:CartesianTensor(1,{}2,{}Integer)\\spad{:=}[\\spad{tv},{}\\spad{tv}] \\spad{X} tn:CartesianTensor(1,{}2,{}Integer)\\spad{:=}[\\spad{tm},{}\\spad{tm}] \\spad{X} elt(\\spad{tn},{}2,{}2,{}2) \\spad{X} \\spad{tn}[2,{}2,{}2]") ((|#3| $ (|Integer|) (|Integer|)) "\\indented{1}{elt(\\spad{t},{}\\spad{i},{}\\spad{j}) gives a component of a rank 2 tensor.} \\blankline \\spad{X} \\spad{v:=}[2,{}3] \\spad{X} tv:CartesianTensor(1,{}2,{}Integer)\\spad{:=v} \\spad{X} tm:CartesianTensor(1,{}2,{}Integer)\\spad{:=}[\\spad{tv},{}\\spad{tv}] \\spad{X} elt(\\spad{tm},{}2,{}2) \\spad{X} \\spad{tm}[2,{}2]") ((|#3| $ (|Integer|)) "\\indented{1}{elt(\\spad{t},{}\\spad{i}) gives a component of a rank 1 tensor.} \\blankline \\spad{X} \\spad{v:=}[2,{}3] \\spad{X} tv:CartesianTensor(1,{}2,{}Integer)\\spad{:=v} \\spad{X} elt(\\spad{tv},{}2) \\spad{X} \\spad{tv}[2]") ((|#3| $) "\\indented{1}{elt(\\spad{t}) gives the component of a rank 0 tensor.} \\blankline \\spad{X} tv:CartesianTensor(1,{}2,{}Integer)\\spad{:=8} \\spad{X} elt(\\spad{tv}) \\spad{X} \\spad{tv}[]")) (|rank| (((|NonNegativeInteger|) $) "\\indented{1}{rank(\\spad{t}) returns the tensorial rank of \\spad{t} (that is,{} the} \\indented{1}{number of indices).\\space{2}This is the same as the graded module} \\indented{1}{degree.} \\blankline \\spad{X} CT:=CARTEN(1,{}2,{}Integer) \\spad{X} \\spad{t0:CT:=8} \\spad{X} rank \\spad{t0}")) (|coerce| (($ (|List| $)) "\\indented{1}{coerce([\\spad{t_1},{}...,{}t_dim]) allows tensors to be constructed} \\indented{1}{using lists.} \\blankline \\spad{X} \\spad{v:=}[2,{}3] \\spad{X} tv:CartesianTensor(1,{}2,{}Integer)\\spad{:=v} \\spad{X} tm:CartesianTensor(1,{}2,{}Integer)\\spad{:=}[\\spad{tv},{}\\spad{tv}]") (($ (|List| |#3|)) "\\indented{1}{coerce([\\spad{r_1},{}...,{}r_dim]) allows tensors to be constructed} \\indented{1}{using lists.} \\blankline \\spad{X} \\spad{v:=}[2,{}3] \\spad{X} tv:CartesianTensor(1,{}2,{}Integer)\\spad{:=v}") (($ (|SquareMatrix| |#2| |#3|)) "\\indented{1}{coerce(\\spad{m}) views a matrix as a rank 2 tensor.} \\blankline \\spad{X} v:SquareMatrix(2,{}Integer)\\spad{:=}[[1,{}2],{}[3,{}4]] \\spad{X} tv:CartesianTensor(1,{}2,{}Integer)\\spad{:=v}") (($ (|DirectProduct| |#2| |#3|)) "\\indented{1}{coerce(\\spad{v}) views a vector as a rank 1 tensor.} \\blankline \\spad{X} v:DirectProduct(2,{}Integer):=directProduct [3,{}4] \\spad{X} tv:CartesianTensor(1,{}2,{}Integer)\\spad{:=v}"))) 
NIL 
NIL 
(-142) 
((|constructor| (NIL "This domain allows classes of characters to be defined and manipulated efficiently.")) (|alphanumeric| (($) "\\spad{alphanumeric()} returns the class of all characters for which alphanumeric? is \\spad{true}.")) (|alphabetic| (($) "\\spad{alphabetic()} returns the class of all characters for which alphabetic? is \\spad{true}.")) (|lowerCase| (($) "\\spad{lowerCase()} returns the class of all characters for which lowerCase? is \\spad{true}.")) (|upperCase| (($) "\\spad{upperCase()} returns the class of all characters for which upperCase? is \\spad{true}.")) (|hexDigit| (($) "\\spad{hexDigit()} returns the class of all characters for which hexDigit? is \\spad{true}.")) (|digit| (($) "\\spad{digit()} returns the class of all characters for which digit? is \\spad{true}.")) (|charClass| (($ (|List| (|Character|))) "\\spad{charClass(l)} creates a character class which contains exactly the characters given in the list \\spad{l}.") (($ (|String|)) "\\spad{charClass(s)} creates a character class which contains exactly the characters given in the string \\spad{s}."))) 
((-4505 . T) (-4495 . T) (-4506 . T)) 
((|HasCategory| (-145) (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| (-145) (QUOTE (-364))) (|HasCategory| (-145) (QUOTE (-834))) (|HasCategory| (-145) (QUOTE (-1082))) (-12 (|HasCategory| (-145) (LIST (QUOTE -298) (QUOTE (-145)))) (|HasCategory| (-145) (QUOTE (-1082)))) (-3322 (-12 (|HasCategory| (-145) (LIST (QUOTE -298) (QUOTE (-145)))) (|HasCategory| (-145) (QUOTE (-364)))) (-12 (|HasCategory| (-145) (LIST (QUOTE -298) (QUOTE (-145)))) (|HasCategory| (-145) (QUOTE (-1082)))))) 
(-143 R Q A) 
((|constructor| (NIL "CommonDenominator provides functions to compute the common denominator of a finite linear aggregate of elements of the quotient field of an integral domain.")) (|splitDenominator| (((|Record| (|:| |num| |#3|) (|:| |den| |#1|)) |#3|) "\\spad{splitDenominator([q1,{}...,{}qn])} returns \\spad{[[p1,{}...,{}pn],{} d]} such that \\spad{\\spad{qi} = pi/d} and \\spad{d} is a common denominator for the \\spad{qi}\\spad{'s}.")) (|clearDenominator| ((|#3| |#3|) "\\spad{clearDenominator([q1,{}...,{}qn])} returns \\spad{[p1,{}...,{}pn]} such that \\spad{\\spad{qi} = pi/d} where \\spad{d} is a common denominator for the \\spad{qi}\\spad{'s}.")) (|commonDenominator| ((|#1| |#3|) "\\spad{commonDenominator([q1,{}...,{}qn])} returns a common denominator \\spad{d} for \\spad{q1},{}...,{}\\spad{qn}."))) 
NIL 
NIL 
(-144) 
((|constructor| (NIL "Category for the usual combinatorial functions.")) (|permutation| (($ $ $) "\\spad{permutation(n,{} m)} returns the number of permutations of \\spad{n} objects taken \\spad{m} at a time. Note that \\spad{permutation(n,{}m) = n!/(n-m)!}.")) (|factorial| (($ $) "\\spad{factorial(n)} computes the factorial of \\spad{n} (denoted in the literature by \\spad{n!}) Note that \\spad{n! = n (n-1)! when n > 0}; also,{} \\spad{0! = 1}.")) (|binomial| (($ $ $) "\\indented{1}{binomial(\\spad{n},{}\\spad{r}) returns the \\spad{(n,{}r)} binomial coefficient} \\indented{1}{(often denoted in the literature by \\spad{C(n,{}r)}).} \\indented{1}{Note that \\spad{C(n,{}r) = n!/(r!(n-r)!)} where \\spad{n >= r >= 0}.} \\blankline \\spad{X} [binomial(5,{}\\spad{i}) for \\spad{i} in 0..5]"))) 
NIL 
NIL 
(-145) 
((|constructor| (NIL "This domain provides the basic character data type.")) (|alphanumeric?| (((|Boolean|) $) "\\indented{1}{alphanumeric?(\\spad{c}) tests if \\spad{c} is either a letter or number,{}} \\indented{1}{\\spadignore{i.e.} one of 0..9,{} a..\\spad{z} or A..\\spad{Z}.} \\blankline \\spad{X} chars \\spad{:=} [char \"a\",{} char \"A\",{} char \\spad{\"X\"},{} char \\spad{\"8\"},{} char \\spad{\"+\"}] \\spad{X} [alphanumeric? \\spad{c} for \\spad{c} in chars]")) (|lowerCase?| (((|Boolean|) $) "\\indented{1}{lowerCase?(\\spad{c}) tests if \\spad{c} is an lower case letter,{}} \\indented{1}{\\spadignore{i.e.} one of a..\\spad{z}.} \\blankline \\spad{X} chars \\spad{:=} [char \"a\",{} char \"A\",{} char \\spad{\"X\"},{} char \\spad{\"8\"},{} char \\spad{\"+\"}] \\spad{X} [lowerCase? \\spad{c} for \\spad{c} in chars]")) (|upperCase?| (((|Boolean|) $) "\\indented{1}{upperCase?(\\spad{c}) tests if \\spad{c} is an upper case letter,{}} \\indented{1}{\\spadignore{i.e.} one of A..\\spad{Z}.} \\blankline \\spad{X} chars \\spad{:=} [char \"a\",{} char \"A\",{} char \\spad{\"X\"},{} char \\spad{\"8\"},{} char \\spad{\"+\"}] \\spad{X} [upperCase? \\spad{c} for \\spad{c} in chars]")) (|alphabetic?| (((|Boolean|) $) "\\indented{1}{alphabetic?(\\spad{c}) tests if \\spad{c} is a letter,{}} \\indented{1}{\\spadignore{i.e.} one of a..\\spad{z} or A..\\spad{Z}.} \\blankline \\spad{X} chars \\spad{:=} [char \"a\",{} char \"A\",{} char \\spad{\"X\"},{} char \\spad{\"8\"},{} char \\spad{\"+\"}] \\spad{X} [alphabetic? \\spad{c} for \\spad{c} in chars]")) (|hexDigit?| (((|Boolean|) $) "\\indented{1}{hexDigit?(\\spad{c}) tests if \\spad{c} is a hexadecimal numeral,{}} \\indented{1}{\\spadignore{i.e.} one of 0..9,{} a..\\spad{f} or A..\\spad{F}.} \\blankline \\spad{X} chars \\spad{:=} [char \"a\",{} char \"A\",{} char \\spad{\"X\"},{} char \\spad{\"8\"},{} char \\spad{\"+\"}] \\spad{X} [hexDigit? \\spad{c} for \\spad{c} in chars]")) (|digit?| (((|Boolean|) $) "\\indented{1}{digit?(\\spad{c}) tests if \\spad{c} is a digit character,{}} \\indented{1}{\\spadignore{i.e.} one of 0..9.} \\blankline \\spad{X} chars \\spad{:=} [char \"a\",{} char \"A\",{} char \\spad{\"X\"},{} char \\spad{\"8\"},{} char \\spad{\"+\"}] \\spad{X} [digit? \\spad{c} for \\spad{c} in chars]")) (|lowerCase| (($ $) "\\indented{1}{lowerCase(\\spad{c}) converts an upper case letter to the corresponding} \\indented{1}{lower case letter.\\space{2}If \\spad{c} is not an upper case letter,{} then} \\indented{1}{it is returned unchanged.} \\blankline \\spad{X} chars \\spad{:=} [char \"a\",{} char \"A\",{} char \\spad{\"X\"},{} char \\spad{\"8\"},{} char \\spad{\"+\"}] \\spad{X} [lowerCase \\spad{c} for \\spad{c} in chars]")) (|upperCase| (($ $) "\\indented{1}{upperCase(\\spad{c}) converts a lower case letter to the corresponding} \\indented{1}{upper case letter.\\space{2}If \\spad{c} is not a lower case letter,{} then} \\indented{1}{it is returned unchanged.} \\blankline \\spad{X} chars \\spad{:=} [char \"a\",{} char \"A\",{} char \\spad{\"X\"},{} char \\spad{\"8\"},{} char \\spad{\"+\"}] \\spad{X} [upperCase \\spad{c} for \\spad{c} in chars]")) (|escape| (($) "\\indented{1}{escape() provides the escape character,{} \\spad{_},{} which} \\indented{1}{is used to allow quotes and other characters within} \\indented{1}{strings.} \\blankline \\spad{X} escape()")) (|quote| (($) "\\indented{1}{quote() provides the string quote character,{} \\spad{\"}.} \\blankline \\spad{X} quote()")) (|space| (($) "\\indented{1}{space() provides the blank character.} \\blankline \\spad{X} space()")) (|char| (($ (|String|)) "\\indented{1}{char(\\spad{s}) provides a character from a string \\spad{s} of length one.} \\blankline \\spad{X} [char \\spad{c} for \\spad{c} in [\"a\",{}\"A\",{}\\spad{\"X\"},{}\\spad{\"8\"},{}\\spad{\"+\"}]]") (($ (|Integer|)) "\\indented{1}{char(\\spad{i}) provides a character corresponding to the integer} \\indented{1}{code \\spad{i}. It is always \\spad{true} that \\spad{ord char i = i}.} \\blankline \\spad{X} [char \\spad{c} for \\spad{c} in [97,{}65,{}88,{}56,{}43]]")) (|ord| (((|Integer|) $) "\\indented{1}{ord(\\spad{c}) provides an integral code corresponding to the} \\indented{1}{character \\spad{c}.\\space{2}It is always \\spad{true} that \\spad{char ord c = c}.} \\blankline \\spad{X} chars \\spad{:=} [char \"a\",{} char \"A\",{} char \\spad{\"X\"},{} char \\spad{\"8\"},{} char \\spad{\"+\"}] \\spad{X} [ord \\spad{c} for \\spad{c} in chars]"))) 
NIL 
NIL 
(-146) 
((|constructor| (NIL "Rings of Characteristic Non Zero")) (|charthRoot| (((|Union| $ "failed") $) "\\spad{charthRoot(x)} returns the \\spad{p}th root of \\spad{x} where \\spad{p} is the characteristic of the ring."))) 
((-4502 . T)) 
NIL 
(-147 R) 
((|constructor| (NIL "This package provides a characteristicPolynomial function for any matrix over a commutative ring.")) (|characteristicPolynomial| ((|#1| (|Matrix| |#1|) |#1|) "\\spad{characteristicPolynomial(m,{}r)} computes the characteristic polynomial of the matrix \\spad{m} evaluated at the point \\spad{r}. In particular,{} if \\spad{r} is the polynomial \\spad{'x},{} then it returns the characteristic polynomial expressed as a polynomial in \\spad{'x}."))) 
NIL 
NIL 
(-148) 
((|constructor| (NIL "Rings of Characteristic Zero."))) 
((-4502 . T)) 
NIL 
(-149 -2262 UP UPUP) 
((|constructor| (NIL "Tools to send a point to infinity on an algebraic curve.")) (|chvar| (((|Record| (|:| |func| |#3|) (|:| |poly| |#3|) (|:| |c1| (|Fraction| |#2|)) (|:| |c2| (|Fraction| |#2|)) (|:| |deg| (|NonNegativeInteger|))) |#3| |#3|) "\\spad{chvar(f(x,{}y),{} p(x,{}y))} returns \\spad{[g(z,{}t),{} q(z,{}t),{} c1(z),{} c2(z),{} n]} such that under the change of variable \\spad{x = c1(z)},{} \\spad{y = t * c2(z)},{} one gets \\spad{f(x,{}y) = g(z,{}t)}. The algebraic relation between \\spad{x} and \\spad{y} is \\spad{p(x,{} y) = 0}. The algebraic relation between \\spad{z} and \\spad{t} is \\spad{q(z,{} t) = 0}.")) (|eval| ((|#3| |#3| (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{eval(p(x,{}y),{} f(x),{} g(x))} returns \\spad{p(f(x),{} y * g(x))}.")) (|goodPoint| ((|#1| |#3| |#3|) "\\spad{goodPoint(p,{} q)} returns an integer a such that a is neither a pole of \\spad{p(x,{}y)} nor a branch point of \\spad{q(x,{}y) = 0}.")) (|rootPoly| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| (|Fraction| |#2|)) (|:| |radicand| |#2|)) (|Fraction| |#2|) (|NonNegativeInteger|)) "\\spad{rootPoly(g,{} n)} returns \\spad{[m,{} c,{} P]} such that \\spad{c * g ** (1/n) = P ** (1/m)} thus if \\spad{y**n = g},{} then \\spad{z**m = P} where \\spad{z = c * y}.")) (|radPoly| (((|Union| (|Record| (|:| |radicand| (|Fraction| |#2|)) (|:| |deg| (|NonNegativeInteger|))) "failed") |#3|) "\\spad{radPoly(p(x,{} y))} returns \\spad{[c(x),{} n]} if \\spad{p} is of the form \\spad{y**n - c(x)},{} \"failed\" otherwise.")) (|mkIntegral| (((|Record| (|:| |coef| (|Fraction| |#2|)) (|:| |poly| |#3|)) |#3|) "\\spad{mkIntegral(p(x,{}y))} returns \\spad{[c(x),{} q(x,{}z)]} such that \\spad{z = c * y} is integral. The algebraic relation between \\spad{x} and \\spad{y} is \\spad{p(x,{} y) = 0}. The algebraic relation between \\spad{x} and \\spad{z} is \\spad{q(x,{} z) = 0}."))) 
NIL 
NIL 
(-150 R CR) 
((|constructor| (NIL "This package provides the generalized euclidean algorithm which is needed as the basic step for factoring polynomials.")) (|solveLinearPolynomialEquation| (((|Union| (|List| (|SparseUnivariatePolynomial| |#2|)) "failed") (|List| (|SparseUnivariatePolynomial| |#2|)) (|SparseUnivariatePolynomial| |#2|)) "\\spad{solveLinearPolynomialEquation([f1,{} ...,{} fn],{} g)} where (\\spad{fi} relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g} = sum \\spad{ai} prod \\spad{fj} (\\spad{j} \\spad{\\=} \\spad{i}) or equivalently g/prod \\spad{fj} = sum (ai/fi) or returns \"failed\" if no such list exists"))) 
NIL 
NIL 
(-151 A S) 
((|constructor| (NIL "A collection is a homogeneous aggregate which can built from list of members. The operation used to build the aggregate is generically named \\spadfun{construct}. However,{} each collection provides its own special function with the same name as the data type,{} except with an initial lower case letter,{} \\spadignore{e.g.} \\spadfun{list} for \\spadtype{List},{} \\spadfun{flexibleArray} for \\spadtype{FlexibleArray},{} and so on.")) (|removeDuplicates| (($ $) "\\spad{removeDuplicates(u)} returns a copy of \\spad{u} with all duplicates removed.")) (|select| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{select(p,{}u)} returns a copy of \\spad{u} containing only those elements such \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. Note that \\axiom{select(\\spad{p},{}\\spad{u}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{u} | \\spad{p}(\\spad{x})]}.")) (|remove| (($ |#2| $) "\\spad{remove(x,{}u)} returns a copy of \\spad{u} with all elements \\axiom{\\spad{y} = \\spad{x}} removed. Note that \\axiom{remove(\\spad{y},{}\\spad{c}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{c} | \\spad{x} \\spad{^=} \\spad{y}]}.") (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{remove(p,{}u)} returns a copy of \\spad{u} removing all elements \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. Note that \\axiom{remove(\\spad{p},{}\\spad{u}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{u} | not \\spad{p}(\\spad{x})]}.")) (|reduce| ((|#2| (|Mapping| |#2| |#2| |#2|) $ |#2| |#2|) "\\spad{reduce(f,{}u,{}x,{}z)} reduces the binary operation \\spad{f} across \\spad{u},{} stopping when an \"absorbing element\" \\spad{z} is encountered. As for \\axiom{reduce(\\spad{f},{}\\spad{u},{}\\spad{x})},{} \\spad{x} is the identity operation of \\spad{f}. Same as \\axiom{reduce(\\spad{f},{}\\spad{u},{}\\spad{x})} when \\spad{u} contains no element \\spad{z}. Thus the third argument \\spad{x} is returned when \\spad{u} is empty.") ((|#2| (|Mapping| |#2| |#2| |#2|) $ |#2|) "\\spad{reduce(f,{}u,{}x)} reduces the binary operation \\spad{f} across \\spad{u},{} where \\spad{x} is the identity operation of \\spad{f}. Same as \\axiom{reduce(\\spad{f},{}\\spad{u})} if \\spad{u} has 2 or more elements. Returns \\axiom{\\spad{f}(\\spad{x},{}\\spad{y})} if \\spad{u} has one element \\spad{y},{} \\spad{x} if \\spad{u} is empty. For example,{} \\axiom{reduce(+,{}\\spad{u},{}0)} returns the sum of the elements of \\spad{u}.") ((|#2| (|Mapping| |#2| |#2| |#2|) $) "\\indented{1}{reduce(\\spad{f},{}\\spad{u}) reduces the binary operation \\spad{f} across \\spad{u}. For example,{}} \\indented{1}{if \\spad{u} is \\axiom{[\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]} then \\axiom{reduce(\\spad{f},{}\\spad{u})}} \\indented{1}{returns \\axiom{\\spad{f}(..\\spad{f}(\\spad{f}(\\spad{x},{}\\spad{y}),{}...),{}\\spad{z})}.} \\indented{1}{Note that if \\spad{u} has one element \\spad{x},{} \\axiom{reduce(\\spad{f},{}\\spad{u})} returns \\spad{x}.} \\indented{1}{Error: if \\spad{u} is empty.} \\blankline \\spad{C} )clear all \\spad{X} reduce(+,{}[\\spad{C}[\\spad{i}]*x**i for \\spad{i} in 1..5])")) (|find| (((|Union| |#2| "failed") (|Mapping| (|Boolean|) |#2|) $) "\\spad{find(p,{}u)} returns the first \\spad{x} in \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true},{} and \"failed\" otherwise.")) (|construct| (($ (|List| |#2|)) "\\axiom{construct(\\spad{x},{}\\spad{y},{}...,{}\\spad{z})} returns the collection of elements \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}} ordered as given. Equivalently written as \\axiom{[\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]\\$\\spad{D}},{} where \\spad{D} is the domain. \\spad{D} may be omitted for those of type List."))) 
NIL 
((|HasCategory| |#2| (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| |#2| (QUOTE (-1082))) (|HasAttribute| |#1| (QUOTE -4505))) 
(-152 S) 
((|constructor| (NIL "A collection is a homogeneous aggregate which can built from list of members. The operation used to build the aggregate is generically named \\spadfun{construct}. However,{} each collection provides its own special function with the same name as the data type,{} except with an initial lower case letter,{} \\spadignore{e.g.} \\spadfun{list} for \\spadtype{List},{} \\spadfun{flexibleArray} for \\spadtype{FlexibleArray},{} and so on.")) (|removeDuplicates| (($ $) "\\spad{removeDuplicates(u)} returns a copy of \\spad{u} with all duplicates removed.")) (|select| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{select(p,{}u)} returns a copy of \\spad{u} containing only those elements such \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. Note that \\axiom{select(\\spad{p},{}\\spad{u}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{u} | \\spad{p}(\\spad{x})]}.")) (|remove| (($ |#1| $) "\\spad{remove(x,{}u)} returns a copy of \\spad{u} with all elements \\axiom{\\spad{y} = \\spad{x}} removed. Note that \\axiom{remove(\\spad{y},{}\\spad{c}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{c} | \\spad{x} \\spad{^=} \\spad{y}]}.") (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{remove(p,{}u)} returns a copy of \\spad{u} removing all elements \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. Note that \\axiom{remove(\\spad{p},{}\\spad{u}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{u} | not \\spad{p}(\\spad{x})]}.")) (|reduce| ((|#1| (|Mapping| |#1| |#1| |#1|) $ |#1| |#1|) "\\spad{reduce(f,{}u,{}x,{}z)} reduces the binary operation \\spad{f} across \\spad{u},{} stopping when an \"absorbing element\" \\spad{z} is encountered. As for \\axiom{reduce(\\spad{f},{}\\spad{u},{}\\spad{x})},{} \\spad{x} is the identity operation of \\spad{f}. Same as \\axiom{reduce(\\spad{f},{}\\spad{u},{}\\spad{x})} when \\spad{u} contains no element \\spad{z}. Thus the third argument \\spad{x} is returned when \\spad{u} is empty.") ((|#1| (|Mapping| |#1| |#1| |#1|) $ |#1|) "\\spad{reduce(f,{}u,{}x)} reduces the binary operation \\spad{f} across \\spad{u},{} where \\spad{x} is the identity operation of \\spad{f}. Same as \\axiom{reduce(\\spad{f},{}\\spad{u})} if \\spad{u} has 2 or more elements. Returns \\axiom{\\spad{f}(\\spad{x},{}\\spad{y})} if \\spad{u} has one element \\spad{y},{} \\spad{x} if \\spad{u} is empty. For example,{} \\axiom{reduce(+,{}\\spad{u},{}0)} returns the sum of the elements of \\spad{u}.") ((|#1| (|Mapping| |#1| |#1| |#1|) $) "\\indented{1}{reduce(\\spad{f},{}\\spad{u}) reduces the binary operation \\spad{f} across \\spad{u}. For example,{}} \\indented{1}{if \\spad{u} is \\axiom{[\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]} then \\axiom{reduce(\\spad{f},{}\\spad{u})}} \\indented{1}{returns \\axiom{\\spad{f}(..\\spad{f}(\\spad{f}(\\spad{x},{}\\spad{y}),{}...),{}\\spad{z})}.} \\indented{1}{Note that if \\spad{u} has one element \\spad{x},{} \\axiom{reduce(\\spad{f},{}\\spad{u})} returns \\spad{x}.} \\indented{1}{Error: if \\spad{u} is empty.} \\blankline \\spad{C} )clear all \\spad{X} reduce(+,{}[\\spad{C}[\\spad{i}]*x**i for \\spad{i} in 1..5])")) (|find| (((|Union| |#1| "failed") (|Mapping| (|Boolean|) |#1|) $) "\\spad{find(p,{}u)} returns the first \\spad{x} in \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true},{} and \"failed\" otherwise.")) (|construct| (($ (|List| |#1|)) "\\axiom{construct(\\spad{x},{}\\spad{y},{}...,{}\\spad{z})} returns the collection of elements \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}} ordered as given. Equivalently written as \\axiom{[\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]\\$\\spad{D}},{} where \\spad{D} is the domain. \\spad{D} may be omitted for those of type List."))) 
((-3576 . T)) 
NIL 
(-153 |n| K Q) 
((|constructor| (NIL "CliffordAlgebra(\\spad{n},{} \\spad{K},{} \\spad{Q}) defines a vector space of dimension \\spad{2**n} over \\spad{K},{} given a quadratic form \\spad{Q} on \\spad{K**n}. \\blankline If \\spad{e[i]},{} \\spad{1<=i<=n} is a basis for \\spad{K**n} then 1,{} \\spad{e[i]} (\\spad{1<=i<=n}),{} \\spad{e[i1]*e[i2]} (\\spad{1<=i1<i2<=n}),{}...,{}\\spad{e[1]*e[2]*..*e[n]} is a basis for the Clifford Algebra. \\blankline The algebra is defined by the relations\\spad{\\br} \\tab{5}\\spad{e[i]*e[j] = -e[j]*e[i]} (\\spad{i \\~~= j}),{}\\spad{\\br} \\tab{5}\\spad{e[i]*e[i] = Q(e[i])} \\blankline Examples of Clifford Algebras are: gaussians,{} quaternions,{} exterior algebras and spin algebras.")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} computes the multiplicative inverse of \\spad{x} or \"failed\" if \\spad{x} is not invertible.")) (|coefficient| ((|#2| $ (|List| (|PositiveInteger|))) "\\spad{coefficient(x,{}[i1,{}i2,{}...,{}iN])} extracts the coefficient of \\spad{e(i1)*e(i2)*...*e(iN)} in \\spad{x}.")) (|monomial| (($ |#2| (|List| (|PositiveInteger|))) "\\spad{monomial(c,{}[i1,{}i2,{}...,{}iN])} produces the value given by \\spad{c*e(i1)*e(i2)*...*e(iN)}.")) (|e| (($ (|PositiveInteger|)) "\\spad{e(n)} produces the appropriate unit element."))) 
((-4500 . T) (-4499 . T) (-4502 . T)) 
NIL 
(-154) 
((|constructor| (NIL "Automatic clipping for 2-dimensional plots The purpose of this package is to provide reasonable plots of functions with singularities.")) (|clipWithRanges| (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|List| (|List| (|Point| (|DoubleFloat|)))) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{clipWithRanges(pointLists,{}xMin,{}xMax,{}yMin,{}yMax)} performs clipping on a list of lists of points,{} \\spad{pointLists}. Clipping is done within the specified ranges of \\spad{xMin},{} \\spad{xMax} and \\spad{yMin},{} \\spad{yMax}. This function is used internally by the \\fakeAxiomFun{iClipParametric} subroutine in this package.")) (|clipParametric| (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|Plot|) (|Fraction| (|Integer|)) (|Fraction| (|Integer|))) "\\spad{clipParametric(p,{}frac,{}sc)} performs two-dimensional clipping on a plot,{} \\spad{p},{} from the domain \\spadtype{Plot} for the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)}; the fraction parameter is specified by \\spad{frac} and the scale parameter is specified by \\spad{sc} for use in the \\fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.") (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|Plot|)) "\\spad{clipParametric(p)} performs two-dimensional clipping on a plot,{} \\spad{p},{} from the domain \\spadtype{Plot} for the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)}; the default parameters \\spad{1/2} for the fraction and \\spad{5/1} for the scale are used in the \\fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.")) (|clip| (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|List| (|List| (|Point| (|DoubleFloat|))))) "\\spad{clip(ll)} performs two-dimensional clipping on a list of lists of points,{} \\spad{ll}; the default parameters \\spad{1/2} for the fraction and \\spad{5/1} for the scale are used in the \\fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.") (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|List| (|Point| (|DoubleFloat|)))) "\\spad{clip(l)} performs two-dimensional clipping on a curve \\spad{l},{} which is a list of points; the default parameters \\spad{1/2} for the fraction and \\spad{5/1} for the scale are used in the \\fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.") (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|Plot|) (|Fraction| (|Integer|)) (|Fraction| (|Integer|))) "\\spad{clip(p,{}frac,{}sc)} performs two-dimensional clipping on a plot,{} \\spad{p},{} from the domain \\spadtype{Plot} for the graph of one variable \\spad{y = f(x)}; the fraction parameter is specified by \\spad{frac} and the scale parameter is specified by \\spad{sc} for use in the \\spadfun{clip} function.") (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|Plot|)) "\\spad{clip(p)} performs two-dimensional clipping on a plot,{} \\spad{p},{} from the domain \\spadtype{Plot} for the graph of one variable,{} \\spad{y = f(x)}; the default parameters \\spad{1/4} for the fraction and \\spad{5/1} for the scale are used in the \\spadfun{clip} function."))) 
NIL 
NIL 
(-155 UP |Par|) 
((|constructor| (NIL "This package provides functions complexZeros for finding the complex zeros of univariate polynomials with complex rational number coefficients. The results are to any user specified precision and are returned as either complex rational number or complex floating point numbers depending on the type of the second argument which specifies the precision.")) (|complexZeros| (((|List| (|Complex| |#2|)) |#1| |#2|) "\\spad{complexZeros(poly,{} eps)} finds the complex zeros of the univariate polynomial \\spad{poly} to precision eps with solutions returned as complex floats or rationals depending on the type of eps."))) 
NIL 
NIL 
(-156) 
((|constructor| (NIL "Color() specifies a domain of 27 colors provided in the \\Language{} system (the colors mix additively).")) (|color| (($ (|Integer|)) "\\spad{color(i)} returns a color of the indicated hue \\spad{i}.")) (|numberOfHues| (((|PositiveInteger|)) "\\spad{numberOfHues()} returns the number of total hues,{} set in totalHues.")) (|hue| (((|Integer|) $) "\\spad{hue(c)} returns the hue index of the indicated color \\spad{c}.")) (|blue| (($) "\\spad{blue()} returns the position of the blue hue from total hues.")) (|green| (($) "\\spad{green()} returns the position of the green hue from total hues.")) (|yellow| (($) "\\spad{yellow()} returns the position of the yellow hue from total hues.")) (|red| (($) "\\spad{red()} returns the position of the red hue from total hues.")) (+ (($ $ $) "\\spad{c1 + c2} additively mixes the two colors \\spad{c1} and \\spad{c2}.")) (* (($ (|DoubleFloat|) $) "\\spad{s * c},{} returns the color \\spad{c},{} whose weighted shade has been scaled by \\spad{s}.") (($ (|PositiveInteger|) $) "\\spad{s * c},{} returns the color \\spad{c},{} whose weighted shade has been scaled by \\spad{s}."))) 
NIL 
NIL 
(-157 R -2262) 
((|constructor| (NIL "Provides combinatorial functions over an integral domain.")) (|ipow| ((|#2| (|List| |#2|)) "\\spad{ipow(l)} should be local but conditional.")) (|iidprod| ((|#2| (|List| |#2|)) "\\spad{iidprod(l)} should be local but conditional.")) (|iidsum| ((|#2| (|List| |#2|)) "\\spad{iidsum(l)} should be local but conditional.")) (|iipow| ((|#2| (|List| |#2|)) "\\spad{iipow(l)} should be local but conditional.")) (|iiperm| ((|#2| (|List| |#2|)) "\\spad{iiperm(l)} should be local but conditional.")) (|iibinom| ((|#2| (|List| |#2|)) "\\spad{iibinom(l)} should be local but conditional.")) (|iifact| ((|#2| |#2|) "\\spad{iifact(x)} should be local but conditional.")) (|product| ((|#2| |#2| (|SegmentBinding| |#2|)) "\\spad{product(f(n),{} n = a..b)} returns \\spad{f}(a) * ... * \\spad{f}(\\spad{b}) as a formal product.") ((|#2| |#2| (|Symbol|)) "\\spad{product(f(n),{} n)} returns the formal product \\spad{P}(\\spad{n}) which verifies \\spad{P}(\\spad{n+1})\\spad{/P}(\\spad{n}) = \\spad{f}(\\spad{n}).")) (|summation| ((|#2| |#2| (|SegmentBinding| |#2|)) "\\spad{summation(f(n),{} n = a..b)} returns \\spad{f}(a) + ... + \\spad{f}(\\spad{b}) as a formal sum.") ((|#2| |#2| (|Symbol|)) "\\spad{summation(f(n),{} n)} returns the formal sum \\spad{S}(\\spad{n}) which verifies \\spad{S}(\\spad{n+1}) - \\spad{S}(\\spad{n}) = \\spad{f}(\\spad{n}).")) (|factorials| ((|#2| |#2| (|Symbol|)) "\\spad{factorials(f,{} x)} rewrites the permutations and binomials in \\spad{f} involving \\spad{x} in terms of factorials.") ((|#2| |#2|) "\\spad{factorials(f)} rewrites the permutations and binomials in \\spad{f} in terms of factorials.")) (|factorial| ((|#2| |#2|) "\\spad{factorial(n)} returns the factorial of \\spad{n},{} \\spadignore{i.e.} \\spad{n!}.")) (|permutation| ((|#2| |#2| |#2|) "\\spad{permutation(n,{} r)} returns the number of permutations of \\spad{n} objects taken \\spad{r} at a time,{} \\spadignore{i.e.} \\spad{n!/}(\\spad{n}-\\spad{r})!.")) (|binomial| ((|#2| |#2| |#2|) "\\indented{1}{binomial(\\spad{n},{} \\spad{r}) returns the number of subsets of \\spad{r} objects} \\indented{1}{taken among \\spad{n} objects,{} \\spadignore{i.e.} \\spad{n!/}(\\spad{r!} * (\\spad{n}-\\spad{r})!);} \\blankline \\spad{X} [binomial(5,{}\\spad{i}) for \\spad{i} in 0..5]")) (** ((|#2| |#2| |#2|) "\\spad{a ** b} is the formal exponential a**b.")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\spad{F}; error if \\spad{op} is not a combinatorial operator.")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} is \\spad{true} if \\spad{op} is a combinatorial operator."))) 
NIL 
NIL 
(-158 I) 
((|constructor| (NIL "The \\spadtype{IntegerCombinatoricFunctions} package provides some standard functions in combinatorics.")) (|stirling2| ((|#1| |#1| |#1|) "\\spad{stirling2(n,{}m)} returns the Stirling number of the second kind denoted \\spad{SS[n,{}m]}.")) (|stirling1| ((|#1| |#1| |#1|) "\\spad{stirling1(n,{}m)} returns the Stirling number of the first kind denoted \\spad{S[n,{}m]}.")) (|permutation| ((|#1| |#1| |#1|) "\\spad{permutation(n)} returns \\spad{!P(n,{}r) = n!/(n-r)!}. This is the number of permutations of \\spad{n} objects taken \\spad{r} at a time.")) (|partition| ((|#1| |#1|) "\\spad{partition(n)} returns the number of partitions of the integer \\spad{n}. This is the number of distinct ways that \\spad{n} can be written as a sum of positive integers.")) (|multinomial| ((|#1| |#1| (|List| |#1|)) "\\spad{multinomial(n,{}[m1,{}m2,{}...,{}mk])} returns the multinomial coefficient \\spad{n!/(m1! m2! ... mk!)}.")) (|factorial| ((|#1| |#1|) "\\spad{factorial(n)} returns \\spad{n!}. this is the product of all integers between 1 and \\spad{n} (inclusive). Note that \\spad{0!} is defined to be 1.")) (|binomial| ((|#1| |#1| |#1|) "\\indented{1}{\\spad{binomial(n,{}r)} returns the binomial coefficient} \\indented{1}{\\spad{C(n,{}r) = n!/(r! (n-r)!)},{} where \\spad{n >= r >= 0}.} \\indented{1}{This is the number of combinations of \\spad{n} objects taken \\spad{r} at a time.} \\blankline \\spad{X} [binomial(5,{}\\spad{i}) for \\spad{i} in 0..5]"))) 
NIL 
NIL 
(-159) 
((|constructor| (NIL "CombinatorialOpsCategory is the category obtaining by adjoining summations and products to the usual combinatorial operations.")) (|product| (($ $ (|SegmentBinding| $)) "\\spad{product(f(n),{} n = a..b)} returns \\spad{f}(a) * ... * \\spad{f}(\\spad{b}) as a formal product.") (($ $ (|Symbol|)) "\\spad{product(f(n),{} n)} returns the formal product \\spad{P}(\\spad{n}) which verifies \\spad{P}(\\spad{n+1})\\spad{/P}(\\spad{n}) = \\spad{f}(\\spad{n}).")) (|summation| (($ $ (|SegmentBinding| $)) "\\spad{summation(f(n),{} n = a..b)} returns \\spad{f}(a) + ... + \\spad{f}(\\spad{b}) as a formal sum.") (($ $ (|Symbol|)) "\\spad{summation(f(n),{} n)} returns the formal sum \\spad{S}(\\spad{n}) which verifies \\spad{S}(\\spad{n+1}) - \\spad{S}(\\spad{n}) = \\spad{f}(\\spad{n}).")) (|factorials| (($ $ (|Symbol|)) "\\spad{factorials(f,{} x)} rewrites the permutations and binomials in \\spad{f} involving \\spad{x} in terms of factorials.") (($ $) "\\spad{factorials(f)} rewrites the permutations and binomials in \\spad{f} in terms of factorials."))) 
NIL 
NIL 
(-160) 
((|constructor| (NIL "A type for basic commutators")) (|mkcomm| (($ $ $) "\\spad{mkcomm(i,{}j)} is not documented") (($ (|Integer|)) "\\spad{mkcomm(i)} is not documented"))) 
NIL 
NIL 
(-161) 
((|constructor| (NIL "This package exports the elementary operators,{} with some semantics already attached to them. The semantics that is attached here is not dependent on the set in which the operators will be applied.")) (|operator| (((|BasicOperator|) (|Symbol|)) "\\spad{operator(s)} returns an operator with name \\spad{s},{} with the appropriate semantics if \\spad{s} is known. If \\spad{s} is not known,{} the result has no semantics."))) 
NIL 
NIL 
(-162 R UP UPUP) 
((|constructor| (NIL "A package for swapping the order of two variables in a tower of two UnivariatePolynomialCategory extensions.")) (|swap| ((|#3| |#3|) "\\spad{swap(p(x,{}y))} returns \\spad{p}(\\spad{y},{}\\spad{x})."))) 
NIL 
NIL 
(-163 S R) 
((|constructor| (NIL "This category represents the extension of a ring by a square root of \\spad{-1}.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(x)} returns \\spad{x} as a rational number,{} or \"failed\" if \\spad{x} is not a rational number.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(x)} returns \\spad{x} as a rational number. Error: if \\spad{x} is not a rational number.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(x)} tests if \\spad{x} is a rational number.")) (|polarCoordinates| (((|Record| (|:| |r| |#2|) (|:| |phi| |#2|)) $) "\\spad{polarCoordinates(x)} returns (\\spad{r},{} phi) such that \\spad{x} = \\spad{r} * exp(\\%\\spad{i} * phi).")) (|argument| ((|#2| $) "\\spad{argument(x)} returns the angle made by (0,{}1) and (0,{}\\spad{x}).")) (|abs| (($ $) "\\spad{abs(x)} returns the absolute value of \\spad{x} = sqrt(norm(\\spad{x})).")) (|exquo| (((|Union| $ "failed") $ |#2|) "\\spad{exquo(x,{} r)} returns the exact quotient of \\spad{x} by \\spad{r},{} or \"failed\" if \\spad{r} does not divide \\spad{x} exactly.")) (|norm| ((|#2| $) "\\spad{norm(x)} returns \\spad{x} * conjugate(\\spad{x})")) (|real| ((|#2| $) "\\spad{real(x)} returns real part of \\spad{x}.")) (|imag| ((|#2| $) "\\spad{imag(x)} returns imaginary part of \\spad{x}.")) (|conjugate| (($ $) "\\spad{conjugate(x + \\%i y)} returns \\spad{x} - \\%\\spad{i} \\spad{y}.")) (|imaginary| (($) "\\spad{imaginary()} = sqrt(\\spad{-1}) = \\%\\spad{i}.")) (|complex| (($ |#2| |#2|) "\\spad{complex(x,{}y)} constructs \\spad{x} + \\%i*y.") ((|attribute|) "indicates that \\% has sqrt(\\spad{-1})"))) 
NIL 
((|HasCategory| |#2| (QUOTE (-896))) (|HasCategory| |#2| (QUOTE (-542))) (|HasCategory| |#2| (QUOTE (-994))) (|HasCategory| |#2| (QUOTE (-1173))) (|HasCategory| |#2| (QUOTE (-1048))) (|HasCategory| |#2| (QUOTE (-1013))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| |#2| (QUOTE (-359))) (|HasAttribute| |#2| (QUOTE -4501)) (|HasAttribute| |#2| (QUOTE -4504)) (|HasCategory| |#2| (QUOTE (-296))) (|HasCategory| |#2| (QUOTE (-550))) (|HasCategory| |#2| (QUOTE (-834)))) 
(-164 R) 
((|constructor| (NIL "This category represents the extension of a ring by a square root of \\spad{-1}.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(x)} returns \\spad{x} as a rational number,{} or \"failed\" if \\spad{x} is not a rational number.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(x)} returns \\spad{x} as a rational number. Error: if \\spad{x} is not a rational number.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(x)} tests if \\spad{x} is a rational number.")) (|polarCoordinates| (((|Record| (|:| |r| |#1|) (|:| |phi| |#1|)) $) "\\spad{polarCoordinates(x)} returns (\\spad{r},{} phi) such that \\spad{x} = \\spad{r} * exp(\\%\\spad{i} * phi).")) (|argument| ((|#1| $) "\\spad{argument(x)} returns the angle made by (0,{}1) and (0,{}\\spad{x}).")) (|abs| (($ $) "\\spad{abs(x)} returns the absolute value of \\spad{x} = sqrt(norm(\\spad{x})).")) (|exquo| (((|Union| $ "failed") $ |#1|) "\\spad{exquo(x,{} r)} returns the exact quotient of \\spad{x} by \\spad{r},{} or \"failed\" if \\spad{r} does not divide \\spad{x} exactly.")) (|norm| ((|#1| $) "\\spad{norm(x)} returns \\spad{x} * conjugate(\\spad{x})")) (|real| ((|#1| $) "\\spad{real(x)} returns real part of \\spad{x}.")) (|imag| ((|#1| $) "\\spad{imag(x)} returns imaginary part of \\spad{x}.")) (|conjugate| (($ $) "\\spad{conjugate(x + \\%i y)} returns \\spad{x} - \\%\\spad{i} \\spad{y}.")) (|imaginary| (($) "\\spad{imaginary()} = sqrt(\\spad{-1}) = \\%\\spad{i}.")) (|complex| (($ |#1| |#1|) "\\spad{complex(x,{}y)} constructs \\spad{x} + \\%i*y.") ((|attribute|) "indicates that \\% has sqrt(\\spad{-1})"))) 
((-4498 -3322 (|has| |#1| (-550)) (-12 (|has| |#1| (-296)) (|has| |#1| (-896)))) (-4503 |has| |#1| (-359)) (-4497 |has| |#1| (-359)) (-4501 |has| |#1| (-6 -4501)) (-4504 |has| |#1| (-6 -4504)) (-3582 . T) (-3576 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-165 RR PR) 
((|constructor| (NIL "This package has no description")) (|factor| (((|Factored| |#2|) |#2|) "\\spad{factor(p)} factorizes the polynomial \\spad{p} with complex coefficients."))) 
NIL 
NIL 
(-166 R S) 
((|constructor| (NIL "This package extends maps from underlying rings to maps between complex over those rings.")) (|map| (((|Complex| |#2|) (|Mapping| |#2| |#1|) (|Complex| |#1|)) "\\spad{map(f,{}u)} maps \\spad{f} onto real and imaginary parts of \\spad{u}."))) 
NIL 
NIL 
(-167 R) 
((|constructor| (NIL "\\spadtype{Complex(R)} creates the domain of elements of the form \\spad{a + b * i} where \\spad{a} and \\spad{b} come from the ring \\spad{R},{} and \\spad{i} is a new element such that \\spad{i**2 = -1}."))) 
((-4498 -3322 (|has| |#1| (-550)) (-12 (|has| |#1| (-296)) (|has| |#1| (-896)))) (-4503 |has| |#1| (-359)) (-4497 |has| |#1| (-359)) (-4501 |has| |#1| (-6 -4501)) (-4504 |has| |#1| (-6 -4504)) (-3582 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-344))) (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-359))) (-3322 (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-344)))) (|HasCategory| |#1| (QUOTE (-364))) (|HasCategory| |#1| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#1| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#1| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#1| (LIST (QUOTE -1029) (QUOTE (-560)))) (-3322 (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-1173))) (-12 (|HasCategory| |#1| (QUOTE (-994))) (|HasCategory| |#1| (QUOTE (-1173)))) (|HasCategory| |#1| (QUOTE (-1013))) (|HasCategory| |#1| (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| |#1| (QUOTE (-834))) (|HasCategory| |#1| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-375))))) (|HasCategory| |#1| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-560))))) (|HasCategory| |#1| (LIST (QUOTE -873) (QUOTE (-560)))) (|HasCategory| |#1| (LIST (QUOTE -873) (QUOTE (-375)))) (|HasCategory| |#1| (LIST (QUOTE -515) (QUOTE (-1153)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -276) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-815))) (|HasCategory| |#1| (QUOTE (-1048))) (-12 (|HasCategory| |#1| (QUOTE (-1048))) (|HasCategory| |#1| (QUOTE (-1173)))) (|HasCategory| |#1| (QUOTE (-542))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#1| (QUOTE (-359)))) (|HasCategory| |#1| (QUOTE (-296))) (-3322 (|HasCategory| |#1| (QUOTE (-296))) (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-344))) (|HasCategory| |#1| (QUOTE (-550)))) (-3322 (|HasCategory| |#1| (QUOTE (-296))) (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-344)))) (|HasCategory| |#1| (QUOTE (-896))) (|HasCategory| |#1| (QUOTE (-221))) (-3322 (-12 (|HasCategory| |#1| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-375))))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-560))))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -276) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -515) (QUOTE (-1153)) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -873) (QUOTE (-375)))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -873) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-344)))) (|HasCategory| |#1| (QUOTE (-221))) (-12 (|HasCategory| |#1| (QUOTE (-296))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-344)))) (-12 (|HasCategory| |#1| (QUOTE (-344))) (|HasCategory| |#1| (QUOTE (-364)))) (-12 (|HasCategory| |#1| (QUOTE (-344))) (|HasCategory| |#1| (QUOTE (-550)))) (-12 (|HasCategory| |#1| (QUOTE (-344))) (|HasCategory| |#1| (QUOTE (-815)))) (-12 (|HasCategory| |#1| (QUOTE (-344))) (|HasCategory| |#1| (QUOTE (-834)))) (-12 (|HasCategory| |#1| (QUOTE (-344))) (|HasCategory| |#1| (QUOTE (-1013)))) (-12 (|HasCategory| |#1| (QUOTE (-344))) (|HasCategory| |#1| (QUOTE (-1173))))) (-12 (|HasCategory| |#1| (QUOTE (-296))) (|HasCategory| |#1| (QUOTE (-896)))) (-3322 (-12 (|HasCategory| |#1| (QUOTE (-296))) (|HasCategory| |#1| (QUOTE (-896)))) (|HasCategory| |#1| (QUOTE (-359))) (-12 (|HasCategory| |#1| (QUOTE (-344))) (|HasCategory| |#1| (QUOTE (-896))))) (-3322 (-12 (|HasCategory| |#1| (QUOTE (-296))) (|HasCategory| |#1| (QUOTE (-896)))) (-12 (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-896)))) (-12 (|HasCategory| |#1| (QUOTE (-344))) (|HasCategory| |#1| (QUOTE (-896))))) (-3322 (-12 (|HasCategory| |#1| (QUOTE (-296))) (|HasCategory| |#1| (QUOTE (-896)))) (|HasCategory| |#1| (QUOTE (-359)))) (-3322 (-12 (|HasCategory| |#1| (QUOTE (-296))) (|HasCategory| |#1| (QUOTE (-896)))) (|HasCategory| |#1| (QUOTE (-550)))) (|HasAttribute| |#1| (QUOTE -4501)) (|HasAttribute| |#1| (QUOTE -4504)) (-12 (|HasCategory| |#1| (QUOTE (-221))) (|HasCategory| |#1| (QUOTE (-359)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#1| (QUOTE (-359)))) (-3322 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-296))) (|HasCategory| |#1| (QUOTE (-896)))) (|HasCategory| |#1| (QUOTE (-146)))) (-3322 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-296))) (|HasCategory| |#1| (QUOTE (-896)))) (|HasCategory| |#1| (QUOTE (-344))))) 
(-168 R S CS) 
((|constructor| (NIL "This package supports converting complex expressions to patterns")) (|convert| (((|Pattern| |#1|) |#3|) "\\spad{convert(cs)} converts the complex expression \\spad{cs} to a pattern"))) 
NIL 
NIL 
(-169) 
((|constructor| (NIL "This domain implements some global properties of subspaces.")) (|copy| (($ $) "\\spad{copy(x)} is not documented")) (|solid| (((|Boolean|) $ (|Boolean|)) "\\spad{solid(x,{}b)} is not documented")) (|close| (((|Boolean|) $ (|Boolean|)) "\\spad{close(x,{}b)} is not documented")) (|solid?| (((|Boolean|) $) "\\spad{solid?(x)} is not documented")) (|closed?| (((|Boolean|) $) "\\spad{closed?(x)} is not documented")) (|new| (($) "\\spad{new()} is not documented"))) 
NIL 
NIL 
(-170) 
((|constructor| (NIL "The category of commutative rings with unity,{} \\spadignore{i.e.} rings where \\spadop{*} is commutative,{} and which have a multiplicative identity element.")) (|commutative| ((|attribute| "*") "multiplication is commutative."))) 
(((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-171 R) 
((|constructor| (NIL "\\spadtype{ContinuedFraction} implements general continued fractions. This version is not restricted to simple,{} finite fractions and uses the \\spadtype{Stream} as a representation. The arithmetic functions assume that the approximants alternate below/above the convergence point. This is enforced by ensuring the partial numerators and partial denominators are greater than 0 in the Euclidean domain view of \\spad{R} (\\spadignore{i.e.} \\spad{sizeLess?(0,{} x)}).")) (|complete| (($ $) "\\spad{complete(x)} causes all entries in \\spadvar{\\spad{x}} to be computed. Normally entries are only computed as needed. If \\spadvar{\\spad{x}} is an infinite continued fraction,{} a user-initiated interrupt is necessary to stop the computation.")) (|extend| (($ $ (|Integer|)) "\\spad{extend(x,{}n)} causes the first \\spadvar{\\spad{n}} entries in the continued fraction \\spadvar{\\spad{x}} to be computed. Normally entries are only computed as needed.")) (|denominators| (((|Stream| |#1|) $) "\\spad{denominators(x)} returns the stream of denominators of the approximants of the continued fraction \\spadvar{\\spad{x}}. If the continued fraction is finite,{} then the stream will be finite.")) (|numerators| (((|Stream| |#1|) $) "\\spad{numerators(x)} returns the stream of numerators of the approximants of the continued fraction \\spadvar{\\spad{x}}. If the continued fraction is finite,{} then the stream will be finite.")) (|convergents| (((|Stream| (|Fraction| |#1|)) $) "\\spad{convergents(x)} returns the stream of the convergents of the continued fraction \\spadvar{\\spad{x}}. If the continued fraction is finite,{} then the stream will be finite.")) (|approximants| (((|Stream| (|Fraction| |#1|)) $) "\\spad{approximants(x)} returns the stream of approximants of the continued fraction \\spadvar{\\spad{x}}. If the continued fraction is finite,{} then the stream will be infinite and periodic with period 1.")) (|reducedForm| (($ $) "\\spad{reducedForm(x)} puts the continued fraction \\spadvar{\\spad{x}} in reduced form,{} \\spadignore{i.e.} the function returns an equivalent continued fraction of the form \\spad{continuedFraction(b0,{}[1,{}1,{}1,{}...],{}[b1,{}b2,{}b3,{}...])}.")) (|wholePart| ((|#1| $) "\\spad{wholePart(x)} extracts the whole part of \\spadvar{\\spad{x}}. That is,{} if \\spad{x = continuedFraction(b0,{} [a1,{}a2,{}a3,{}...],{} [b1,{}b2,{}b3,{}...])},{} then \\spad{wholePart(x) = b0}.")) (|partialQuotients| (((|Stream| |#1|) $) "\\spad{partialQuotients(x)} extracts the partial quotients in \\spadvar{\\spad{x}}. That is,{} if \\spad{x = continuedFraction(b0,{} [a1,{}a2,{}a3,{}...],{} [b1,{}b2,{}b3,{}...])},{} then \\spad{partialQuotients(x) = [b0,{}b1,{}b2,{}b3,{}...]}.")) (|partialDenominators| (((|Stream| |#1|) $) "\\spad{partialDenominators(x)} extracts the denominators in \\spadvar{\\spad{x}}. That is,{} if \\spad{x = continuedFraction(b0,{} [a1,{}a2,{}a3,{}...],{} [b1,{}b2,{}b3,{}...])},{} then \\spad{partialDenominators(x) = [b1,{}b2,{}b3,{}...]}.")) (|partialNumerators| (((|Stream| |#1|) $) "\\spad{partialNumerators(x)} extracts the numerators in \\spadvar{\\spad{x}}. That is,{} if \\spad{x = continuedFraction(b0,{} [a1,{}a2,{}a3,{}...],{} [b1,{}b2,{}b3,{}...])},{} then \\spad{partialNumerators(x) = [a1,{}a2,{}a3,{}...]}.")) (|reducedContinuedFraction| (($ |#1| (|Stream| |#1|)) "\\spad{reducedContinuedFraction(b0,{}b)} constructs a continued fraction in the following way: if \\spad{b = [b1,{}b2,{}...]} then the result is the continued fraction \\spad{b0 + 1/(b1 + 1/(b2 + ...))}. That is,{} the result is the same as \\spad{continuedFraction(b0,{}[1,{}1,{}1,{}...],{}[b1,{}b2,{}b3,{}...])}.")) (|continuedFraction| (($ |#1| (|Stream| |#1|) (|Stream| |#1|)) "\\spad{continuedFraction(b0,{}a,{}b)} constructs a continued fraction in the following way: if \\spad{a = [a1,{}a2,{}...]} and \\spad{b = [b1,{}b2,{}...]} then the result is the continued fraction \\spad{b0 + a1/(b1 + a2/(b2 + ...))}.") (($ (|Fraction| |#1|)) "\\spad{continuedFraction(r)} converts the fraction \\spadvar{\\spad{r}} with components of type \\spad{R} to a continued fraction over \\spad{R}."))) 
(((-4507 "*") . T) (-4498 . T) (-4503 . T) (-4497 . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-172 R) 
((|constructor| (NIL "CoordinateSystems provides coordinate transformation functions for plotting. Functions in this package return conversion functions which take points expressed in other coordinate systems and return points with the corresponding Cartesian coordinates.")) (|conical| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1| |#1|) "\\spad{conical(a,{}b)} transforms from conical coordinates to Cartesian coordinates: \\spad{conical(a,{}b)} is a function which will map the point \\spad{(lambda,{}mu,{}nu)} to \\spad{x = lambda*mu*nu/(a*b)},{} \\spad{y = lambda/a*sqrt((mu**2-a**2)*(nu**2-a**2)/(a**2-b**2))},{} \\spad{z = lambda/b*sqrt((mu**2-b**2)*(nu**2-b**2)/(b**2-a**2))}.")) (|toroidal| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1|) "\\spad{toroidal(a)} transforms from toroidal coordinates to Cartesian coordinates: \\spad{toroidal(a)} is a function which will map the point \\spad{(u,{}v,{}phi)} to \\spad{x = a*sinh(v)*cos(phi)/(cosh(v)-cos(u))},{} \\spad{y = a*sinh(v)*sin(phi)/(cosh(v)-cos(u))},{} \\spad{z = a*sin(u)/(cosh(v)-cos(u))}.")) (|bipolarCylindrical| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1|) "\\spad{bipolarCylindrical(a)} transforms from bipolar cylindrical coordinates to Cartesian coordinates: \\spad{bipolarCylindrical(a)} is a function which will map the point \\spad{(u,{}v,{}z)} to \\spad{x = a*sinh(v)/(cosh(v)-cos(u))},{} \\spad{y = a*sin(u)/(cosh(v)-cos(u))},{} \\spad{z}.")) (|bipolar| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1|) "\\spad{bipolar(a)} transforms from bipolar coordinates to Cartesian coordinates: \\spad{bipolar(a)} is a function which will map the point \\spad{(u,{}v)} to \\spad{x = a*sinh(v)/(cosh(v)-cos(u))},{} \\spad{y = a*sin(u)/(cosh(v)-cos(u))}.")) (|oblateSpheroidal| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1|) "\\spad{oblateSpheroidal(a)} transforms from oblate spheroidal coordinates to Cartesian coordinates: \\spad{oblateSpheroidal(a)} is a function which will map the point \\spad{(\\spad{xi},{}eta,{}phi)} to \\spad{x = a*sinh(\\spad{xi})*sin(eta)*cos(phi)},{} \\spad{y = a*sinh(\\spad{xi})*sin(eta)*sin(phi)},{} \\spad{z = a*cosh(\\spad{xi})*cos(eta)}.")) (|prolateSpheroidal| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1|) "\\spad{prolateSpheroidal(a)} transforms from prolate spheroidal coordinates to Cartesian coordinates: \\spad{prolateSpheroidal(a)} is a function which will map the point \\spad{(\\spad{xi},{}eta,{}phi)} to \\spad{x = a*sinh(\\spad{xi})*sin(eta)*cos(phi)},{} \\spad{y = a*sinh(\\spad{xi})*sin(eta)*sin(phi)},{} \\spad{z = a*cosh(\\spad{xi})*cos(eta)}.")) (|ellipticCylindrical| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1|) "\\spad{ellipticCylindrical(a)} transforms from elliptic cylindrical coordinates to Cartesian coordinates: \\spad{ellipticCylindrical(a)} is a function which will map the point \\spad{(u,{}v,{}z)} to \\spad{x = a*cosh(u)*cos(v)},{} \\spad{y = a*sinh(u)*sin(v)},{} \\spad{z}.")) (|elliptic| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1|) "\\spad{elliptic(a)} transforms from elliptic coordinates to Cartesian coordinates: \\spad{elliptic(a)} is a function which will map the point \\spad{(u,{}v)} to \\spad{x = a*cosh(u)*cos(v)},{} \\spad{y = a*sinh(u)*sin(v)}.")) (|paraboloidal| (((|Point| |#1|) (|Point| |#1|)) "\\spad{paraboloidal(pt)} transforms \\spad{pt} from paraboloidal coordinates to Cartesian coordinates: the function produced will map the point \\spad{(u,{}v,{}phi)} to \\spad{x = u*v*cos(phi)},{} \\spad{y = u*v*sin(phi)},{} \\spad{z = 1/2 * (u**2 - v**2)}.")) (|parabolicCylindrical| (((|Point| |#1|) (|Point| |#1|)) "\\spad{parabolicCylindrical(pt)} transforms \\spad{pt} from parabolic cylindrical coordinates to Cartesian coordinates: the function produced will map the point \\spad{(u,{}v,{}z)} to \\spad{x = 1/2*(u**2 - v**2)},{} \\spad{y = u*v},{} \\spad{z}.")) (|parabolic| (((|Point| |#1|) (|Point| |#1|)) "\\spad{parabolic(pt)} transforms \\spad{pt} from parabolic coordinates to Cartesian coordinates: the function produced will map the point \\spad{(u,{}v)} to \\spad{x = 1/2*(u**2 - v**2)},{} \\spad{y = u*v}.")) (|spherical| (((|Point| |#1|) (|Point| |#1|)) "\\spad{spherical(pt)} transforms \\spad{pt} from spherical coordinates to Cartesian coordinates: the function produced will map the point \\spad{(r,{}theta,{}phi)} to \\spad{x = r*sin(phi)*cos(theta)},{} \\spad{y = r*sin(phi)*sin(theta)},{} \\spad{z = r*cos(phi)}.")) (|cylindrical| (((|Point| |#1|) (|Point| |#1|)) "\\spad{cylindrical(pt)} transforms \\spad{pt} from polar coordinates to Cartesian coordinates: the function produced will map the point \\spad{(r,{}theta,{}z)} to \\spad{x = r * cos(theta)},{} \\spad{y = r * sin(theta)},{} \\spad{z}.")) (|polar| (((|Point| |#1|) (|Point| |#1|)) "\\spad{polar(pt)} transforms \\spad{pt} from polar coordinates to Cartesian coordinates: the function produced will map the point \\spad{(r,{}theta)} to \\spad{x = r * cos(theta)} ,{} \\spad{y = r * sin(theta)}.")) (|cartesian| (((|Point| |#1|) (|Point| |#1|)) "\\spad{cartesian(pt)} returns the Cartesian coordinates of point \\spad{pt}."))) 
NIL 
NIL 
(-173 R |PolR| E) 
((|constructor| (NIL "This package implements characteristicPolynomials for monogenic algebras using resultants")) (|characteristicPolynomial| ((|#2| |#3|) "\\spad{characteristicPolynomial(e)} returns the characteristic polynomial of \\spad{e} using resultants"))) 
NIL 
NIL 
(-174 R S CS) 
((|constructor| (NIL "This package supports matching patterns involving complex expressions")) (|patternMatch| (((|PatternMatchResult| |#1| |#3|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|)) "\\spad{patternMatch(cexpr,{} pat,{} res)} matches the pattern \\spad{pat} to the complex expression \\spad{cexpr}. res contains the variables of \\spad{pat} which are already matched and their matches."))) 
NIL 
((|HasCategory| (-945 |#2|) (LIST (QUOTE -873) (|devaluate| |#1|)))) 
(-175 R) 
((|constructor| (NIL "This package has no documentation")) (|multiEuclideanTree| (((|List| |#1|) (|List| |#1|) |#1|) "\\spad{multiEuclideanTree(l,{}r)} \\undocumented{}")) (|chineseRemainder| (((|List| |#1|) (|List| (|List| |#1|)) (|List| |#1|)) "\\spad{chineseRemainder(llv,{}lm)} returns a list of values,{} each of which corresponds to the Chinese remainder of the associated element of \\axiom{\\spad{llv}} and axiom{\\spad{lm}}. This is more efficient than applying chineseRemainder several times.") ((|#1| (|List| |#1|) (|List| |#1|)) "\\spad{chineseRemainder(lv,{}lm)} returns a value \\axiom{\\spad{v}} such that,{} if \\spad{x} is \\axiom{\\spad{lv}.\\spad{i}} modulo \\axiom{\\spad{lm}.\\spad{i}} for all \\axiom{\\spad{i}},{} then \\spad{x} is \\axiom{\\spad{v}} modulo \\axiom{\\spad{lm}(1)\\spad{*lm}(2)*...\\spad{*lm}(\\spad{n})}.")) (|modTree| (((|List| |#1|) |#1| (|List| |#1|)) "\\spad{modTree(r,{}l)} \\undocumented{}"))) 
NIL 
NIL 
(-176 R UP) 
((|constructor| (NIL "\\spadtype{ComplexRootFindingPackage} provides functions to find all roots of a polynomial \\spad{p} over the complex number by using Plesken\\spad{'s} idea to calculate in the polynomial ring modulo \\spad{f} and employing the Chinese Remainder Theorem. In this first version,{} the precision (see digits) is not increased when this is necessary to avoid rounding errors. Hence it is the user\\spad{'s} responsibility to increase the precision if necessary. Note also,{} if this package is called with \\spadignore{e.g.} \\spadtype{Fraction Integer},{} the precise calculations could require a lot of time. Also note that evaluating the zeros is not necessarily a good check whether the result is correct: already evaluation can cause rounding errors.")) (|startPolynomial| (((|Record| (|:| |start| |#2|) (|:| |factors| (|Factored| |#2|))) |#2|) "\\spad{startPolynomial(p)} uses the ideas of Schoenhage\\spad{'s} variant of Graeffe\\spad{'s} method to construct circles which separate roots to get a good start polynomial,{} \\spadignore{i.e.} one whose image under the Chinese Remainder Isomorphism has both entries of norm smaller and greater or equal to 1. In case the roots are found during internal calculations. The corresponding factors are in factors which are otherwise 1.")) (|setErrorBound| ((|#1| |#1|) "\\spad{setErrorBound(eps)} changes the internal error bound,{} by default being 10 \\spad{**} (\\spad{-3}) to \\spad{eps},{} if \\spad{R} is a member in the category \\spadtype{QuotientFieldCategory Integer}. The internal globalDigits is set to \\em ceiling(1/r)\\spad{**2*10} being 10**7 by default.")) (|schwerpunkt| (((|Complex| |#1|) |#2|) "\\spad{schwerpunkt(p)} determines the 'Schwerpunkt' of the roots of the polynomial \\spad{p} of degree \\spad{n},{} \\spadignore{i.e.} the center of gravity,{} which is coeffient of \\spad{x**(n-1)} divided by \\spad{n} times coefficient of \\spad{x**n}.")) (|rootRadius| ((|#1| |#2|) "\\spad{rootRadius(p)} calculates the root radius of \\spad{p} with a maximal error quotient of 1+globalEps,{} where globalEps is the internal error bound,{} which can be set by setErrorBound.") ((|#1| |#2| |#1|) "\\spad{rootRadius(p,{}errQuot)} calculates the root radius of \\spad{p} with a maximal error quotient of \\spad{errQuot}.")) (|reciprocalPolynomial| ((|#2| |#2|) "\\spad{reciprocalPolynomial(p)} calulates a polynomial which has exactly the inverses of the non-zero roots of \\spad{p} as roots,{} and the same number of 0-roots.")) (|pleskenSplit| (((|Factored| |#2|) |#2| |#1|) "\\spad{pleskenSplit(poly,{} eps)} determines a start polynomial start by using \"startPolynomial then it increases the exponent \\spad{n} of start \\spad{**} \\spad{n} mod \\spad{poly} to get an approximate factor of \\spad{poly},{} in general of degree \"degree \\spad{poly} \\spad{-1\"}. Then a divisor cascade is calculated and the best splitting is chosen,{} as soon as the error is small enough.") (((|Factored| |#2|) |#2| |#1| (|Boolean|)) "\\spad{pleskenSplit(poly,{}eps,{}info)} determines a start polynomial start by using \"startPolynomial then it increases the exponent \\spad{n} of start \\spad{**} \\spad{n} mod \\spad{poly} to get an approximate factor of \\spad{poly},{} in general of degree \"degree \\spad{poly} \\spad{-1\"}. Then a divisor cascade is calculated and the best splitting is chosen,{} as soon as the error is small enough. If \\spad{info} is \\spad{true},{} then information messages are issued.")) (|norm| ((|#1| |#2|) "\\spad{norm(p)} determines sum of absolute values of coefficients Note that this function depends on abs.")) (|graeffe| ((|#2| |#2|) "\\spad{graeffe p} determines \\spad{q} such that \\spad{q(-z**2) = p(z)*p(-z)}. Note that the roots of \\spad{q} are the squares of the roots of \\spad{p}.")) (|factor| (((|Factored| |#2|) |#2|) "\\spad{factor(p)} tries to factor \\spad{p} into linear factors with error atmost globalEps,{} the internal error bound,{} which can be set by setErrorBound. An overall error bound \\spad{eps0} is determined and iterated tree-like calls to pleskenSplit are used to get the factorization.") (((|Factored| |#2|) |#2| |#1|) "\\spad{factor(p,{} eps)} tries to factor \\spad{p} into linear factors with error atmost eps. An overall error bound \\spad{eps0} is determined and iterated tree-like calls to pleskenSplit are used to get the factorization.") (((|Factored| |#2|) |#2| |#1| (|Boolean|)) "\\spad{factor(p,{} eps,{} info)} tries to factor \\spad{p} into linear factors with error atmost \\spad{eps}. An overall error bound \\spad{eps0} is determined and iterated tree-like calls to pleskenSplit are used to get the factorization. If info is \\spad{true},{} then information messages are given.")) (|divisorCascade| (((|List| (|Record| (|:| |factors| (|List| |#2|)) (|:| |error| |#1|))) |#2| |#2|) "\\spad{divisorCascade(p,{}tp)} assumes that degree of polynomial \\spad{tp} is smaller than degree of polynomial \\spad{p},{} both monic. A sequence of divisions is calculated using the remainder,{} made monic,{} as divisor for the the next division. The result contains also the error of the factorizations,{} \\spadignore{i.e.} the norm of the remainder polynomial.") (((|List| (|Record| (|:| |factors| (|List| |#2|)) (|:| |error| |#1|))) |#2| |#2| (|Boolean|)) "\\spad{divisorCascade(p,{}tp)} assumes that degree of polynomial \\spad{tp} is smaller than degree of polynomial \\spad{p},{} both monic. A sequence of divisions are calculated using the remainder,{} made monic,{} as divisor for the the next division. The result contains also the error of the factorizations,{} \\spadignore{i.e.} the norm of the remainder polynomial. If info is \\spad{true},{} then information messages are issued.")) (|complexZeros| (((|List| (|Complex| |#1|)) |#2| |#1|) "\\spad{complexZeros(p,{} eps)} tries to determine all complex zeros of the polynomial \\spad{p} with accuracy given by eps.") (((|List| (|Complex| |#1|)) |#2|) "\\spad{complexZeros(p)} tries to determine all complex zeros of the polynomial \\spad{p} with accuracy given by the package constant globalEps which you may change by setErrorBound."))) 
NIL 
NIL 
(-177 S ST) 
((|constructor| (NIL "This package provides tools for working with cyclic streams.")) (|computeCycleEntry| ((|#2| |#2| |#2|) "\\indented{1}{computeCycleEntry(\\spad{x},{}cycElt),{} where cycElt is a pointer to a} \\indented{1}{node in the cyclic part of the cyclic stream \\spad{x},{} returns a} \\indented{1}{pointer to the first node in the cycle} \\blankline \\spad{X} p:=repeating([1,{}2,{}3]) \\spad{X} q:=cons(4,{}\\spad{p}) \\spad{X} computeCycleEntry(\\spad{q},{}cycleElt(\\spad{q}))")) (|computeCycleLength| (((|NonNegativeInteger|) |#2|) "\\indented{1}{computeCycleLength(\\spad{s}) returns the length of the cycle of a} \\indented{1}{cyclic stream \\spad{t},{} where \\spad{s} is a pointer to a node in the} \\indented{1}{cyclic part of \\spad{t}.} \\blankline \\spad{X} p:=repeating([1,{}2,{}3]) \\spad{X} q:=cons(4,{}\\spad{p}) \\spad{X} computeCycleLength(cycleElt(\\spad{q}))")) (|cycleElt| (((|Union| |#2| "failed") |#2|) "\\indented{1}{cycleElt(\\spad{s}) returns a pointer to a node in the cycle if the stream} \\indented{1}{\\spad{s} is cyclic and returns \"failed\" if \\spad{s} is not cyclic} \\blankline \\spad{X} p:=repeating([1,{}2,{}3]) \\spad{X} q:=cons(4,{}\\spad{p}) \\spad{X} cycleElt \\spad{q} \\spad{X} \\spad{r:=}[1,{}2,{}3]::Stream(Integer) \\spad{X} cycleElt \\spad{r}"))) 
NIL 
NIL 
(-178 R -2262) 
((|constructor| (NIL "\\spadtype{ComplexTrigonometricManipulations} provides function that compute the real and imaginary parts of complex functions.")) (|complexForm| (((|Complex| (|Expression| |#1|)) |#2|) "\\spad{complexForm(f)} returns \\spad{[real f,{} imag f]}.")) (|trigs| ((|#2| |#2|) "\\spad{trigs(f)} rewrites all the complex logs and exponentials appearing in \\spad{f} in terms of trigonometric functions.")) (|real?| (((|Boolean|) |#2|) "\\spad{real?(f)} returns \\spad{true} if \\spad{f = real f}.")) (|imag| (((|Expression| |#1|) |#2|) "\\spad{imag(f)} returns the imaginary part of \\spad{f} where \\spad{f} is a complex function.")) (|real| (((|Expression| |#1|) |#2|) "\\spad{real(f)} returns the real part of \\spad{f} where \\spad{f} is a complex function.")) (|complexElementary| ((|#2| |#2| (|Symbol|)) "\\spad{complexElementary(f,{} x)} rewrites the kernels of \\spad{f} involving \\spad{x} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log,{} exp}.") ((|#2| |#2|) "\\spad{complexElementary(f)} rewrites \\spad{f} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log,{} exp}.")) (|complexNormalize| ((|#2| |#2| (|Symbol|)) "\\spad{complexNormalize(f,{} x)} rewrites \\spad{f} using the least possible number of complex independent kernels involving \\spad{x}.") ((|#2| |#2|) "\\spad{complexNormalize(f)} rewrites \\spad{f} using the least possible number of complex independent kernels."))) 
NIL 
NIL 
(-179 R) 
((|constructor| (NIL "CoerceVectorMatrixPackage is an unexposed,{} technical package for data conversions")) (|coerce| (((|Vector| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|Vector| (|Matrix| |#1|))) "\\spad{coerce(v)} coerces a vector \\spad{v} with entries in \\spadtype{Matrix R} as vector over \\spadtype{Matrix Fraction Polynomial R}")) (|coerceP| (((|Vector| (|Matrix| (|Polynomial| |#1|))) (|Vector| (|Matrix| |#1|))) "\\spad{coerceP(v)} coerces a vector \\spad{v} with entries in \\spadtype{Matrix R} as vector over \\spadtype{Matrix Polynomial R}"))) 
NIL 
NIL 
(-180) 
((|constructor| (NIL "Polya-Redfield enumeration by cycle indices.")) (|skewSFunction| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{skewSFunction(li1,{}li2)} is the \\spad{S}-function \\indented{1}{of the partition difference \\spad{li1 - li2}} \\indented{1}{expressed in terms of power sum symmetric functions.}")) (|SFunction| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|List| (|Integer|))) "\\spad{SFunction(\\spad{li})} is the \\spad{S}-function of the partition \\spad{\\spad{li}} \\indented{1}{expressed in terms of power sum symmetric functions.}")) (|wreath| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|SymmetricPolynomial| (|Fraction| (|Integer|))) (|SymmetricPolynomial| (|Fraction| (|Integer|)))) "\\spad{wreath(s1,{}s2)} is the cycle index of the wreath product \\indented{1}{of the two groups whose cycle indices are \\spad{s1} and} \\indented{1}{\\spad{s2}.}")) (|eval| (((|Fraction| (|Integer|)) (|SymmetricPolynomial| (|Fraction| (|Integer|)))) "\\spad{eval s} is the sum of the coefficients of a cycle index.")) (|cup| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|SymmetricPolynomial| (|Fraction| (|Integer|))) (|SymmetricPolynomial| (|Fraction| (|Integer|)))) "\\spad{cup(s1,{}s2)},{} introduced by Redfield,{} \\indented{1}{is the scalar product of two cycle indices,{} in which the} \\indented{1}{power sums are retained to produce a cycle index.}")) (|cap| (((|Fraction| (|Integer|)) (|SymmetricPolynomial| (|Fraction| (|Integer|))) (|SymmetricPolynomial| (|Fraction| (|Integer|)))) "\\spad{cap(s1,{}s2)},{} introduced by Redfield,{} \\indented{1}{is the scalar product of two cycle indices.}")) (|graphs| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|Integer|)) "\\spad{graphs n} is the cycle index of the group induced on \\indented{1}{the edges of a graph by applying the symmetric function to the} \\indented{1}{\\spad{n} nodes.}")) (|dihedral| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|Integer|)) "\\spad{dihedral n} is the cycle index of the \\indented{1}{dihedral group of degree \\spad{n}.}")) (|cyclic| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|Integer|)) "\\spad{cyclic n} is the cycle index of the \\indented{1}{cyclic group of degree \\spad{n}.}")) (|alternating| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|Integer|)) "\\spad{alternating n} is the cycle index of the \\indented{1}{alternating group of degree \\spad{n}.}")) (|elementary| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|Integer|)) "\\spad{elementary n} is the \\spad{n} th elementary symmetric \\indented{1}{function expressed in terms of power sums.}")) (|powerSum| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|Integer|)) "\\spad{powerSum n} is the \\spad{n} th power sum symmetric \\indented{1}{function.}")) (|complete| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|Integer|)) "\\spad{complete n} is the \\spad{n} th complete homogeneous \\indented{1}{symmetric function expressed in terms of power sums.} \\indented{1}{Alternatively it is the cycle index of the symmetric} \\indented{1}{group of degree \\spad{n}.}"))) 
NIL 
NIL 
(-181) 
((|constructor| (NIL "This package has no description")) (|cyclotomicFactorization| (((|Factored| (|SparseUnivariatePolynomial| (|Integer|))) (|Integer|)) "\\spad{cyclotomicFactorization(n)} \\undocumented{}")) (|cyclotomic| (((|SparseUnivariatePolynomial| (|Integer|)) (|Integer|)) "\\spad{cyclotomic(n)} \\undocumented{}")) (|cyclotomicDecomposition| (((|List| (|SparseUnivariatePolynomial| (|Integer|))) (|Integer|)) "\\spad{cyclotomicDecomposition(n)} \\undocumented{}"))) 
NIL 
NIL 
(-182) 
((|constructor| (NIL "\\axiomType{d01AgentsPackage} is a package of numerical agents to be used to investigate attributes of an input function so as to decide the \\axiomFun{measure} of an appropriate numerical integration routine. It contains functions \\axiomFun{rangeIsFinite} to test the input range and \\axiomFun{functionIsContinuousAtEndPoints} to check for continuity at the end points of the range.")) (|changeName| (((|Result|) (|Symbol|) (|Symbol|) (|Result|)) "\\spad{changeName(s,{}t,{}r)} changes the name of item \\axiom{\\spad{s}} in \\axiom{\\spad{r}} to \\axiom{\\spad{t}}.")) (|commaSeparate| (((|String|) (|List| (|String|))) "\\spad{commaSeparate(l)} produces a comma separated string from a list of strings.")) (|sdf2lst| (((|List| (|String|)) (|Stream| (|DoubleFloat|))) "\\spad{sdf2lst(ln)} coerces a Stream of \\axiomType{DoubleFloat} to \\axiomType{List String}")) (|ldf2lst| (((|List| (|String|)) (|List| (|DoubleFloat|))) "\\spad{ldf2lst(ln)} coerces a List of \\axiomType{DoubleFloat} to \\axiomType{List String}")) (|df2st| (((|String|) (|DoubleFloat|)) "\\spad{df2st(n)} coerces a \\axiomType{DoubleFloat} to \\axiomType{String}")) (|singularitiesOf| (((|Stream| (|DoubleFloat|)) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{singularitiesOf(args)} returns a list of potential singularities of the function within the given range")) (|problemPoints| (((|List| (|DoubleFloat|)) (|Expression| (|DoubleFloat|)) (|Symbol|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{problemPoints(f,{}var,{}range)} returns a list of possible problem points by looking at the zeros of the denominator of the function if it can be retracted to \\axiomType{Polynomial DoubleFloat}.")) (|functionIsOscillatory| (((|Float|) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{functionIsOscillatory(a)} tests whether the function \\spad{a.fn} has many zeros of its derivative.")) (|gethi| (((|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{gethi(x)} gets the \\axiomType{DoubleFloat} equivalent of the second endpoint of the range \\axiom{\\spad{x}}")) (|getlo| (((|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{getlo(x)} gets the \\axiomType{DoubleFloat} equivalent of the first endpoint of the range \\axiom{\\spad{x}}")) (|functionIsContinuousAtEndPoints| (((|Union| (|:| |continuous| "Continuous at the end points") (|:| |lowerSingular| "There is a singularity at the lower end point") (|:| |upperSingular| "There is a singularity at the upper end point") (|:| |bothSingular| "There are singularities at both end points") (|:| |notEvaluated| "End point continuity not yet evaluated")) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{functionIsContinuousAtEndPoints(args)} uses power series limits to check for problems at the end points of the range of \\spad{args}.")) (|rangeIsFinite| (((|Union| (|:| |finite| "The range is finite") (|:| |lowerInfinite| "The bottom of range is infinite") (|:| |upperInfinite| "The top of range is infinite") (|:| |bothInfinite| "Both top and bottom points are infinite") (|:| |notEvaluated| "Range not yet evaluated")) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{rangeIsFinite(args)} tests the endpoints of \\spad{args.range} for infinite end points."))) 
NIL 
NIL 
(-183) 
((|constructor| (NIL "\\axiomType{d01ajfAnnaType} is a domain of \\axiomType{NumericalIntegrationCategory} for the NAG routine D01AJF,{} a general numerical integration routine which can handle some singularities in the input function. The function \\axiomFun{measure} measures the usefulness of the routine D01AJF for the given problem. The function \\axiomFun{numericalIntegration} performs the integration by using \\axiomType{NagIntegrationPackage}."))) 
NIL 
NIL 
(-184) 
((|constructor| (NIL "\\axiomType{d01akfAnnaType} is a domain of \\axiomType{NumericalIntegrationCategory} for the NAG routine D01AKF,{} a numerical integration routine which is is suitable for oscillating,{} non-singular functions. The function \\axiomFun{measure} measures the usefulness of the routine D01AKF for the given problem. The function \\axiomFun{numericalIntegration} performs the integration by using \\axiomType{NagIntegrationPackage}."))) 
NIL 
NIL 
(-185) 
((|constructor| (NIL "\\axiomType{d01alfAnnaType} is a domain of \\axiomType{NumericalIntegrationCategory} for the NAG routine D01ALF,{} a general numerical integration routine which can handle a list of singularities. The function \\axiomFun{measure} measures the usefulness of the routine D01ALF for the given problem. The function \\axiomFun{numericalIntegration} performs the integration by using \\axiomType{NagIntegrationPackage}."))) 
NIL 
NIL 
(-186) 
((|constructor| (NIL "\\axiomType{d01amfAnnaType} is a domain of \\axiomType{NumericalIntegrationCategory} for the NAG routine D01AMF,{} a general numerical integration routine which can handle infinite or semi-infinite range of the input function. The function \\axiomFun{measure} measures the usefulness of the routine D01AMF for the given problem. The function \\axiomFun{numericalIntegration} performs the integration by using \\axiomType{NagIntegrationPackage}."))) 
NIL 
NIL 
(-187) 
((|constructor| (NIL "\\axiomType{d01anfAnnaType} is a domain of \\axiomType{NumericalIntegrationCategory} for the NAG routine D01ANF,{} a numerical integration routine which can handle weight functions of the form cos(\\omega \\spad{x}) or sin(\\omega \\spad{x}). The function \\axiomFun{measure} measures the usefulness of the routine D01ANF for the given problem. The function \\axiomFun{numericalIntegration} performs the integration by using \\axiomType{NagIntegrationPackage}."))) 
NIL 
NIL 
(-188) 
((|constructor| (NIL "\\axiomType{d01apfAnnaType} is a domain of \\axiomType{NumericalIntegrationCategory} for the NAG routine D01APF,{} a general numerical integration routine which can handle end point singularities of the algebraico-logarithmic form \\spad{w}(\\spad{x}) = (\\spad{x}-a)\\spad{^c} * (\\spad{b}-\\spad{x})\\spad{^d}. The function \\axiomFun{measure} measures the usefulness of the routine D01APF for the given problem. The function \\axiomFun{numericalIntegration} performs the integration by using \\axiomType{NagIntegrationPackage}."))) 
NIL 
NIL 
(-189) 
((|constructor| (NIL "\\axiomType{d01aqfAnnaType} is a domain of \\axiomType{NumericalIntegrationCategory} for the NAG routine D01AQF,{} a general numerical integration routine which can solve an integral of the form /home/bjd/Axiom/anna/hypertex/bitmaps/d01aqf.\\spad{xbm} The function \\axiomFun{measure} measures the usefulness of the routine D01AQF for the given problem. The function \\axiomFun{numericalIntegration} performs the integration by using \\axiomType{NagIntegrationPackage}."))) 
NIL 
NIL 
(-190) 
((|constructor| (NIL "\\axiomType{d01asfAnnaType} is a domain of \\axiomType{NumericalIntegrationCategory} for the NAG routine D01ASF,{} a numerical integration routine which can handle weight functions of the form cos(\\omega \\spad{x}) or sin(\\omega \\spad{x}) on an semi-infinite range. The function \\axiomFun{measure} measures the usefulness of the routine D01ASF for the given problem. The function \\axiomFun{numericalIntegration} performs the integration by using \\axiomType{NagIntegrationPackage}."))) 
NIL 
NIL 
(-191) 
((|constructor| (NIL "\\axiomType{d01fcfAnnaType} is a domain of \\axiomType{NumericalIntegrationCategory} for the NAG routine D01FCF,{} a numerical integration routine which can handle multi-dimensional quadrature over a finite region. The function \\axiomFun{measure} measures the usefulness of the routine D01GBF for the given problem. The function \\axiomFun{numericalIntegration} performs the integration by using \\axiomType{NagIntegrationPackage}."))) 
NIL 
NIL 
(-192) 
((|constructor| (NIL "\\axiomType{d01gbfAnnaType} is a domain of \\axiomType{NumericalIntegrationCategory} for the NAG routine D01GBF,{} a numerical integration routine which can handle multi-dimensional quadrature over a finite region. The function \\axiomFun{measure} measures the usefulness of the routine D01GBF for the given problem. The function \\axiomFun{numericalIntegration} performs the integration by using \\axiomType{NagIntegrationPackage}."))) 
NIL 
NIL 
(-193) 
NIL 
NIL 
NIL 
(-194) 
((|constructor| (NIL "\\axiom{d01WeightsPackage} is a package for functions used to investigate whether a function can be divided into a simpler function and a weight function. The types of weights investigated are those giving rise to end-point singularities of the algebraico-logarithmic type,{} and trigonometric weights.")) (|exprHasLogarithmicWeights| (((|Integer|) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\axiom{exprHasLogarithmicWeights} looks for logarithmic weights giving rise to singularities of the function at the end-points.")) (|exprHasAlgebraicWeight| (((|Union| (|List| (|DoubleFloat|)) "failed") (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\axiom{exprHasAlgebraicWeight} looks for algebraic weights giving rise to singularities of the function at the end-points.")) (|exprHasWeightCosWXorSinWX| (((|Union| (|Record| (|:| |op| (|BasicOperator|)) (|:| |w| (|DoubleFloat|))) "failed") (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\axiom{exprHasWeightCosWXorSinWX} looks for trigonometric weights in an expression of the form \\axiom{cos \\omega \\spad{x}} or \\axiom{sin \\omega \\spad{x}},{} returning the value of \\omega (\\notequal 1) and the operator."))) 
NIL 
NIL 
(-195) 
((|constructor| (NIL "\\axiom{d02AgentsPackage} contains a set of computational agents for use with Ordinary Differential Equation solvers.")) (|intermediateResultsIF| (((|Float|) (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{intermediateResultsIF(o)} returns a value corresponding to the required number of intermediate results required and,{} therefore,{} an indication of how much this would affect the step-length of the calculation. It returns a value in the range [0,{}1].")) (|accuracyIF| (((|Float|) (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{accuracyIF(o)} returns the intensity value of the accuracy requirements of the input ODE. A request of accuracy of 10^-6 corresponds to the neutral intensity. It returns a value in the range [0,{}1].")) (|expenseOfEvaluationIF| (((|Float|) (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{expenseOfEvaluationIF(o)} returns the intensity value of the cost of evaluating the input ODE. This is in terms of the number of ``operational units\\spad{''}. It returns a value in the range [0,{}1].\\indent{20} 400 ``operation units\\spad{''} \\spad{->} 0.75 200 ``operation units\\spad{''} \\spad{->} 0.5 83 ``operation units\\spad{''} \\spad{->} 0.25 \\indent{15} exponentiation = 4 units ,{} function calls = 10 units.")) (|systemSizeIF| (((|Float|) (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{systemSizeIF(ode)} returns the intensity value of the size of the system of ODEs. 20 equations corresponds to the neutral value. It returns a value in the range [0,{}1].")) (|stiffnessAndStabilityOfODEIF| (((|Record| (|:| |stiffnessFactor| (|Float|)) (|:| |stabilityFactor| (|Float|))) (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{stiffnessAndStabilityOfODEIF(ode)} calculates the intensity values of stiffness of a system of first-order differential equations (by evaluating the maximum difference in the real parts of the negative eigenvalues of the jacobian of the system for which \\spad{O}(10) equates to mildly stiff wheras stiffness ratios of \\spad{O}(10^6) are not uncommon) and whether the system is likely to show any oscillations (identified by the closeness to the imaginary axis of the complex eigenvalues of the jacobian). \\blankline It returns two values in the range [0,{}1].")) (|stiffnessAndStabilityFactor| (((|Record| (|:| |stiffnessFactor| (|Float|)) (|:| |stabilityFactor| (|Float|))) (|Matrix| (|Expression| (|DoubleFloat|)))) "\\spad{stiffnessAndStabilityFactor(me)} calculates the stability and stiffness factor of a system of first-order differential equations (by evaluating the maximum difference in the real parts of the negative eigenvalues of the jacobian of the system for which \\spad{O}(10) equates to mildly stiff wheras stiffness ratios of \\spad{O}(10^6) are not uncommon) and whether the system is likely to show any oscillations (identified by the closeness to the imaginary axis of the complex eigenvalues of the jacobian).")) (|eval| (((|Matrix| (|Expression| (|DoubleFloat|))) (|Matrix| (|Expression| (|DoubleFloat|))) (|List| (|Symbol|)) (|Vector| (|Expression| (|DoubleFloat|)))) "\\spad{eval(mat,{}symbols,{}values)} evaluates a multivariable matrix at given \\spad{values} for each of a list of variables")) (|jacobian| (((|Matrix| (|Expression| (|DoubleFloat|))) (|Vector| (|Expression| (|DoubleFloat|))) (|List| (|Symbol|))) "\\spad{jacobian(v,{}w)} is a local function to make a jacobian matrix")) (|sparsityIF| (((|Float|) (|Matrix| (|Expression| (|DoubleFloat|)))) "\\spad{sparsityIF(m)} calculates the sparsity of a jacobian matrix")) (|combineFeatureCompatibility| (((|Float|) (|Float|) (|List| (|Float|))) "\\spad{combineFeatureCompatibility(C1,{}L)} is for interacting attributes") (((|Float|) (|Float|) (|Float|)) "\\spad{combineFeatureCompatibility(C1,{}C2)} is for interacting attributes"))) 
NIL 
NIL 
(-196) 
((|constructor| (NIL "\\axiomType{d02bbfAnnaType} is a domain of \\axiomType{OrdinaryDifferentialEquationsInitialValueProblemSolverCategory} for the NAG routine D02BBF,{} a ODE routine which uses an Runge-Kutta method to solve a system of differential equations. The function \\axiomFun{measure} measures the usefulness of the routine D02BBF for the given problem. The function \\axiomFun{ODESolve} performs the integration by using \\axiomType{NagOrdinaryDifferentialEquationsPackage}."))) 
NIL 
NIL 
(-197) 
((|constructor| (NIL "\\axiomType{d02bhfAnnaType} is a domain of \\axiomType{OrdinaryDifferentialEquationsInitialValueProblemSolverCategory} for the NAG routine D02BHF,{} a ODE routine which uses an Runge-Kutta method to solve a system of differential equations. The function \\axiomFun{measure} measures the usefulness of the routine D02BHF for the given problem. The function \\axiomFun{ODESolve} performs the integration by using \\axiomType{NagOrdinaryDifferentialEquationsPackage}."))) 
NIL 
NIL 
(-198) 
((|constructor| (NIL "\\axiomType{d02cjfAnnaType} is a domain of \\axiomType{OrdinaryDifferentialEquationsInitialValueProblemSolverCategory} for the NAG routine D02CJF,{} a ODE routine which uses an Adams-Moulton-Bashworth method to solve a system of differential equations. The function \\axiomFun{measure} measures the usefulness of the routine D02CJF for the given problem. The function \\axiomFun{ODESolve} performs the integration by using \\axiomType{NagOrdinaryDifferentialEquationsPackage}."))) 
NIL 
NIL 
(-199) 
((|constructor| (NIL "\\axiomType{d02ejfAnnaType} is a domain of \\axiomType{OrdinaryDifferentialEquationsInitialValueProblemSolverCategory} for the NAG routine D02EJF,{} a ODE routine which uses a backward differentiation formulae method to handle a stiff system of differential equations. The function \\axiomFun{measure} measures the usefulness of the routine D02EJF for the given problem. The function \\axiomFun{ODESolve} performs the integration by using \\axiomType{NagOrdinaryDifferentialEquationsPackage}."))) 
NIL 
NIL 
(-200) 
((|constructor| (NIL "\\axiom{d03AgentsPackage} contains a set of computational agents for use with Partial Differential Equation solvers.")) (|elliptic?| (((|Boolean|) (|Record| (|:| |pde| (|List| (|Expression| (|DoubleFloat|)))) (|:| |constraints| (|List| (|Record| (|:| |start| (|DoubleFloat|)) (|:| |finish| (|DoubleFloat|)) (|:| |grid| (|NonNegativeInteger|)) (|:| |boundaryType| (|Integer|)) (|:| |dStart| (|Matrix| (|DoubleFloat|))) (|:| |dFinish| (|Matrix| (|DoubleFloat|)))))) (|:| |f| (|List| (|List| (|Expression| (|DoubleFloat|))))) (|:| |st| (|String|)) (|:| |tol| (|DoubleFloat|)))) "\\spad{elliptic?(r)} \\undocumented{}")) (|central?| (((|Boolean|) (|DoubleFloat|) (|DoubleFloat|) (|List| (|Expression| (|DoubleFloat|)))) "\\spad{central?(f,{}g,{}l)} \\undocumented{}")) (|subscriptedVariables| (((|Expression| (|DoubleFloat|)) (|Expression| (|DoubleFloat|))) "\\spad{subscriptedVariables(e)} \\undocumented{}")) (|varList| (((|List| (|Symbol|)) (|Symbol|) (|NonNegativeInteger|)) "\\spad{varList(s,{}n)} \\undocumented{}"))) 
NIL 
NIL 
(-201) 
((|constructor| (NIL "\\axiomType{d03eefAnnaType} is a domain of \\axiomType{PartialDifferentialEquationsSolverCategory} for the NAG routines D03EEF/D03EDF."))) 
NIL 
NIL 
(-202) 
((|constructor| (NIL "\\axiomType{d03fafAnnaType} is a domain of \\axiomType{PartialDifferentialEquationsSolverCategory} for the NAG routine D03FAF."))) 
NIL 
NIL 
(-203 S) 
((|constructor| (NIL "This domain implements a simple view of a database whose fields are indexed by symbols")) (|coerce| (($ (|List| |#1|)) "\\spad{coerce(l)} makes a database out of a list")) (- (($ $ $) "\\spad{db1-db2} returns the difference of databases \\spad{db1} and \\spad{db2} \\spadignore{i.e.} consisting of elements in \\spad{db1} but not in \\spad{db2}")) (+ (($ $ $) "\\spad{db1+db2} returns the merge of databases \\spad{db1} and \\spad{db2}")) (|fullDisplay| (((|Void|) $ (|PositiveInteger|) (|PositiveInteger|)) "\\spad{fullDisplay(db,{}start,{}end )} prints full details of entries in the range \\axiom{\\spad{start}..end} in \\axiom{\\spad{db}}.") (((|Void|) $) "\\spad{fullDisplay(db)} prints full details of each entry in \\axiom{\\spad{db}}.") (((|Void|) $) "\\spad{fullDisplay(x)} displays \\spad{x} in detail")) (|display| (((|Void|) $) "\\spad{display(db)} prints a summary line for each entry in \\axiom{\\spad{db}}.") (((|Void|) $) "\\spad{display(x)} displays \\spad{x} in some form")) (|elt| (((|DataList| (|String|)) $ (|Symbol|)) "\\spad{elt(db,{}s)} returns the \\axiom{\\spad{s}} field of each element of \\axiom{\\spad{db}}.") (($ $ (|QueryEquation|)) "\\spad{elt(db,{}q)} returns all elements of \\axiom{\\spad{db}} which satisfy \\axiom{\\spad{q}}.") (((|String|) $ (|Symbol|)) "\\spad{elt(x,{}s)} returns an element of \\spad{x} indexed by \\spad{s}"))) 
NIL 
NIL 
(-204 -2262 UP UPUP R) 
((|constructor| (NIL "This package provides functions for computing the residues of a function on an algebraic curve.")) (|doubleResultant| ((|#2| |#4| (|Mapping| |#2| |#2|)) "\\spad{doubleResultant(f,{} ')} returns \\spad{p}(\\spad{x}) whose roots are rational multiples of the residues of \\spad{f} at all its finite poles. Argument ' is the derivation to use."))) 
NIL 
NIL 
(-205 -2262 FP) 
((|constructor| (NIL "Package for the factorization of a univariate polynomial with coefficients in a finite field. The algorithm used is the \"distinct degree\" algorithm of Cantor-Zassenhaus,{} modified to use trace instead of the norm and a table for computing Frobenius as suggested by Naudin and Quitte .")) (|irreducible?| (((|Boolean|) |#2|) "\\spad{irreducible?(p)} tests whether the polynomial \\spad{p} is irreducible.")) (|tracePowMod| ((|#2| |#2| (|NonNegativeInteger|) |#2|) "\\spad{tracePowMod(u,{}k,{}v)} produces the sum of \\spad{u**(q**i)} for \\spad{i} running and \\spad{q=} size \\spad{F}")) (|trace2PowMod| ((|#2| |#2| (|NonNegativeInteger|) |#2|) "\\spad{trace2PowMod(u,{}k,{}v)} produces the sum of u**(2**i) for \\spad{i} running from 1 to \\spad{k} all computed modulo the polynomial \\spad{v}.")) (|exptMod| ((|#2| |#2| (|NonNegativeInteger|) |#2|) "\\spad{exptMod(u,{}k,{}v)} raises the polynomial \\spad{u} to the \\spad{k}th power modulo the polynomial \\spad{v}.")) (|separateFactors| (((|List| |#2|) (|List| (|Record| (|:| |deg| (|NonNegativeInteger|)) (|:| |prod| |#2|)))) "\\spad{separateFactors(lfact)} takes the list produced by separateDegrees and produces the complete list of factors.")) (|separateDegrees| (((|List| (|Record| (|:| |deg| (|NonNegativeInteger|)) (|:| |prod| |#2|))) |#2|) "\\spad{separateDegrees(p)} splits the square free polynomial \\spad{p} into factors each of which is a product of irreducibles of the same degree.")) (|distdfact| (((|Record| (|:| |cont| |#1|) (|:| |factors| (|List| (|Record| (|:| |irr| |#2|) (|:| |pow| (|Integer|)))))) |#2| (|Boolean|)) "\\spad{distdfact(p,{}sqfrflag)} produces the complete factorization of the polynomial \\spad{p} returning an internal data structure. If argument \\spad{sqfrflag} is \\spad{true},{} the polynomial is assumed square free.")) (|factorSquareFree| (((|Factored| |#2|) |#2|) "\\spad{factorSquareFree(p)} produces the complete factorization of the square free polynomial \\spad{p}.")) (|factor| (((|Factored| |#2|) |#2|) "\\spad{factor(p)} produces the complete factorization of the polynomial \\spad{p}."))) 
NIL 
NIL 
(-206) 
((|constructor| (NIL "This domain allows rational numbers to be presented as repeating decimal expansions.")) (|decimal| (($ (|Fraction| (|Integer|))) "\\spad{decimal(r)} converts a rational number to a decimal expansion.")) (|fractionPart| (((|Fraction| (|Integer|)) $) "\\spad{fractionPart(d)} returns the fractional part of a decimal expansion.")) (|coerce| (((|RadixExpansion| 10) $) "\\spad{coerce(d)} converts a decimal expansion to a radix expansion with base 10.") (((|Fraction| (|Integer|)) $) "\\spad{coerce(d)} converts a decimal expansion to a rational number."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| (-560) (QUOTE (-896))) (|HasCategory| (-560) (LIST (QUOTE -1029) (QUOTE (-1153)))) (|HasCategory| (-560) (QUOTE (-146))) (|HasCategory| (-560) (QUOTE (-148))) (|HasCategory| (-560) (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| (-560) (QUOTE (-1013))) (|HasCategory| (-560) (QUOTE (-807))) (|HasCategory| (-560) (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| (-560) (QUOTE (-1128))) (|HasCategory| (-560) (LIST (QUOTE -873) (QUOTE (-560)))) (|HasCategory| (-560) (LIST (QUOTE -873) (QUOTE (-375)))) (|HasCategory| (-560) (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-375))))) (|HasCategory| (-560) (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-560))))) (|HasCategory| (-560) (QUOTE (-221))) (|HasCategory| (-560) (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| (-560) (LIST (QUOTE -515) (QUOTE (-1153)) (QUOTE (-560)))) (|HasCategory| (-560) (LIST (QUOTE -298) (QUOTE (-560)))) (|HasCategory| (-560) (LIST (QUOTE -276) (QUOTE (-560)) (QUOTE (-560)))) (|HasCategory| (-560) (QUOTE (-296))) (|HasCategory| (-560) (QUOTE (-542))) (|HasCategory| (-560) (QUOTE (-834))) (-3322 (|HasCategory| (-560) (QUOTE (-807))) (|HasCategory| (-560) (QUOTE (-834)))) (|HasCategory| (-560) (LIST (QUOTE -622) (QUOTE (-560)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-560) (QUOTE (-896)))) (-3322 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-560) (QUOTE (-896)))) (|HasCategory| (-560) (QUOTE (-146))))) 
(-207 R -2262) 
((|constructor| (NIL "\\spadtype{ElementaryFunctionDefiniteIntegration} provides functions to compute definite integrals of elementary functions.")) (|innerint| (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| "failed") (|:| |pole| "potentialPole")) |#2| (|Symbol|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{innerint(f,{} x,{} a,{} b,{} ignore?)} should be local but conditional")) (|integrate| (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| "failed") (|:| |pole| "potentialPole")) |#2| (|SegmentBinding| (|OrderedCompletion| |#2|)) (|String|)) "\\spad{integrate(f,{} x = a..b,{} \"noPole\")} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. If it is not possible to check whether \\spad{f} has a pole for \\spad{x} between a and \\spad{b} (because of parameters),{} then this function will assume that \\spad{f} has no such pole. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b} or if the last argument is not \"noPole\".") (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| "failed") (|:| |pole| "potentialPole")) |#2| (|SegmentBinding| (|OrderedCompletion| |#2|))) "\\spad{integrate(f,{} x = a..b)} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b}."))) 
NIL 
NIL 
(-208 R) 
((|constructor| (NIL "Definite integration of rational functions. \\spadtype{RationalFunctionDefiniteIntegration} provides functions to compute definite integrals of rational functions.")) (|integrate| (((|Union| (|:| |f1| (|OrderedCompletion| (|Expression| |#1|))) (|:| |f2| (|List| (|OrderedCompletion| (|Expression| |#1|)))) (|:| |fail| "failed") (|:| |pole| "potentialPole")) (|Fraction| (|Polynomial| |#1|)) (|SegmentBinding| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|)))) (|String|)) "\\spad{integrate(f,{} x = a..b,{} \"noPole\")} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. If it is not possible to check whether \\spad{f} has a pole for \\spad{x} between a and \\spad{b} (because of parameters),{} then this function will assume that \\spad{f} has no such pole. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b} or if the last argument is not \"noPole\".") (((|Union| (|:| |f1| (|OrderedCompletion| (|Expression| |#1|))) (|:| |f2| (|List| (|OrderedCompletion| (|Expression| |#1|)))) (|:| |fail| "failed") (|:| |pole| "potentialPole")) (|Fraction| (|Polynomial| |#1|)) (|SegmentBinding| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))))) "\\spad{integrate(f,{} x = a..b)} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b}.") (((|Union| (|:| |f1| (|OrderedCompletion| (|Expression| |#1|))) (|:| |f2| (|List| (|OrderedCompletion| (|Expression| |#1|)))) (|:| |fail| "failed") (|:| |pole| "potentialPole")) (|Fraction| (|Polynomial| |#1|)) (|SegmentBinding| (|OrderedCompletion| (|Expression| |#1|))) (|String|)) "\\spad{integrate(f,{} x = a..b,{} \"noPole\")} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. If it is not possible to check whether \\spad{f} has a pole for \\spad{x} between a and \\spad{b} (because of parameters),{} then this function will assume that \\spad{f} has no such pole. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b} or if the last argument is not \"noPole\".") (((|Union| (|:| |f1| (|OrderedCompletion| (|Expression| |#1|))) (|:| |f2| (|List| (|OrderedCompletion| (|Expression| |#1|)))) (|:| |fail| "failed") (|:| |pole| "potentialPole")) (|Fraction| (|Polynomial| |#1|)) (|SegmentBinding| (|OrderedCompletion| (|Expression| |#1|)))) "\\spad{integrate(f,{} x = a..b)} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b}."))) 
NIL 
NIL 
(-209 R1 R2) 
((|constructor| (NIL "This package has no description")) (|expand| (((|List| (|Expression| |#2|)) (|Expression| |#2|) (|PositiveInteger|)) "\\spad{expand(f,{}n)} \\undocumented{}")) (|reduce| (((|Record| (|:| |pol| (|SparseUnivariatePolynomial| |#1|)) (|:| |deg| (|PositiveInteger|))) (|SparseUnivariatePolynomial| |#1|)) "\\spad{reduce(p)} \\undocumented{}"))) 
NIL 
NIL 
(-210 S) 
((|constructor| (NIL "Linked list implementation of a Dequeue")) (|member?| (((|Boolean|) |#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} member?(3,{}a)")) (|members| (((|List| |#1|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} members a")) (|parts| (((|List| |#1|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} parts a")) (|#| (((|NonNegativeInteger|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} \\#a")) (|count| (((|NonNegativeInteger|) |#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} count(4,{}a)") (((|NonNegativeInteger|) (|Mapping| (|Boolean|) |#1|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} count(\\spad{x+}->(\\spad{x>2}),{}a)")) (|any?| (((|Boolean|) (|Mapping| (|Boolean|) |#1|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} any?(\\spad{x+}->(\\spad{x=4}),{}a)")) (|every?| (((|Boolean|) (|Mapping| (|Boolean|) |#1|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} every?(\\spad{x+}->(\\spad{x=4}),{}a)")) (~= (((|Boolean|) $ $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} b:=copy a \\spad{X} (a~=b)")) (= (((|Boolean|) $ $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} b:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} (a=b)@Boolean")) (|coerce| (((|OutputForm|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} coerce a")) (|hash| (((|SingleInteger|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} hash a")) (|latex| (((|String|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} latex a")) (|map!| (($ (|Mapping| |#1| |#1|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} map!(\\spad{x+}-\\spad{>x+10},{}a) \\spad{X} a")) (|top!| ((|#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} top! a \\spad{X} a")) (|reverse!| (($ $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} reverse! a \\spad{X} a")) (|push!| ((|#1| |#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} push! a \\spad{X} a")) (|pop!| ((|#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} pop! a \\spad{X} a")) (|insertTop!| ((|#1| |#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} insertTop! a \\spad{X} a")) (|insertBottom!| ((|#1| |#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} insertBottom! a \\spad{X} a")) (|extractTop!| ((|#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} extractTop! a \\spad{X} a")) (|extractBottom!| ((|#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} extractBottom! a \\spad{X} a")) (|bottom!| ((|#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} bottom! a \\spad{X} a")) (|top| ((|#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} top a")) (|height| (((|NonNegativeInteger|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} height a")) (|depth| (((|NonNegativeInteger|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} depth a")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} map(\\spad{x+}-\\spad{>x+10},{}a) \\spad{X} a")) (|eq?| (((|Boolean|) $ $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} b:=copy a \\spad{X} eq?(a,{}\\spad{b})")) (|copy| (($ $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} copy a")) (|sample| (($) "\\blankline \\spad{X} sample()\\$Dequeue(INT)")) (|empty| (($) "\\blankline \\spad{X} b:=empty()\\$(Dequeue INT)")) (|empty?| (((|Boolean|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} empty? a")) (|bag| (($ (|List| |#1|)) "\\blankline \\spad{X} bag([1,{}2,{}3,{}4,{}5])\\$Dequeue(INT)")) (|size?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} size?(a,{}5)")) (|more?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} more?(a,{}9)")) (|less?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} less?(a,{}9)")) (|length| (((|NonNegativeInteger|) $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} length a")) (|rotate!| (($ $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} rotate! a")) (|back| ((|#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} back a")) (|front| ((|#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} front a")) (|inspect| ((|#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} inspect a")) (|insert!| (($ |#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} insert! (8,{}a) \\spad{X} a")) (|enqueue!| ((|#1| |#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} enqueue! (9,{}a) \\spad{X} a")) (|extract!| ((|#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} extract! a \\spad{X} a")) (|dequeue!| ((|#1| $) "\\blankline \\spad{X} a:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5] \\spad{X} dequeue! a \\spad{X} a")) (|dequeue| (($) "\\blankline \\spad{X} a:Dequeue INT:= dequeue ()") (($ (|List| |#1|)) "\\indented{1}{dequeue([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) creates a dequeue with first (top or front)} \\indented{1}{element \\spad{x},{} second element \\spad{y},{}...,{}and last (bottom or back) element \\spad{z}.} \\blankline \\spad{E} g:Dequeue INT:= dequeue [1,{}2,{}3,{}4,{}5]"))) 
((-4505 . T) (-4506 . T)) 
((|HasCategory| |#1| (QUOTE (-1082))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082))))) 
(-211 |CoefRing| |listIndVar|) 
((|constructor| (NIL "The deRham complex of Euclidean space,{} that is,{} the class of differential forms of arbitary degree over a coefficient ring. See Flanders,{} Harley,{} Differential Forms,{} With Applications to the Physical Sciences,{} New York,{} Academic Press,{} 1963.")) (|exteriorDifferential| (($ $) "\\spad{exteriorDifferential(df)} returns the exterior derivative (gradient,{} curl,{} divergence,{} ...) of the differential form \\spad{df}.")) (|totalDifferential| (($ (|Expression| |#1|)) "\\spad{totalDifferential(x)} returns the total differential (gradient) form for element \\spad{x}.")) (|map| (($ (|Mapping| (|Expression| |#1|) (|Expression| |#1|)) $) "\\spad{map(f,{}df)} replaces each coefficient \\spad{x} of differential form \\spad{df} by \\spad{f(x)}.")) (|degree| (((|Integer|) $) "\\spad{degree(df)} returns the homogeneous degree of differential form \\spad{df}.")) (|retractable?| (((|Boolean|) $) "\\spad{retractable?(df)} tests if differential form \\spad{df} is a 0-form,{} \\spadignore{i.e.} if degree(\\spad{df}) = 0.")) (|homogeneous?| (((|Boolean|) $) "\\spad{homogeneous?(df)} tests if all of the terms of differential form \\spad{df} have the same degree.")) (|generator| (($ (|NonNegativeInteger|)) "\\spad{generator(n)} returns the \\spad{n}th basis term for a differential form.")) (|coefficient| (((|Expression| |#1|) $ $) "\\spad{coefficient(df,{}u)},{} where \\spad{df} is a differential form,{} returns the coefficient of \\spad{df} containing the basis term \\spad{u} if such a term exists,{} and 0 otherwise.")) (|reductum| (($ $) "\\spad{reductum(df)},{} where \\spad{df} is a differential form,{} returns \\spad{df} minus the leading term of \\spad{df} if \\spad{df} has two or more terms,{} and 0 otherwise.")) (|leadingBasisTerm| (($ $) "\\spad{leadingBasisTerm(df)} returns the leading basis term of differential form \\spad{df}.")) (|leadingCoefficient| (((|Expression| |#1|) $) "\\spad{leadingCoefficient(df)} returns the leading coefficient of differential form \\spad{df}."))) 
((-4502 . T)) 
NIL 
(-212 R -2262) 
((|constructor| (NIL "\\spadtype{DefiniteIntegrationTools} provides common tools used by the definite integration of both rational and elementary functions.")) (|checkForZero| (((|Union| (|Boolean|) "failed") (|SparseUnivariatePolynomial| |#2|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{checkForZero(p,{} a,{} b,{} incl?)} is \\spad{true} if \\spad{p} has a zero between a and \\spad{b},{} \\spad{false} otherwise,{} \"failed\" if this cannot be determined. Check for a and \\spad{b} inclusive if incl? is \\spad{true},{} exclusive otherwise.") (((|Union| (|Boolean|) "failed") (|Polynomial| |#1|) (|Symbol|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{checkForZero(p,{} x,{} a,{} b,{} incl?)} is \\spad{true} if \\spad{p} has a zero for \\spad{x} between a and \\spad{b},{} \\spad{false} otherwise,{} \"failed\" if this cannot be determined. Check for a and \\spad{b} inclusive if incl? is \\spad{true},{} exclusive otherwise.")) (|computeInt| (((|Union| (|OrderedCompletion| |#2|) "failed") (|Kernel| |#2|) |#2| (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{computeInt(x,{} g,{} a,{} b,{} eval?)} returns the integral of \\spad{f} for \\spad{x} between a and \\spad{b},{} assuming that \\spad{g} is an indefinite integral of \\spad{f} and \\spad{f} has no pole between a and \\spad{b}. If \\spad{eval?} is \\spad{true},{} then \\spad{g} can be evaluated safely at \\spad{a} and \\spad{b},{} provided that they are finite values. Otherwise,{} limits must be computed.")) (|ignore?| (((|Boolean|) (|String|)) "\\spad{ignore?(s)} is \\spad{true} if \\spad{s} is the string that tells the integrator to assume that the function has no pole in the integration interval."))) 
NIL 
NIL 
(-213) 
((|constructor| (NIL "\\spadtype{DoubleFloat} is intended to make accessible hardware floating point arithmetic in Axiom,{} either native double precision,{} or IEEE. On most machines,{} there will be hardware support for the arithmetic operations: \\spad{++} +,{} *,{} / and possibly also the sqrt operation. The operations exp,{} log,{} sin,{} cos,{} atan are normally coded in software based on minimax polynomial/rational approximations. \\blankline Some general comments about the accuracy of the operations: the operations +,{} *,{} / and sqrt are expected to be fully accurate. The operations exp,{} log,{} sin,{} cos and atan are not expected to be fully accurate. In particular,{} sin and cos will lose all precision for large arguments. \\blankline The Float domain provides an alternative to the DoubleFloat domain. It provides an arbitrary precision model of floating point arithmetic. This means that accuracy problems like those above are eliminated by increasing the working precision where necessary. \\spadtype{Float} provides some special functions such as erf,{} the error function in addition to the elementary functions. The disadvantage of Float is that it is much more expensive than small floats when the latter can be used.")) (|integerDecode| (((|List| (|Integer|)) $) "\\indented{1}{integerDecode(\\spad{x}) returns the multiple values of the\\space{2}common} \\indented{1}{lisp integer-decode-float function.} \\indented{1}{See Steele,{} ISBN 0-13-152414-3 \\spad{p354}. This function can be used} \\indented{1}{to ensure that the results are bit-exact and do not depend on} \\indented{1}{the binary-to-decimal conversions.} \\blankline \\spad{X} a:DFLOAT:=-1.0/3.0 \\spad{X} integerDecode a")) (|machineFraction| (((|Fraction| (|Integer|)) $) "\\indented{1}{machineFraction(\\spad{x}) returns a bit-exact fraction of the machine} \\indented{1}{floating point number using the common lisp integer-decode-float} \\indented{1}{function. See Steele,{} ISBN 0-13-152414-3 \\spad{p354}} \\indented{1}{This function can be used to print results which do not depend} \\indented{1}{on binary-to-decimal conversions} \\blankline \\spad{X} a:DFLOAT:=-1.0/3.0 \\spad{X} machineFraction a")) (|rationalApproximation| (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{rationalApproximation(f,{} n,{} b)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< b**(-n)} (that is,{} \\spad{|(r-f)/f| < b**(-n)}).") (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|)) "\\spad{rationalApproximation(f,{} n)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< 10**(-n)}.")) (|doubleFloatFormat| (((|String|) (|String|)) "change the output format for doublefloats using lisp format strings")) (|Beta| (($ $ $) "\\spad{Beta(x,{}y)} is \\spad{Gamma(x) * Gamma(y)/Gamma(x+y)}.")) (|Gamma| (($ $) "\\spad{Gamma(x)} is the Euler Gamma function.")) (|atan| (($ $ $) "\\spad{atan(x,{}y)} computes the arc tangent from \\spad{x} with phase \\spad{y}.")) (|log10| (($ $) "\\spad{log10(x)} computes the logarithm with base 10 for \\spad{x}.")) (|log2| (($ $) "\\spad{log2(x)} computes the logarithm with base 2 for \\spad{x}.")) (|hash| (((|Integer|) $) "\\spad{hash(x)} returns the hash key for \\spad{x}")) (|exp1| (($) "\\spad{exp1()} returns the natural log base \\spad{2.718281828...}.")) (** (($ $ $) "\\spad{x ** y} returns the \\spad{y}th power of \\spad{x} (equal to \\spad{exp(y log x)}).")) (/ (($ $ (|Integer|)) "\\spad{x / i} computes the division from \\spad{x} by an integer \\spad{i}."))) 
((-3580 . T) (-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-214) 
((|constructor| (NIL "This package provides special functions for double precision real and complex floating point.")) (|hypergeometric0F1| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{hypergeometric0F1(c,{}z)} is the hypergeometric function \\spad{0F1(; c; z)}.") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{hypergeometric0F1(c,{}z)} is the hypergeometric function \\spad{0F1(; c; z)}.")) (|airyBi| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{airyBi(x)} is the Airy function \\spad{\\spad{Bi}(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{\\spad{Bi}''(x) - x * \\spad{Bi}(x) = 0}.}") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{airyBi(x)} is the Airy function \\spad{\\spad{Bi}(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{\\spad{Bi}''(x) - x * \\spad{Bi}(x) = 0}.}")) (|airyAi| (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{airyAi(x)} is the Airy function \\spad{\\spad{Ai}(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{\\spad{Ai}''(x) - x * \\spad{Ai}(x) = 0}.}") (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{airyAi(x)} is the Airy function \\spad{\\spad{Ai}(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{\\spad{Ai}''(x) - x * \\spad{Ai}(x) = 0}.}")) (|besselK| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselK(v,{}x)} is the modified Bessel function of the second kind,{} \\spad{K(v,{}x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.} Note that the default implementation uses the relation \\indented{2}{\\spad{K(v,{}x) = \\%pi/2*(I(-v,{}x) - I(v,{}x))/sin(v*\\%\\spad{pi})}} so is not valid for integer values of \\spad{v}.") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselK(v,{}x)} is the modified Bessel function of the second kind,{} \\spad{K(v,{}x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.} Note that the default implementation uses the relation \\indented{2}{\\spad{K(v,{}x) = \\%pi/2*(I(-v,{}x) - I(v,{}x))/sin(v*\\%\\spad{pi})}.} so is not valid for integer values of \\spad{v}.")) (|besselI| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselI(v,{}x)} is the modified Bessel function of the first kind,{} \\spad{I(v,{}x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.}") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselI(v,{}x)} is the modified Bessel function of the first kind,{} \\spad{I(v,{}x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.}")) (|besselY| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselY(v,{}x)} is the Bessel function of the second kind,{} \\spad{Y(v,{}x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.} Note that the default implementation uses the relation \\indented{2}{\\spad{Y(v,{}x) = (J(v,{}x) cos(v*\\%\\spad{pi}) - J(-v,{}x))/sin(v*\\%\\spad{pi})}} so is not valid for integer values of \\spad{v}.") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselY(v,{}x)} is the Bessel function of the second kind,{} \\spad{Y(v,{}x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.} Note that the default implementation uses the relation \\indented{2}{\\spad{Y(v,{}x) = (J(v,{}x) cos(v*\\%\\spad{pi}) - J(-v,{}x))/sin(v*\\%\\spad{pi})}} so is not valid for integer values of \\spad{v}.")) (|besselJ| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselJ(v,{}x)} is the Bessel function of the first kind,{} \\spad{J(v,{}x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.}") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselJ(v,{}x)} is the Bessel function of the first kind,{} \\spad{J(v,{}x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.}")) (|polygamma| (((|Complex| (|DoubleFloat|)) (|NonNegativeInteger|) (|Complex| (|DoubleFloat|))) "\\spad{polygamma(n,{} x)} is the \\spad{n}-th derivative of \\spad{digamma(x)}.") (((|DoubleFloat|) (|NonNegativeInteger|) (|DoubleFloat|)) "\\spad{polygamma(n,{} x)} is the \\spad{n}-th derivative of \\spad{digamma(x)}.")) (|digamma| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{digamma(x)} is the function,{} \\spad{psi(x)},{} defined by \\indented{2}{\\spad{psi(x) = Gamma'(x)/Gamma(x)}.}") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{digamma(x)} is the function,{} \\spad{psi(x)},{} defined by \\indented{2}{\\spad{psi(x) = Gamma'(x)/Gamma(x)}.}")) (|logGamma| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{logGamma(x)} is the natural log of \\spad{Gamma(x)}. This can often be computed even if \\spad{Gamma(x)} cannot.") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{logGamma(x)} is the natural log of \\spad{Gamma(x)}. This can often be computed even if \\spad{Gamma(x)} cannot.")) (|Beta| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{Beta(x,{} y)} is the Euler beta function,{} \\spad{B(x,{}y)},{} defined by \\indented{2}{\\spad{Beta(x,{}y) = integrate(t^(x-1)*(1-t)^(y-1),{} t=0..1)}.} This is related to \\spad{Gamma(x)} by \\indented{2}{\\spad{Beta(x,{}y) = Gamma(x)*Gamma(y) / Gamma(x + y)}.}") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{Beta(x,{} y)} is the Euler beta function,{} \\spad{B(x,{}y)},{} defined by \\indented{2}{\\spad{Beta(x,{}y) = integrate(t^(x-1)*(1-t)^(y-1),{} t=0..1)}.} This is related to \\spad{Gamma(x)} by \\indented{2}{\\spad{Beta(x,{}y) = Gamma(x)*Gamma(y) / Gamma(x + y)}.}")) (|Ei6| (((|OnePointCompletion| (|DoubleFloat|)) (|OnePointCompletion| (|DoubleFloat|))) "\\spad{Ei6} is the first approximation of \\spad{Ei} where the result is \\spad{x*}\\%e^-x*Ei(\\spad{x}) from 32 to infinity (preserves digits)")) (|Ei5| (((|OnePointCompletion| (|DoubleFloat|)) (|OnePointCompletion| (|DoubleFloat|))) "\\spad{Ei5} is the first approximation of \\spad{Ei} where the result is \\spad{x*}\\%e^-x*Ei(\\spad{x}) from 12 to 32 (preserves digits)")) (|Ei4| (((|OnePointCompletion| (|DoubleFloat|)) (|OnePointCompletion| (|DoubleFloat|))) "\\spad{Ei4} is the first approximation of \\spad{Ei} where the result is \\spad{x*}\\%e^-x*Ei(\\spad{x}) from 4 to 12 (preserves digits)")) (|Ei3| (((|OnePointCompletion| (|DoubleFloat|)) (|OnePointCompletion| (|DoubleFloat|))) "\\spad{Ei3} is the first approximation of \\spad{Ei} where the result is (\\spad{Ei}(\\spad{x})-log \\spad{|x|} - gamma)\\spad{/x} from \\spad{-4} to 4 (preserves digits)")) (|Ei2| (((|OnePointCompletion| (|DoubleFloat|)) (|OnePointCompletion| (|DoubleFloat|))) "\\spad{Ei2} is the first approximation of \\spad{Ei} where the result is \\spad{x*}\\%e^-x*Ei(\\spad{x}) from \\spad{-10} to \\spad{-4} (preserves digits)")) (|Ei1| (((|OnePointCompletion| (|DoubleFloat|)) (|OnePointCompletion| (|DoubleFloat|))) "\\spad{Ei1} is the first approximation of \\spad{Ei} where the result is \\spad{x*}\\%e^-x*Ei(\\spad{x}) from -infinity to \\spad{-10} (preserves digits)")) (|Ei| (((|OnePointCompletion| (|DoubleFloat|)) (|OnePointCompletion| (|DoubleFloat|))) "\\spad{Ei} is the Exponential Integral function This is computed using a 6 part piecewise approximation. DoubleFloat can only preserve about 16 digits but the Chebyshev approximation used can give 30 digits.")) (|En| (((|OnePointCompletion| (|DoubleFloat|)) (|Integer|) (|DoubleFloat|)) "\\spad{En(n,{}x)} is the \\spad{n}th Exponential Integral Function")) (E1 (((|OnePointCompletion| (|DoubleFloat|)) (|DoubleFloat|)) "\\spad{E1(x)} is the Exponential Integral function The current implementation is a piecewise approximation involving one poly from \\spad{-4}..4 and a second poly for \\spad{x} > 4")) (|Gamma| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{Gamma(x)} is the Euler gamma function,{} \\spad{Gamma(x)},{} defined by \\indented{2}{\\spad{Gamma(x) = integrate(t^(x-1)*exp(-t),{} t=0..\\%infinity)}.}") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{Gamma(x)} is the Euler gamma function,{} \\spad{Gamma(x)},{} defined by \\indented{2}{\\spad{Gamma(x) = integrate(t^(x-1)*exp(-t),{} t=0..\\%infinity)}.}"))) 
NIL 
NIL 
(-215 R) 
((|constructor| (NIL "4x4 Matrices for coordinate transformations\\spad{\\br} This package contains functions to create 4x4 matrices useful for rotating and transforming coordinate systems. These matrices are useful for graphics and robotics. (Reference: Robot Manipulators Richard Paul MIT Press 1981) \\blankline A Denavit-Hartenberg Matrix is a 4x4 Matrix of the form:\\spad{\\br} \\tab{5}\\spad{nx ox ax px}\\spad{\\br} \\tab{5}\\spad{ny oy ay py}\\spad{\\br} \\tab{5}\\spad{nz oz az pz}\\spad{\\br} \\tab{5}\\spad{0 0 0 1}\\spad{\\br} (\\spad{n},{} \\spad{o},{} and a are the direction cosines)")) (|translate| (($ |#1| |#1| |#1|) "\\spad{translate(x,{}y,{}z)} returns a dhmatrix for translation by \\spad{x},{} \\spad{y},{} and \\spad{z}")) (|scale| (($ |#1| |#1| |#1|) "\\spad{scale(sx,{}sy,{}sz)} returns a dhmatrix for scaling in the \\spad{x},{} \\spad{y} and \\spad{z} directions")) (|rotatez| (($ |#1|) "\\spad{rotatez(r)} returns a dhmatrix for rotation about axis \\spad{z} for \\spad{r} degrees")) (|rotatey| (($ |#1|) "\\spad{rotatey(r)} returns a dhmatrix for rotation about axis \\spad{y} for \\spad{r} degrees")) (|rotatex| (($ |#1|) "\\spad{rotatex(r)} returns a dhmatrix for rotation about axis \\spad{x} for \\spad{r} degrees")) (|identity| (($) "\\spad{identity()} create the identity dhmatrix")) (* (((|Point| |#1|) $ (|Point| |#1|)) "\\spad{t*p} applies the dhmatrix \\spad{t} to point \\spad{p}"))) 
((-4505 . T) (-4506 . T)) 
((|HasCategory| |#1| (QUOTE (-1082))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082)))) (|HasCategory| |#1| (QUOTE (-296))) (|HasCategory| |#1| (QUOTE (-550))) (|HasAttribute| |#1| (QUOTE (-4507 "*"))) (|HasCategory| |#1| (QUOTE (-170))) (|HasCategory| |#1| (QUOTE (-359)))) 
(-216 A S) 
((|constructor| (NIL "A dictionary is an aggregate in which entries can be inserted,{} searched for and removed. Duplicates are thrown away on insertion. This category models the usual notion of dictionary which involves large amounts of data where copying is impractical. Principal operations are thus destructive (non-copying) ones."))) 
NIL 
NIL 
(-217 S) 
((|constructor| (NIL "A dictionary is an aggregate in which entries can be inserted,{} searched for and removed. Duplicates are thrown away on insertion. This category models the usual notion of dictionary which involves large amounts of data where copying is impractical. Principal operations are thus destructive (non-copying) ones."))) 
((-4506 . T) (-3576 . T)) 
NIL 
(-218 S R) 
((|constructor| (NIL "Differential extensions of a ring \\spad{R}. Given a differentiation on \\spad{R},{} extend it to a differentiation on \\%.")) (D (($ $ (|Mapping| |#2| |#2|) (|NonNegativeInteger|)) "\\spad{D(x,{} deriv,{} n)} differentiate \\spad{x} \\spad{n} times using a derivation which extends \\spad{deriv} on \\spad{R}.") (($ $ (|Mapping| |#2| |#2|)) "\\spad{D(x,{} deriv)} differentiates \\spad{x} extending the derivation deriv on \\spad{R}.")) (|differentiate| (($ $ (|Mapping| |#2| |#2|) (|NonNegativeInteger|)) "\\spad{differentiate(x,{} deriv,{} n)} differentiate \\spad{x} \\spad{n} times using a derivation which extends \\spad{deriv} on \\spad{R}.") (($ $ (|Mapping| |#2| |#2|)) "\\spad{differentiate(x,{} deriv)} differentiates \\spad{x} extending the derivation deriv on \\spad{R}."))) 
NIL 
((|HasCategory| |#2| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#2| (QUOTE (-221)))) 
(-219 R) 
((|constructor| (NIL "Differential extensions of a ring \\spad{R}. Given a differentiation on \\spad{R},{} extend it to a differentiation on \\%.")) (D (($ $ (|Mapping| |#1| |#1|) (|NonNegativeInteger|)) "\\spad{D(x,{} deriv,{} n)} differentiate \\spad{x} \\spad{n} times using a derivation which extends \\spad{deriv} on \\spad{R}.") (($ $ (|Mapping| |#1| |#1|)) "\\spad{D(x,{} deriv)} differentiates \\spad{x} extending the derivation deriv on \\spad{R}.")) (|differentiate| (($ $ (|Mapping| |#1| |#1|) (|NonNegativeInteger|)) "\\spad{differentiate(x,{} deriv,{} n)} differentiate \\spad{x} \\spad{n} times using a derivation which extends \\spad{deriv} on \\spad{R}.") (($ $ (|Mapping| |#1| |#1|)) "\\spad{differentiate(x,{} deriv)} differentiates \\spad{x} extending the derivation deriv on \\spad{R}."))) 
((-4502 . T)) 
NIL 
(-220 S) 
((|constructor| (NIL "An ordinary differential ring,{} that is,{} a ring with an operation \\spadfun{differentiate}. \\blankline Axioms\\spad{\\br} \\tab{5}\\spad{differentiate(x+y) = differentiate(x)+differentiate(y)}\\spad{\\br} \\tab{5}\\spad{differentiate(x*y) = x*differentiate(y) + differentiate(x)*y}")) (D (($ $ (|NonNegativeInteger|)) "\\spad{D(x,{} n)} returns the \\spad{n}-th derivative of \\spad{x}.") (($ $) "\\spad{D(x)} returns the derivative of \\spad{x}. This function is a simple differential operator where no variable needs to be specified.")) (|differentiate| (($ $ (|NonNegativeInteger|)) "\\spad{differentiate(x,{} n)} returns the \\spad{n}-th derivative of \\spad{x}.") (($ $) "\\spad{differentiate(x)} returns the derivative of \\spad{x}. This function is a simple differential operator where no variable needs to be specified."))) 
NIL 
NIL 
(-221) 
((|constructor| (NIL "An ordinary differential ring,{} that is,{} a ring with an operation \\spadfun{differentiate}. \\blankline Axioms\\spad{\\br} \\tab{5}\\spad{differentiate(x+y) = differentiate(x)+differentiate(y)}\\spad{\\br} \\tab{5}\\spad{differentiate(x*y) = x*differentiate(y) + differentiate(x)*y}")) (D (($ $ (|NonNegativeInteger|)) "\\spad{D(x,{} n)} returns the \\spad{n}-th derivative of \\spad{x}.") (($ $) "\\spad{D(x)} returns the derivative of \\spad{x}. This function is a simple differential operator where no variable needs to be specified.")) (|differentiate| (($ $ (|NonNegativeInteger|)) "\\spad{differentiate(x,{} n)} returns the \\spad{n}-th derivative of \\spad{x}.") (($ $) "\\spad{differentiate(x)} returns the derivative of \\spad{x}. This function is a simple differential operator where no variable needs to be specified."))) 
((-4502 . T)) 
NIL 
(-222 A S) 
((|constructor| (NIL "This category is a collection of operations common to both categories \\spadtype{Dictionary} and \\spadtype{MultiDictionary}")) (|select!| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{select!(p,{}d)} destructively changes dictionary \\spad{d} by removing all entries \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is not \\spad{true}.")) (|remove!| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{remove!(p,{}d)} destructively changes dictionary \\spad{d} by removeing all entries \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}.") (($ |#2| $) "\\spad{remove!(x,{}d)} destructively changes dictionary \\spad{d} by removing all entries \\spad{y} such that \\axiom{\\spad{y} = \\spad{x}}.")) (|dictionary| (($ (|List| |#2|)) "\\spad{dictionary([x,{}y,{}...,{}z])} creates a dictionary consisting of entries \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}}.") (($) "\\spad{dictionary()}\\$\\spad{D} creates an empty dictionary of type \\spad{D}."))) 
NIL 
((|HasAttribute| |#1| (QUOTE -4505))) 
(-223 S) 
((|constructor| (NIL "This category is a collection of operations common to both categories \\spadtype{Dictionary} and \\spadtype{MultiDictionary}")) (|select!| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{select!(p,{}d)} destructively changes dictionary \\spad{d} by removing all entries \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is not \\spad{true}.")) (|remove!| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{remove!(p,{}d)} destructively changes dictionary \\spad{d} by removeing all entries \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}.") (($ |#1| $) "\\spad{remove!(x,{}d)} destructively changes dictionary \\spad{d} by removing all entries \\spad{y} such that \\axiom{\\spad{y} = \\spad{x}}.")) (|dictionary| (($ (|List| |#1|)) "\\spad{dictionary([x,{}y,{}...,{}z])} creates a dictionary consisting of entries \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}}.") (($) "\\spad{dictionary()}\\$\\spad{D} creates an empty dictionary of type \\spad{D}."))) 
((-4506 . T) (-3576 . T)) 
NIL 
(-224) 
((|constructor| (NIL "Any solution of a homogeneous linear Diophantine equation can be represented as a sum of minimal solutions,{} which form a \"basis\" (a minimal solution cannot be represented as a nontrivial sum of solutions) in the case of an inhomogeneous linear Diophantine equation,{} each solution is the sum of a inhomogeneous solution and any number of homogeneous solutions therefore,{} it suffices to compute two sets:\\spad{\\br} \\tab{5}1. all minimal inhomogeneous solutions\\spad{\\br} \\tab{5}2. all minimal homogeneous solutions\\spad{\\br} the algorithm implemented is a completion procedure,{} which enumerates all solutions in a recursive depth-first-search it can be seen as finding monotone paths in a graph for more details see Reference")) (|dioSolve| (((|Record| (|:| |varOrder| (|List| (|Symbol|))) (|:| |inhom| (|Union| (|List| (|Vector| (|NonNegativeInteger|))) "failed")) (|:| |hom| (|List| (|Vector| (|NonNegativeInteger|))))) (|Equation| (|Polynomial| (|Integer|)))) "\\spad{dioSolve(u)} computes a basis of all minimal solutions for linear homogeneous Diophantine equation \\spad{u},{} then all minimal solutions of inhomogeneous equation"))) 
NIL 
NIL 
(-225 S -3780 R) 
((|constructor| (NIL "This category represents a finite cartesian product of a given type. Many categorical properties are preserved under this construction.")) (|dot| ((|#3| $ $) "\\spad{dot(x,{}y)} computes the inner product of the vectors \\spad{x} and \\spad{y}.")) (|unitVector| (($ (|PositiveInteger|)) "\\spad{unitVector(n)} produces a vector with 1 in position \\spad{n} and zero elsewhere.")) (|directProduct| (($ (|Vector| |#3|)) "\\spad{directProduct(v)} converts the vector \\spad{v} to become a direct product. Error: if the length of \\spad{v} is different from dim.")) (|finiteAggregate| ((|attribute|) "attribute to indicate an aggregate of finite size"))) 
NIL 
((|HasCategory| |#3| (QUOTE (-359))) (|HasCategory| |#3| (QUOTE (-780))) (|HasCategory| |#3| (QUOTE (-832))) (|HasAttribute| |#3| (QUOTE -4502)) (|HasCategory| |#3| (QUOTE (-170))) (|HasCategory| |#3| (QUOTE (-364))) (|HasCategory| |#3| (QUOTE (-708))) (|HasCategory| |#3| (QUOTE (-137))) (|HasCategory| |#3| (QUOTE (-25))) (|HasCategory| |#3| (QUOTE (-1039))) (|HasCategory| |#3| (QUOTE (-1082)))) 
(-226 -3780 R) 
((|constructor| (NIL "This category represents a finite cartesian product of a given type. Many categorical properties are preserved under this construction.")) (|dot| ((|#2| $ $) "\\spad{dot(x,{}y)} computes the inner product of the vectors \\spad{x} and \\spad{y}.")) (|unitVector| (($ (|PositiveInteger|)) "\\spad{unitVector(n)} produces a vector with 1 in position \\spad{n} and zero elsewhere.")) (|directProduct| (($ (|Vector| |#2|)) "\\spad{directProduct(v)} converts the vector \\spad{v} to become a direct product. Error: if the length of \\spad{v} is different from dim.")) (|finiteAggregate| ((|attribute|) "attribute to indicate an aggregate of finite size"))) 
((-4499 |has| |#2| (-1039)) (-4500 |has| |#2| (-1039)) (-4502 |has| |#2| (-6 -4502)) ((-4507 "*") |has| |#2| (-170)) (-4505 . T) (-3576 . T)) 
NIL 
(-227 -3780 A B) 
((|constructor| (NIL "This package provides operations which all take as arguments direct products of elements of some type \\spad{A} and functions from \\spad{A} to another type \\spad{B}. The operations all iterate over their vector argument and either return a value of type \\spad{B} or a direct product over \\spad{B}.")) (|map| (((|DirectProduct| |#1| |#3|) (|Mapping| |#3| |#2|) (|DirectProduct| |#1| |#2|)) "\\spad{map(f,{} v)} applies the function \\spad{f} to every element of the vector \\spad{v} producing a new vector containing the values.")) (|reduce| ((|#3| (|Mapping| |#3| |#2| |#3|) (|DirectProduct| |#1| |#2|) |#3|) "\\spad{reduce(func,{}vec,{}ident)} combines the elements in \\spad{vec} using the binary function \\spad{func}. Argument \\spad{ident} is returned if the vector is empty.")) (|scan| (((|DirectProduct| |#1| |#3|) (|Mapping| |#3| |#2| |#3|) (|DirectProduct| |#1| |#2|) |#3|) "\\spad{scan(func,{}vec,{}ident)} creates a new vector whose elements are the result of applying reduce to the binary function \\spad{func},{} increasing initial subsequences of the vector \\spad{vec},{} and the element \\spad{ident}."))) 
NIL 
NIL 
(-228 -3780 R) 
((|constructor| (NIL "This type represents the finite direct or cartesian product of an underlying component type. This contrasts with simple vectors in that the members can be viewed as having constant length. Thus many categorical properties can by lifted from the underlying component type. Component extraction operations are provided but no updating operations. Thus new direct product elements can either be created by converting vector elements using the \\spadfun{directProduct} function or by taking appropriate linear combinations of basis vectors provided by the \\spad{unitVector} operation."))) 
((-4499 |has| |#2| (-1039)) (-4500 |has| |#2| (-1039)) (-4502 |has| |#2| (-6 -4502)) ((-4507 "*") |has| |#2| (-170)) (-4505 . T)) 
((|HasCategory| |#2| (QUOTE (-1082))) (|HasCategory| |#2| (QUOTE (-359))) (|HasCategory| |#2| (QUOTE (-1039))) (|HasCategory| |#2| (QUOTE (-780))) (|HasCategory| |#2| (QUOTE (-832))) (-3322 (|HasCategory| |#2| (QUOTE (-780))) (|HasCategory| |#2| (QUOTE (-832)))) (|HasCategory| |#2| (QUOTE (-708))) (|HasCategory| |#2| (QUOTE (-170))) (-3322 (|HasCategory| |#2| (QUOTE (-170))) (|HasCategory| |#2| (QUOTE (-359))) (|HasCategory| |#2| (QUOTE (-1039)))) (-3322 (|HasCategory| |#2| (QUOTE (-170))) (|HasCategory| |#2| (QUOTE (-359)))) (-3322 (|HasCategory| |#2| (QUOTE (-170))) (|HasCategory| |#2| (QUOTE (-1039)))) (|HasCategory| |#2| (QUOTE (-364))) (|HasCategory| |#2| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#2| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#2| (QUOTE (-221))) (-3322 (|HasCategory| |#2| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#2| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#2| (QUOTE (-170))) (|HasCategory| |#2| (QUOTE (-221))) (|HasCategory| |#2| (QUOTE (-359))) (|HasCategory| |#2| (QUOTE (-1039)))) (-3322 (|HasCategory| |#2| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#2| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#2| (QUOTE (-170))) (|HasCategory| |#2| (QUOTE (-221))) (|HasCategory| |#2| (QUOTE (-1039)))) (|HasCategory| (-560) (QUOTE (-834))) (-12 (|HasCategory| |#2| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-1039)))) (-12 (|HasCategory| |#2| (QUOTE (-221))) (|HasCategory| |#2| (QUOTE (-1039)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#2| (QUOTE (-1039)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-1082)))) (-3322 (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-1082)))) (|HasCategory| |#2| (QUOTE (-1039)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (QUOTE (-1082)))) (|HasAttribute| |#2| (QUOTE -4502)) (|HasCategory| |#2| (QUOTE (-137))) (-3322 (|HasCategory| |#2| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#2| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#2| (QUOTE (-137))) (|HasCategory| |#2| (QUOTE (-170))) (|HasCategory| |#2| (QUOTE (-221))) (|HasCategory| |#2| (QUOTE (-359))) (|HasCategory| |#2| (QUOTE (-1039)))) (|HasCategory| |#2| (QUOTE (-25))) (-3322 (|HasCategory| |#2| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#2| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (QUOTE (-137))) (|HasCategory| |#2| (QUOTE (-170))) (|HasCategory| |#2| (QUOTE (-221))) (|HasCategory| |#2| (QUOTE (-359))) (|HasCategory| |#2| (QUOTE (-364))) (|HasCategory| |#2| (QUOTE (-708))) (|HasCategory| |#2| (QUOTE (-780))) (|HasCategory| |#2| (QUOTE (-832))) (|HasCategory| |#2| (QUOTE (-1039))) (|HasCategory| |#2| (QUOTE (-1082)))) (-3322 (|HasCategory| |#2| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#2| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (QUOTE (-137))) (|HasCategory| |#2| (QUOTE (-170))) (|HasCategory| |#2| (QUOTE (-221))) (|HasCategory| |#2| (QUOTE (-359))) (|HasCategory| |#2| (QUOTE (-1039)))) (-3322 (-12 (|HasCategory| |#2| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (QUOTE (-25)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (QUOTE (-137)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (QUOTE (-170)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (QUOTE (-221)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (QUOTE (-359)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (QUOTE (-364)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (QUOTE (-708)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (QUOTE (-780)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (QUOTE (-832)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (QUOTE (-1039)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (QUOTE (-1082))))) (-3322 (-12 (|HasCategory| |#2| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-25)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-137)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-170)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-221)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-359)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-364)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-708)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-780)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-832)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-1039)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-1082))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-1082)))) (-3322 (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -622) (QUOTE (-560))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -887) (QUOTE (-1153))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-25)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-137)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-170)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-221)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-359)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-364)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-708)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-780)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-832)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-1039)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-1082)))))) 
(-229) 
((|constructor| (NIL "DisplayPackage allows one to print strings in a nice manner,{} including highlighting substrings.")) (|sayLength| (((|Integer|) (|List| (|String|))) "\\spad{sayLength(l)} returns the length of a list of strings \\spad{l} as an integer.") (((|Integer|) (|String|)) "\\spad{sayLength(s)} returns the length of a string \\spad{s} as an integer.")) (|say| (((|Void|) (|List| (|String|))) "\\spad{say(l)} sends a list of strings \\spad{l} to output.") (((|Void|) (|String|)) "\\spad{say(s)} sends a string \\spad{s} to output.")) (|center| (((|List| (|String|)) (|List| (|String|)) (|Integer|) (|String|)) "\\spad{center(l,{}i,{}s)} takes a list of strings \\spad{l},{} and centers them within a list of strings which is \\spad{i} characters long,{} in which the remaining spaces are filled with strings composed of as many repetitions as possible of the last string parameter \\spad{s}.") (((|String|) (|String|) (|Integer|) (|String|)) "\\spad{center(s,{}i,{}s)} takes the first string \\spad{s},{} and centers it within a string of length \\spad{i},{} in which the other elements of the string are composed of as many replications as possible of the second indicated string,{} \\spad{s} which must have a length greater than that of an empty string.")) (|copies| (((|String|) (|Integer|) (|String|)) "\\spad{copies(i,{}s)} will take a string \\spad{s} and create a new string composed of \\spad{i} copies of \\spad{s}.")) (|newLine| (((|String|)) "\\spad{newLine()} sends a new line command to output.")) (|bright| (((|List| (|String|)) (|List| (|String|))) "\\spad{bright(l)} sets the font property of a list of strings,{} \\spad{l},{} to bold-face type.") (((|List| (|String|)) (|String|)) "\\spad{bright(s)} sets the font property of the string \\spad{s} to bold-face type."))) 
NIL 
NIL 
(-230 S) 
((|constructor| (NIL "This category exports the function for domains")) (|divOfPole| (($ $) "\\spad{divOfPole(d)} returns the negative part of \\spad{d}.")) (|divOfZero| (($ $) "\\spad{divOfZero(d)} returns the positive part of \\spad{d}.")) (|suppOfPole| (((|List| |#1|) $) "suppOfZero(\\spad{d}) returns the elements of the support of \\spad{d} that have a negative coefficient.")) (|suppOfZero| (((|List| |#1|) $) "\\spad{suppOfZero(d)} returns the elements of the support of \\spad{d} that have a positive coefficient.")) (|supp| (((|List| |#1|) $) "\\spad{supp(d)} returns the support of the divisor \\spad{d}.")) (|effective?| (((|Boolean|) $) "\\spad{effective?(d)} returns \\spad{true} if \\spad{d} \\spad{>=} 0.")) (|concat| (($ $ $) "\\spad{concat(a,{}b)} concats the divisor a and \\spad{b} without collecting the duplicative points.")) (|collect| (($ $) "\\spad{collect collects} the duplicative points in the divisor.")) (|split| (((|List| $) $) "\\spad{split(d)} splits the divisor \\spad{d}. For example,{} split( 2 \\spad{p1} + 3p2 ) returns the list [ 2 \\spad{p1},{} 3 \\spad{p2} ].")) (|degree| (((|Integer|) $) "\\spad{degree(d)} returns the degree of the divisor \\spad{d}"))) 
((-4500 . T) (-4499 . T)) 
NIL 
(-231 S) 
((|constructor| (NIL "The following is part of the PAFF package"))) 
((-4500 . T) (-4499 . T)) 
((|HasCategory| (-560) (QUOTE (-779)))) 
(-232 S) 
((|constructor| (NIL "A division ring (sometimes called a skew field),{} \\spadignore{i.e.} a not necessarily commutative ring where all non-zero elements have multiplicative inverses.")) (|inv| (($ $) "\\spad{inv x} returns the multiplicative inverse of \\spad{x}. Error: if \\spad{x} is 0.")) (^ (($ $ (|Integer|)) "\\spad{x^n} returns \\spad{x} raised to the integer power \\spad{n}.")) (** (($ $ (|Integer|)) "\\spad{x**n} returns \\spad{x} raised to the integer power \\spad{n}."))) 
NIL 
NIL 
(-233) 
((|constructor| (NIL "A division ring (sometimes called a skew field),{} \\spadignore{i.e.} a not necessarily commutative ring where all non-zero elements have multiplicative inverses.")) (|inv| (($ $) "\\spad{inv x} returns the multiplicative inverse of \\spad{x}. Error: if \\spad{x} is 0.")) (^ (($ $ (|Integer|)) "\\spad{x^n} returns \\spad{x} raised to the integer power \\spad{n}.")) (** (($ $ (|Integer|)) "\\spad{x**n} returns \\spad{x} raised to the integer power \\spad{n}."))) 
((-4498 . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-234 S) 
((|constructor| (NIL "A doubly-linked aggregate serves as a model for a doubly-linked list,{} that is,{} a list which can has links to both next and previous nodes and thus can be efficiently traversed in both directions.")) (|setnext!| (($ $ $) "\\spad{setnext!(u,{}v)} destructively sets the next node of doubly-linked aggregate \\spad{u} to \\spad{v},{} returning \\spad{v}.")) (|setprevious!| (($ $ $) "\\spad{setprevious!(u,{}v)} destructively sets the previous node of doubly-linked aggregate \\spad{u} to \\spad{v},{} returning \\spad{v}.")) (|concat!| (($ $ $) "\\spad{concat!(u,{}v)} destructively concatenates doubly-linked aggregate \\spad{v} to the end of doubly-linked aggregate \\spad{u}.")) (|next| (($ $) "\\spad{next(l)} returns the doubly-linked aggregate beginning with its next element. Error: if \\spad{l} has no next element. Note that \\axiom{next(\\spad{l}) = rest(\\spad{l})} and \\axiom{previous(next(\\spad{l})) = \\spad{l}}.")) (|previous| (($ $) "\\spad{previous(l)} returns the doubly-link list beginning with its previous element. Error: if \\spad{l} has no previous element. Note that \\axiom{next(previous(\\spad{l})) = \\spad{l}}.")) (|tail| (($ $) "\\spad{tail(l)} returns the doubly-linked aggregate \\spad{l} starting at its second element. Error: if \\spad{l} is empty.")) (|head| (($ $) "\\spad{head(l)} returns the first element of a doubly-linked aggregate \\spad{l}. Error: if \\spad{l} is empty.")) (|last| ((|#1| $) "\\spad{last(l)} returns the last element of a doubly-linked aggregate \\spad{l}. Error: if \\spad{l} is empty."))) 
((-3576 . T)) 
NIL 
(-235 S) 
((|constructor| (NIL "This domain provides some nice functions on lists")) (|elt| (((|NonNegativeInteger|) $ "count") "\\axiom{\\spad{l}.\"count\"} returns the number of elements in \\axiom{\\spad{l}}.") (($ $ "sort") "\\axiom{\\spad{l}.sort} returns \\axiom{\\spad{l}} with elements sorted. Note: \\axiom{\\spad{l}.sort = sort(\\spad{l})}") (($ $ "unique") "\\axiom{\\spad{l}.unique} returns \\axiom{\\spad{l}} with duplicates removed. Note: \\axiom{\\spad{l}.unique = removeDuplicates(\\spad{l})}.")) (|datalist| (($ (|List| |#1|)) "\\spad{datalist(l)} creates a datalist from \\spad{l}")) (|coerce| (((|List| |#1|) $) "\\spad{coerce(x)} returns the list of elements in \\spad{x}") (($ (|List| |#1|)) "\\spad{coerce(l)} creates a datalist from \\spad{l}"))) 
((-4506 . T) (-4505 . T)) 
((|HasCategory| |#1| (QUOTE (-1082))) (|HasCategory| |#1| (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| |#1| (QUOTE (-834))) (-3322 (|HasCategory| |#1| (QUOTE (-834))) (|HasCategory| |#1| (QUOTE (-1082)))) (|HasCategory| (-560) (QUOTE (-834))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082)))) (-3322 (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-834)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082)))))) 
(-236 M) 
((|constructor| (NIL "DiscreteLogarithmPackage implements help functions for discrete logarithms in monoids using small cyclic groups.")) (|shanksDiscLogAlgorithm| (((|Union| (|NonNegativeInteger|) "failed") |#1| |#1| (|NonNegativeInteger|)) "\\spad{shanksDiscLogAlgorithm(b,{}a,{}p)} computes \\spad{s} with \\spad{b**s = a} for assuming that \\spad{a} and \\spad{b} are elements in a 'small' cyclic group of order \\spad{p} by Shank\\spad{'s} algorithm. Note that this is a subroutine of the function \\spadfun{discreteLog}.")) (** ((|#1| |#1| (|Integer|)) "\\spad{x ** n} returns \\spad{x} raised to the integer power \\spad{n}"))) 
NIL 
NIL 
(-237 |vl| R) 
((|constructor| (NIL "This type supports distributed multivariate polynomials whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is lexicographic specified by the variable list parameter with the most significant variable first in the list.")) (|reorder| (($ $ (|List| (|Integer|))) "\\spad{reorder(p,{} perm)} applies the permutation perm to the variables in a polynomial and returns the new correctly ordered polynomial"))) 
(((-4507 "*") |has| |#2| (-170)) (-4498 |has| |#2| (-550)) (-4503 |has| |#2| (-6 -4503)) (-4500 . T) (-4499 . T) (-4502 . T)) 
((|HasCategory| |#2| (QUOTE (-896))) (|HasCategory| |#2| (QUOTE (-550))) (|HasCategory| |#2| (QUOTE (-170))) (-3322 (|HasCategory| |#2| (QUOTE (-170))) (|HasCategory| |#2| (QUOTE (-550)))) (-12 (|HasCategory| (-844 |#1|) (LIST (QUOTE -873) (QUOTE (-375)))) (|HasCategory| |#2| (LIST (QUOTE -873) (QUOTE (-375))))) (-12 (|HasCategory| (-844 |#1|) (LIST (QUOTE -873) (QUOTE (-560)))) (|HasCategory| |#2| (LIST (QUOTE -873) (QUOTE (-560))))) (-12 (|HasCategory| (-844 |#1|) (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-375))))) (|HasCategory| |#2| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-375)))))) (-12 (|HasCategory| (-844 |#1|) (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-560))))) (|HasCategory| |#2| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-560)))))) (-12 (|HasCategory| (-844 |#1|) (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| |#2| (LIST (QUOTE -601) (QUOTE (-533))))) (|HasCategory| |#2| (QUOTE (-834))) (|HasCategory| |#2| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -43) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (QUOTE (-359))) (-3322 (|HasCategory| |#2| (LIST (QUOTE -43) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#2| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560)))))) (|HasAttribute| |#2| (QUOTE -4503)) (|HasCategory| |#2| (QUOTE (-447))) (-3322 (|HasCategory| |#2| (QUOTE (-170))) (|HasCategory| |#2| (QUOTE (-447))) (|HasCategory| |#2| (QUOTE (-550))) (|HasCategory| |#2| (QUOTE (-896)))) (-3322 (|HasCategory| |#2| (QUOTE (-447))) (|HasCategory| |#2| (QUOTE (-550))) (|HasCategory| |#2| (QUOTE (-896)))) (-3322 (|HasCategory| |#2| (QUOTE (-447))) (|HasCategory| |#2| (QUOTE (-896)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-896)))) (-3322 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-896)))) (|HasCategory| |#2| (QUOTE (-146))))) 
(-238 |n| R M S) 
((|constructor| (NIL "This constructor provides a direct product type with a left matrix-module view."))) 
((-4502 -3322 (-1367 (|has| |#4| (-1039)) (|has| |#4| (-221))) (-1367 (|has| |#4| (-1039)) (|has| |#4| (-887 (-1153)))) (|has| |#4| (-6 -4502)) (-1367 (|has| |#4| (-1039)) (|has| |#4| (-622 (-560))))) (-4499 |has| |#4| (-1039)) (-4500 |has| |#4| (-1039)) ((-4507 "*") |has| |#4| (-170)) (-4505 . T)) 
((|HasCategory| |#4| (QUOTE (-359))) (|HasCategory| |#4| (QUOTE (-1039))) (|HasCategory| |#4| (QUOTE (-780))) (|HasCategory| |#4| (QUOTE (-832))) (-3322 (|HasCategory| |#4| (QUOTE (-780))) (|HasCategory| |#4| (QUOTE (-832)))) (|HasCategory| |#4| (QUOTE (-708))) (|HasCategory| |#4| (QUOTE (-170))) (-3322 (|HasCategory| |#4| (QUOTE (-170))) (|HasCategory| |#4| (QUOTE (-359))) (|HasCategory| |#4| (QUOTE (-1039)))) (-3322 (|HasCategory| |#4| (QUOTE (-170))) (|HasCategory| |#4| (QUOTE (-359)))) (-3322 (|HasCategory| |#4| (QUOTE (-170))) (|HasCategory| |#4| (QUOTE (-1039)))) (|HasCategory| |#4| (QUOTE (-364))) (|HasCategory| |#4| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#4| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#4| (QUOTE (-221))) (-3322 (|HasCategory| |#4| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#4| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#4| (QUOTE (-170))) (|HasCategory| |#4| (QUOTE (-221))) (|HasCategory| |#4| (QUOTE (-1039)))) (|HasCategory| |#4| (QUOTE (-1082))) (|HasCategory| (-560) (QUOTE (-834))) (-12 (|HasCategory| |#4| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#4| (QUOTE (-1039)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#4| (QUOTE (-1039)))) (-12 (|HasCategory| |#4| (QUOTE (-221))) (|HasCategory| |#4| (QUOTE (-1039)))) (-3322 (-12 (|HasCategory| |#4| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#4| (QUOTE (-1039)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#4| (QUOTE (-1039)))) (-12 (|HasCategory| |#4| (QUOTE (-221))) (|HasCategory| |#4| (QUOTE (-1039)))) (|HasCategory| |#4| (QUOTE (-708)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#4| (QUOTE (-1082)))) (-3322 (-12 (|HasCategory| |#4| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#4| (LIST (QUOTE -1029) (QUOTE (-560))))) (-12 (|HasCategory| |#4| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#4| (LIST (QUOTE -1029) (QUOTE (-560))))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#4| (QUOTE (-170)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#4| (QUOTE (-221)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#4| (QUOTE (-359)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#4| (QUOTE (-364)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#4| (QUOTE (-708)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#4| (QUOTE (-780)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#4| (QUOTE (-832)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#4| (QUOTE (-1039)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#4| (QUOTE (-1082))))) (-3322 (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#4| (QUOTE (-1082)))) (|HasCategory| |#4| (QUOTE (-1039)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#4| (QUOTE (-1082)))) (-3322 (-12 (|HasCategory| |#4| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#4| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560)))))) (-12 (|HasCategory| |#4| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#4| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560)))))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#4| (QUOTE (-170)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#4| (QUOTE (-221)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#4| (QUOTE (-359)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#4| (QUOTE (-364)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#4| (QUOTE (-708)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#4| (QUOTE (-780)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#4| (QUOTE (-832)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#4| (QUOTE (-1039)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#4| (QUOTE (-1082))))) (-3322 (|HasAttribute| |#4| (QUOTE -4502)) (-12 (|HasCategory| |#4| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#4| (QUOTE (-1039)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#4| (QUOTE (-1039)))) (-12 (|HasCategory| |#4| (QUOTE (-221))) (|HasCategory| |#4| (QUOTE (-1039))))) (|HasCategory| |#4| (QUOTE (-137))) (|HasCategory| |#4| (QUOTE (-25))) (-12 (|HasCategory| |#4| (LIST (QUOTE -298) (|devaluate| |#4|))) (|HasCategory| |#4| (QUOTE (-1082)))) (-3322 (-12 (|HasCategory| |#4| (LIST (QUOTE -298) (|devaluate| |#4|))) (|HasCategory| |#4| (LIST (QUOTE -622) (QUOTE (-560))))) (-12 (|HasCategory| |#4| (LIST (QUOTE -298) (|devaluate| |#4|))) (|HasCategory| |#4| (LIST (QUOTE -887) (QUOTE (-1153))))) (-12 (|HasCategory| |#4| (LIST (QUOTE -298) (|devaluate| |#4|))) (|HasCategory| |#4| (QUOTE (-170)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -298) (|devaluate| |#4|))) (|HasCategory| |#4| (QUOTE (-221)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -298) (|devaluate| |#4|))) (|HasCategory| |#4| (QUOTE (-359)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -298) (|devaluate| |#4|))) (|HasCategory| |#4| (QUOTE (-364)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -298) (|devaluate| |#4|))) (|HasCategory| |#4| (QUOTE (-708)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -298) (|devaluate| |#4|))) (|HasCategory| |#4| (QUOTE (-780)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -298) (|devaluate| |#4|))) (|HasCategory| |#4| (QUOTE (-832)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -298) (|devaluate| |#4|))) (|HasCategory| |#4| (QUOTE (-1039)))) (-12 (|HasCategory| |#4| (LIST (QUOTE -298) (|devaluate| |#4|))) (|HasCategory| |#4| (QUOTE (-1082)))))) 
(-239 |n| R S) 
((|constructor| (NIL "This constructor provides a direct product of \\spad{R}-modules with an \\spad{R}-module view."))) 
((-4502 -3322 (-1367 (|has| |#3| (-1039)) (|has| |#3| (-221))) (-1367 (|has| |#3| (-1039)) (|has| |#3| (-887 (-1153)))) (|has| |#3| (-6 -4502)) (-1367 (|has| |#3| (-1039)) (|has| |#3| (-622 (-560))))) (-4499 |has| |#3| (-1039)) (-4500 |has| |#3| (-1039)) ((-4507 "*") |has| |#3| (-170)) (-4505 . T)) 
((|HasCategory| |#3| (QUOTE (-359))) (|HasCategory| |#3| (QUOTE (-1039))) (|HasCategory| |#3| (QUOTE (-780))) (|HasCategory| |#3| (QUOTE (-832))) (-3322 (|HasCategory| |#3| (QUOTE (-780))) (|HasCategory| |#3| (QUOTE (-832)))) (|HasCategory| |#3| (QUOTE (-708))) (|HasCategory| |#3| (QUOTE (-170))) (-3322 (|HasCategory| |#3| (QUOTE (-170))) (|HasCategory| |#3| (QUOTE (-359))) (|HasCategory| |#3| (QUOTE (-1039)))) (-3322 (|HasCategory| |#3| (QUOTE (-170))) (|HasCategory| |#3| (QUOTE (-359)))) (-3322 (|HasCategory| |#3| (QUOTE (-170))) (|HasCategory| |#3| (QUOTE (-1039)))) (|HasCategory| |#3| (QUOTE (-364))) (|HasCategory| |#3| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#3| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#3| (QUOTE (-221))) (-3322 (|HasCategory| |#3| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#3| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#3| (QUOTE (-170))) (|HasCategory| |#3| (QUOTE (-221))) (|HasCategory| |#3| (QUOTE (-1039)))) (|HasCategory| |#3| (QUOTE (-1082))) (|HasCategory| (-560) (QUOTE (-834))) (-12 (|HasCategory| |#3| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#3| (QUOTE (-1039)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#3| (QUOTE (-1039)))) (-12 (|HasCategory| |#3| (QUOTE (-221))) (|HasCategory| |#3| (QUOTE (-1039)))) (-3322 (-12 (|HasCategory| |#3| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#3| (QUOTE (-1039)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#3| (QUOTE (-1039)))) (-12 (|HasCategory| |#3| (QUOTE (-221))) (|HasCategory| |#3| (QUOTE (-1039)))) (|HasCategory| |#3| (QUOTE (-708)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#3| (QUOTE (-1082)))) (-3322 (-12 (|HasCategory| |#3| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-560))))) (-12 (|HasCategory| |#3| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-560))))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#3| (QUOTE (-170)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#3| (QUOTE (-221)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#3| (QUOTE (-359)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#3| (QUOTE (-364)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#3| (QUOTE (-708)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#3| (QUOTE (-780)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#3| (QUOTE (-832)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#3| (QUOTE (-1039)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#3| (QUOTE (-1082))))) (-3322 (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#3| (QUOTE (-1082)))) (|HasCategory| |#3| (QUOTE (-1039)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#3| (QUOTE (-1082)))) (-3322 (-12 (|HasCategory| |#3| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#3| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560)))))) (-12 (|HasCategory| |#3| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#3| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560)))))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#3| (QUOTE (-170)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#3| (QUOTE (-221)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#3| (QUOTE (-359)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#3| (QUOTE (-364)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#3| (QUOTE (-708)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#3| (QUOTE (-780)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#3| (QUOTE (-832)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#3| (QUOTE (-1039)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#3| (QUOTE (-1082))))) (-3322 (|HasAttribute| |#3| (QUOTE -4502)) (-12 (|HasCategory| |#3| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#3| (QUOTE (-1039)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#3| (QUOTE (-1039)))) (-12 (|HasCategory| |#3| (QUOTE (-221))) (|HasCategory| |#3| (QUOTE (-1039))))) (|HasCategory| |#3| (QUOTE (-137))) (|HasCategory| |#3| (QUOTE (-25))) (-12 (|HasCategory| |#3| (LIST (QUOTE -298) (|devaluate| |#3|))) (|HasCategory| |#3| (QUOTE (-1082)))) (-3322 (-12 (|HasCategory| |#3| (LIST (QUOTE -298) (|devaluate| |#3|))) (|HasCategory| |#3| (LIST (QUOTE -622) (QUOTE (-560))))) (-12 (|HasCategory| |#3| (LIST (QUOTE -298) (|devaluate| |#3|))) (|HasCategory| |#3| (LIST (QUOTE -887) (QUOTE (-1153))))) (-12 (|HasCategory| |#3| (LIST (QUOTE -298) (|devaluate| |#3|))) (|HasCategory| |#3| (QUOTE (-170)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -298) (|devaluate| |#3|))) (|HasCategory| |#3| (QUOTE (-221)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -298) (|devaluate| |#3|))) (|HasCategory| |#3| (QUOTE (-359)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -298) (|devaluate| |#3|))) (|HasCategory| |#3| (QUOTE (-364)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -298) (|devaluate| |#3|))) (|HasCategory| |#3| (QUOTE (-708)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -298) (|devaluate| |#3|))) (|HasCategory| |#3| (QUOTE (-780)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -298) (|devaluate| |#3|))) (|HasCategory| |#3| (QUOTE (-832)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -298) (|devaluate| |#3|))) (|HasCategory| |#3| (QUOTE (-1039)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -298) (|devaluate| |#3|))) (|HasCategory| |#3| (QUOTE (-1082)))))) 
(-240 A R S V E) 
((|constructor| (NIL "\\spadtype{DifferentialPolynomialCategory} is a category constructor specifying basic functions in an ordinary differential polynomial ring with a given ordered set of differential indeterminates. In addition,{} it implements defaults for the basic functions. The functions \\spadfun{order} and \\spadfun{weight} are extended from the set of derivatives of differential indeterminates to the set of differential polynomials. Other operations provided on differential polynomials are \\spadfun{leader},{} \\spadfun{initial},{} \\spadfun{separant},{} \\spadfun{differentialVariables},{} and \\spadfun{isobaric?}. Furthermore,{} if the ground ring is a differential ring,{} then evaluation (substitution of differential indeterminates by elements of the ground ring or by differential polynomials) is provided by \\spadfun{eval}. A convenient way of referencing derivatives is provided by the functions \\spadfun{makeVariable}. \\blankline To construct a domain using this constructor,{} one needs to provide a ground ring \\spad{R},{} an ordered set \\spad{S} of differential indeterminates,{} a ranking \\spad{V} on the set of derivatives of the differential indeterminates,{} and a set \\spad{E} of exponents in bijection with the set of differential monomials in the given differential indeterminates.")) (|separant| (($ $) "\\spad{separant(p)} returns the partial derivative of the differential polynomial \\spad{p} with respect to its leader.")) (|initial| (($ $) "\\spad{initial(p)} returns the leading coefficient when the differential polynomial \\spad{p} is written as a univariate polynomial in its leader.")) (|leader| ((|#4| $) "\\spad{leader(p)} returns the derivative of the highest rank appearing in the differential polynomial \\spad{p} Note that an error occurs if \\spad{p} is in the ground ring.")) (|isobaric?| (((|Boolean|) $) "\\spad{isobaric?(p)} returns \\spad{true} if every differential monomial appearing in the differential polynomial \\spad{p} has same weight,{} and returns \\spad{false} otherwise.")) (|weight| (((|NonNegativeInteger|) $ |#3|) "\\spad{weight(p,{} s)} returns the maximum weight of all differential monomials appearing in the differential polynomial \\spad{p} when \\spad{p} is viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.") (((|NonNegativeInteger|) $) "\\spad{weight(p)} returns the maximum weight of all differential monomials appearing in the differential polynomial \\spad{p}.")) (|weights| (((|List| (|NonNegativeInteger|)) $ |#3|) "\\spad{weights(p,{} s)} returns a list of weights of differential monomials appearing in the differential polynomial \\spad{p} when \\spad{p} is viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.") (((|List| (|NonNegativeInteger|)) $) "\\spad{weights(p)} returns a list of weights of differential monomials appearing in differential polynomial \\spad{p}.")) (|degree| (((|NonNegativeInteger|) $ |#3|) "\\spad{degree(p,{} s)} returns the maximum degree of the differential polynomial \\spad{p} viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(p)} returns the order of the differential polynomial \\spad{p},{} which is the maximum number of differentiations of a differential indeterminate,{} among all those appearing in \\spad{p}.") (((|NonNegativeInteger|) $ |#3|) "\\spad{order(p,{}s)} returns the order of the differential polynomial \\spad{p} in differential indeterminate \\spad{s}.")) (|differentialVariables| (((|List| |#3|) $) "\\spad{differentialVariables(p)} returns a list of differential indeterminates occurring in a differential polynomial \\spad{p}.")) (|makeVariable| (((|Mapping| $ (|NonNegativeInteger|)) $) "\\spad{makeVariable(p)} views \\spad{p} as an element of a differential ring,{} in such a way that the \\spad{n}-th derivative of \\spad{p} may be simply referenced as \\spad{z}.\\spad{n} where \\spad{z} \\spad{:=} makeVariable(\\spad{p}). Note that In the interpreter,{} \\spad{z} is given as an internal map,{} which may be ignored.") (((|Mapping| $ (|NonNegativeInteger|)) |#3|) "\\spad{makeVariable(s)} views \\spad{s} as a differential indeterminate,{} in such a way that the \\spad{n}-th derivative of \\spad{s} may be simply referenced as \\spad{z}.\\spad{n} where \\spad{z} :=makeVariable(\\spad{s}). Note that In the interpreter,{} \\spad{z} is given as an internal map,{} which may be ignored."))) 
NIL 
((|HasCategory| |#2| (QUOTE (-221)))) 
(-241 R S V E) 
((|constructor| (NIL "\\spadtype{DifferentialPolynomialCategory} is a category constructor specifying basic functions in an ordinary differential polynomial ring with a given ordered set of differential indeterminates. In addition,{} it implements defaults for the basic functions. The functions \\spadfun{order} and \\spadfun{weight} are extended from the set of derivatives of differential indeterminates to the set of differential polynomials. Other operations provided on differential polynomials are \\spadfun{leader},{} \\spadfun{initial},{} \\spadfun{separant},{} \\spadfun{differentialVariables},{} and \\spadfun{isobaric?}. Furthermore,{} if the ground ring is a differential ring,{} then evaluation (substitution of differential indeterminates by elements of the ground ring or by differential polynomials) is provided by \\spadfun{eval}. A convenient way of referencing derivatives is provided by the functions \\spadfun{makeVariable}. \\blankline To construct a domain using this constructor,{} one needs to provide a ground ring \\spad{R},{} an ordered set \\spad{S} of differential indeterminates,{} a ranking \\spad{V} on the set of derivatives of the differential indeterminates,{} and a set \\spad{E} of exponents in bijection with the set of differential monomials in the given differential indeterminates.")) (|separant| (($ $) "\\spad{separant(p)} returns the partial derivative of the differential polynomial \\spad{p} with respect to its leader.")) (|initial| (($ $) "\\spad{initial(p)} returns the leading coefficient when the differential polynomial \\spad{p} is written as a univariate polynomial in its leader.")) (|leader| ((|#3| $) "\\spad{leader(p)} returns the derivative of the highest rank appearing in the differential polynomial \\spad{p} Note that an error occurs if \\spad{p} is in the ground ring.")) (|isobaric?| (((|Boolean|) $) "\\spad{isobaric?(p)} returns \\spad{true} if every differential monomial appearing in the differential polynomial \\spad{p} has same weight,{} and returns \\spad{false} otherwise.")) (|weight| (((|NonNegativeInteger|) $ |#2|) "\\spad{weight(p,{} s)} returns the maximum weight of all differential monomials appearing in the differential polynomial \\spad{p} when \\spad{p} is viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.") (((|NonNegativeInteger|) $) "\\spad{weight(p)} returns the maximum weight of all differential monomials appearing in the differential polynomial \\spad{p}.")) (|weights| (((|List| (|NonNegativeInteger|)) $ |#2|) "\\spad{weights(p,{} s)} returns a list of weights of differential monomials appearing in the differential polynomial \\spad{p} when \\spad{p} is viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.") (((|List| (|NonNegativeInteger|)) $) "\\spad{weights(p)} returns a list of weights of differential monomials appearing in differential polynomial \\spad{p}.")) (|degree| (((|NonNegativeInteger|) $ |#2|) "\\spad{degree(p,{} s)} returns the maximum degree of the differential polynomial \\spad{p} viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(p)} returns the order of the differential polynomial \\spad{p},{} which is the maximum number of differentiations of a differential indeterminate,{} among all those appearing in \\spad{p}.") (((|NonNegativeInteger|) $ |#2|) "\\spad{order(p,{}s)} returns the order of the differential polynomial \\spad{p} in differential indeterminate \\spad{s}.")) (|differentialVariables| (((|List| |#2|) $) "\\spad{differentialVariables(p)} returns a list of differential indeterminates occurring in a differential polynomial \\spad{p}.")) (|makeVariable| (((|Mapping| $ (|NonNegativeInteger|)) $) "\\spad{makeVariable(p)} views \\spad{p} as an element of a differential ring,{} in such a way that the \\spad{n}-th derivative of \\spad{p} may be simply referenced as \\spad{z}.\\spad{n} where \\spad{z} \\spad{:=} makeVariable(\\spad{p}). Note that In the interpreter,{} \\spad{z} is given as an internal map,{} which may be ignored.") (((|Mapping| $ (|NonNegativeInteger|)) |#2|) "\\spad{makeVariable(s)} views \\spad{s} as a differential indeterminate,{} in such a way that the \\spad{n}-th derivative of \\spad{s} may be simply referenced as \\spad{z}.\\spad{n} where \\spad{z} :=makeVariable(\\spad{s}). Note that In the interpreter,{} \\spad{z} is given as an internal map,{} which may be ignored."))) 
(((-4507 "*") |has| |#1| (-170)) (-4498 |has| |#1| (-550)) (-4503 |has| |#1| (-6 -4503)) (-4500 . T) (-4499 . T) (-4502 . T)) 
NIL 
(-242 S) 
((|constructor| (NIL "A dequeue is a doubly ended stack,{} that is,{} a bag where first items inserted are the first items extracted,{} at either the front or the back end of the data structure.")) (|reverse!| (($ $) "\\spad{reverse!(d)} destructively replaces \\spad{d} by its reverse dequeue,{} \\spadignore{i.e.} the top (front) element is now the bottom (back) element,{} and so on.")) (|extractBottom!| ((|#1| $) "\\spad{extractBottom!(d)} destructively extracts the bottom (back) element from the dequeue \\spad{d}. Error: if \\spad{d} is empty.")) (|extractTop!| ((|#1| $) "\\spad{extractTop!(d)} destructively extracts the top (front) element from the dequeue \\spad{d}. Error: if \\spad{d} is empty.")) (|insertBottom!| ((|#1| |#1| $) "\\spad{insertBottom!(x,{}d)} destructively inserts \\spad{x} into the dequeue \\spad{d} at the bottom (back) of the dequeue.")) (|insertTop!| ((|#1| |#1| $) "\\spad{insertTop!(x,{}d)} destructively inserts \\spad{x} into the dequeue \\spad{d},{} that is,{} at the top (front) of the dequeue. The element previously at the top of the dequeue becomes the second in the dequeue,{} and so on.")) (|bottom!| ((|#1| $) "\\spad{bottom!(d)} returns the element at the bottom (back) of the dequeue.")) (|top!| ((|#1| $) "\\spad{top!(d)} returns the element at the top (front) of the dequeue.")) (|height| (((|NonNegativeInteger|) $) "\\spad{height(d)} returns the number of elements in dequeue \\spad{d}. Note that \\axiom{height(\\spad{d}) = \\# \\spad{d}}.")) (|dequeue| (($ (|List| |#1|)) "\\spad{dequeue([x,{}y,{}...,{}z])} creates a dequeue with first (top or front) element \\spad{x},{} second element \\spad{y},{}...,{}and last (bottom or back) element \\spad{z}.") (($) "\\spad{dequeue()}\\$\\spad{D} creates an empty dequeue of type \\spad{D}."))) 
((-4505 . T) (-4506 . T) (-3576 . T)) 
NIL 
(-243) 
((|constructor| (NIL "TopLevelDrawFunctionsForCompiledFunctions provides top level functions for drawing graphics of expressions.")) (|recolor| (((|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) "\\spad{recolor()},{} uninteresting to top level user; exported in order to compile package.")) (|makeObject| (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSurface| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{makeObject(surface(f,{}g,{}h),{}a..b,{}c..d,{}l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{x = f(u,{}v)},{} \\spad{y = g(u,{}v)},{} \\spad{z = h(u,{}v)} as \\spad{u} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{v} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSurface| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(surface(f,{}g,{}h),{}a..b,{}c..d,{}l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{x = f(u,{}v)},{} \\spad{y = g(u,{}v)},{} \\spad{z = h(u,{}v)} as \\spad{u} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{v} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{makeObject(f,{}a..b,{}c..d,{}l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{f(u,{}v)} as \\spad{u} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{v} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(f,{}a..b,{}c..d,{}l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{f(u,{}v)} as \\spad{u} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{v} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}; The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{makeObject(f,{}a..b,{}c..d)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of \\spad{z = f(x,{}y)} as \\spad{x} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{y} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(f,{}a..b,{}c..d,{}l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of \\spad{z = f(x,{}y)} as \\spad{x} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{y} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)},{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|Float|))) "\\spad{makeObject(sp,{}curve(f,{}g,{}h),{}a..b)} returns the space \\spad{sp} of the domain \\spadtype{ThreeSpace} with the addition of the graph of the parametric curve \\spad{x = f(t),{} y = g(t),{} z = h(t)} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(curve(f,{}g,{}h),{}a..b,{}l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric curve \\spad{x = f(t),{} y = g(t),{} z = h(t)} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSpaceCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|))) "\\spad{makeObject(sp,{}curve(f,{}g,{}h),{}a..b)} returns the space \\spad{sp} of the domain \\spadtype{ThreeSpace} with the addition of the graph of the parametric curve \\spad{x = f(t),{} y = g(t),{} z = h(t)} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSpaceCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(curve(f,{}g,{}h),{}a..b,{}l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric curve \\spad{x = f(t),{} y = g(t),{} z = h(t)} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}; The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.")) (|draw| (((|ThreeDimensionalViewport|) (|ParametricSurface| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{draw(surface(f,{}g,{}h),{}a..b,{}c..d)} draws the graph of the parametric surface \\spad{x = f(u,{}v)},{} \\spad{y = g(u,{}v)},{} \\spad{z = h(u,{}v)} as \\spad{u} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{v} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}.") (((|ThreeDimensionalViewport|) (|ParametricSurface| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(surface(f,{}g,{}h),{}a..b,{}c..d)} draws the graph of the parametric surface \\spad{x = f(u,{}v)},{} \\spad{y = g(u,{}v)},{} \\spad{z = h(u,{}v)} as \\spad{u} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{v} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}; The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{draw(f,{}a..b,{}c..d)} draws the graph of the parametric surface \\spad{f(u,{}v)} as \\spad{u} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{v} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)} The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f,{}a..b,{}c..d)} draws the graph of the parametric surface \\spad{f(u,{}v)} as \\spad{u} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{v} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{draw(f,{}a..b,{}c..d)} draws the graph of \\spad{z = f(x,{}y)} as \\spad{x} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{y} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}.") (((|ThreeDimensionalViewport|) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f,{}a..b,{}c..d,{}l)} draws the graph of \\spad{z = f(x,{}y)} as \\spad{x} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{y} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}. and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|Float|))) "\\spad{draw(f,{}a..b,{}l)} draws the graph of the parametric curve \\spad{f} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}.") (((|ThreeDimensionalViewport|) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f,{}a..b,{}l)} draws the graph of the parametric curve \\spad{f} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|ParametricSpaceCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|))) "\\spad{draw(curve(f,{}g,{}h),{}a..b,{}l)} draws the graph of the parametric curve \\spad{x = f(t),{} y = g(t),{} z = h(t)} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}.") (((|ThreeDimensionalViewport|) (|ParametricSpaceCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(curve(f,{}g,{}h),{}a..b,{}l)} draws the graph of the parametric curve \\spad{x = f(t),{} y = g(t),{} z = h(t)} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) (|ParametricPlaneCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|))) "\\spad{draw(curve(f,{}g),{}a..b)} draws the graph of the parametric curve \\spad{x = f(t),{} y = g(t)} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}.") (((|TwoDimensionalViewport|) (|ParametricPlaneCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(curve(f,{}g),{}a..b,{}l)} draws the graph of the parametric curve \\spad{x = f(t),{} y = g(t)} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|))) "\\spad{draw(f,{}a..b)} draws the graph of \\spad{y = f(x)} as \\spad{x} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}.") (((|TwoDimensionalViewport|) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f,{}a..b,{}l)} draws the graph of \\spad{y = f(x)} as \\spad{x} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied."))) 
NIL 
NIL 
(-244 R |Ex|) 
((|constructor| (NIL "TopLevelDrawFunctionsForAlgebraicCurves provides top level functions for drawing non-singular algebraic curves.")) (|draw| (((|TwoDimensionalViewport|) (|Equation| |#2|) (|Symbol|) (|Symbol|) (|List| (|DrawOption|))) "\\spad{draw(f(x,{}y) = g(x,{}y),{}x,{}y,{}l)} draws the graph of a polynomial equation. The list \\spad{l} of draw options must specify a region in the plane in which the curve is to sketched."))) 
NIL 
NIL 
(-245) 
((|constructor| (NIL "\\axiomType{DrawComplex} provides some facilities for drawing complex functions.")) (|setClipValue| (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{setClipValue(x)} sets to \\spad{x} the maximum value to plot when drawing complex functions. Returns \\spad{x}.")) (|setImagSteps| (((|Integer|) (|Integer|)) "\\spad{setImagSteps(i)} sets to \\spad{i} the number of steps to use in the imaginary direction when drawing complex functions. Returns \\spad{i}.")) (|setRealSteps| (((|Integer|) (|Integer|)) "\\spad{setRealSteps(i)} sets to \\spad{i} the number of steps to use in the real direction when drawing complex functions. Returns \\spad{i}.")) (|drawComplexVectorField| (((|ThreeDimensionalViewport|) (|Mapping| (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{drawComplexVectorField(f,{}rRange,{}iRange)} draws a complex vector field using arrows on the \\spad{x--y} plane. These vector fields should be viewed from the top by pressing the \"XY\" translate button on the 3-\\spad{d} viewport control panel. Sample call: \\indented{3}{\\spad{f z == sin z}} \\indented{3}{\\spad{drawComplexVectorField(f,{} -2..2,{} -2..2)}} Parameter descriptions: \\indented{2}{\\spad{f} : the function to draw} \\indented{2}{\\spad{rRange} : the range of the real values} \\indented{2}{\\spad{iRange} : the range of the imaginary values} Call the functions \\axiomFunFrom{setRealSteps}{DrawComplex} and \\axiomFunFrom{setImagSteps}{DrawComplex} to change the number of steps used in each direction.")) (|drawComplex| (((|ThreeDimensionalViewport|) (|Mapping| (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Boolean|)) "\\spad{drawComplex(f,{}rRange,{}iRange,{}arrows?)} draws a complex function as a height field. It uses the complex norm as the height and the complex argument as the color. It will optionally draw arrows on the surface indicating the direction of the complex value. Sample call: \\indented{2}{\\spad{f z == exp(1/z)}} \\indented{2}{\\spad{drawComplex(f,{} 0.3..3,{} 0..2*\\%\\spad{pi},{} false)}} Parameter descriptions: \\indented{2}{\\spad{f:}\\space{2}the function to draw} \\indented{2}{\\spad{rRange} : the range of the real values} \\indented{2}{\\spad{iRange} : the range of imaginary values} \\indented{2}{\\spad{arrows?} : a flag indicating whether to draw the phase arrows for \\spad{f}} Call the functions \\axiomFunFrom{setRealSteps}{DrawComplex} and \\axiomFunFrom{setImagSteps}{DrawComplex} to change the number of steps used in each direction."))) 
NIL 
NIL 
(-246 R) 
((|constructor| (NIL "Hack for the draw interface. DrawNumericHack provides a \"coercion\" from something of the form \\spad{x = a..b} where \\spad{a} and \\spad{b} are formal expressions to a binding of the form \\spad{x = c..d} where \\spad{c} and \\spad{d} are the numerical values of \\spad{a} and \\spad{b}. This \"coercion\" fails if \\spad{a} and \\spad{b} contains symbolic variables,{} but is meant for expressions involving \\%\\spad{pi}. Note that this package is meant for internal use only.")) (|coerce| (((|SegmentBinding| (|Float|)) (|SegmentBinding| (|Expression| |#1|))) "\\spad{coerce(x = a..b)} returns \\spad{x = c..d} where \\spad{c} and \\spad{d} are the numerical values of \\spad{a} and \\spad{b}."))) 
NIL 
NIL 
(-247 |Ex|) 
((|constructor| (NIL "TopLevelDrawFunctions provides top level functions for drawing graphics of expressions.")) (|makeObject| (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSurface| |#1|) (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|))) "\\spad{makeObject(surface(f(u,{}v),{}g(u,{}v),{}h(u,{}v)),{}u = a..b,{}v = c..d)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{x = f(u,{}v)},{} \\spad{y = g(u,{}v)},{} \\spad{z = h(u,{}v)} as \\spad{u} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{v} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}; \\spad{h(t)} is the default title.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSurface| |#1|) (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(surface(f(u,{}v),{}g(u,{}v),{}h(u,{}v)),{}u = a..b,{}v = c..d,{}l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{x = f(u,{}v)},{} \\spad{y = g(u,{}v)},{} \\spad{z = h(u,{}v)} as \\spad{u} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{v} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}; \\spad{h(t)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) |#1| (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|))) "\\spad{makeObject(f(x,{}y),{}x = a..b,{}y = c..d)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of \\spad{z = f(x,{}y)} as \\spad{x} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{y} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}; \\spad{f(x,{}y)} appears as the default title.") (((|ThreeSpace| (|DoubleFloat|)) |#1| (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(f(x,{}y),{}x = a..b,{}y = c..d,{}l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of \\spad{z = f(x,{}y)} as \\spad{x} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{y} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}; \\spad{f(x,{}y)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSpaceCurve| |#1|) (|SegmentBinding| (|Float|))) "\\spad{makeObject(curve(f(t),{}g(t),{}h(t)),{}t = a..b)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}; \\spad{h(t)} is the default title.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSpaceCurve| |#1|) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(curve(f(t),{}g(t),{}h(t)),{}t = a..b,{}l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}; \\spad{h(t)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.")) (|draw| (((|ThreeDimensionalViewport|) (|ParametricSurface| |#1|) (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|))) "\\spad{draw(surface(f(u,{}v),{}g(u,{}v),{}h(u,{}v)),{}u = a..b,{}v = c..d)} draws the graph of the parametric surface \\spad{x = f(u,{}v)},{} \\spad{y = g(u,{}v)},{} \\spad{z = h(u,{}v)} as \\spad{u} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{v} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}; \\spad{h(t)} is the default title.") (((|ThreeDimensionalViewport|) (|ParametricSurface| |#1|) (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(surface(f(u,{}v),{}g(u,{}v),{}h(u,{}v)),{}u = a..b,{}v = c..d,{}l)} draws the graph of the parametric surface \\spad{x = f(u,{}v)},{} \\spad{y = g(u,{}v)},{} \\spad{z = h(u,{}v)} as \\spad{u} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{v} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}; \\spad{h(t)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) |#1| (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|))) "\\spad{draw(f(x,{}y),{}x = a..b,{}y = c..d)} draws the graph of \\spad{z = f(x,{}y)} as \\spad{x} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{y} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}; \\spad{f(x,{}y)} appears in the title bar.") (((|ThreeDimensionalViewport|) |#1| (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f(x,{}y),{}x = a..b,{}y = c..d,{}l)} draws the graph of \\spad{z = f(x,{}y)} as \\spad{x} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)} and \\spad{y} ranges from \\spad{min(c,{}d)} to \\spad{max(c,{}d)}; \\spad{f(x,{}y)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|ParametricSpaceCurve| |#1|) (|SegmentBinding| (|Float|))) "\\spad{draw(curve(f(t),{}g(t),{}h(t)),{}t = a..b)} draws the graph of the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}; \\spad{h(t)} is the default title.") (((|ThreeDimensionalViewport|) (|ParametricSpaceCurve| |#1|) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(curve(f(t),{}g(t),{}h(t)),{}t = a..b,{}l)} draws the graph of the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}; \\spad{h(t)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) (|ParametricPlaneCurve| |#1|) (|SegmentBinding| (|Float|))) "\\spad{draw(curve(f(t),{}g(t)),{}t = a..b)} draws the graph of the parametric curve \\spad{x = f(t),{} y = g(t)} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}; \\spad{(f(t),{}g(t))} appears in the title bar.") (((|TwoDimensionalViewport|) (|ParametricPlaneCurve| |#1|) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(curve(f(t),{}g(t)),{}t = a..b,{}l)} draws the graph of the parametric curve \\spad{x = f(t),{} y = g(t)} as \\spad{t} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}; \\spad{(f(t),{}g(t))} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) |#1| (|SegmentBinding| (|Float|))) "\\spad{draw(f(x),{}x = a..b)} draws the graph of \\spad{y = f(x)} as \\spad{x} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}; \\spad{f(x)} appears in the title bar.") (((|TwoDimensionalViewport|) |#1| (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f(x),{}x = a..b,{}l)} draws the graph of \\spad{y = f(x)} as \\spad{x} ranges from \\spad{min(a,{}b)} to \\spad{max(a,{}b)}; \\spad{f(x)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied."))) 
NIL 
NIL 
(-248) 
((|constructor| (NIL "TopLevelDrawFunctionsForPoints provides top level functions for drawing curves and surfaces described by sets of points.")) (|draw| (((|ThreeDimensionalViewport|) (|List| (|DoubleFloat|)) (|List| (|DoubleFloat|)) (|List| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{draw(lx,{}ly,{}lz,{}l)} draws the surface constructed by projecting the values in the \\axiom{\\spad{lz}} list onto the rectangular grid formed by the The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|List| (|DoubleFloat|)) (|List| (|DoubleFloat|)) (|List| (|DoubleFloat|))) "\\spad{draw(lx,{}ly,{}lz)} draws the surface constructed by projecting the values in the \\axiom{\\spad{lz}} list onto the rectangular grid formed by the \\axiom{\\spad{lx} \\spad{x} \\spad{ly}}.") (((|TwoDimensionalViewport|) (|List| (|Point| (|DoubleFloat|))) (|List| (|DrawOption|))) "\\spad{draw(lp,{}l)} plots the curve constructed from the list of points \\spad{lp}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) (|List| (|Point| (|DoubleFloat|)))) "\\spad{draw(lp)} plots the curve constructed from the list of points \\spad{lp}.") (((|TwoDimensionalViewport|) (|List| (|DoubleFloat|)) (|List| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{draw(lx,{}ly,{}l)} plots the curve constructed of points (\\spad{x},{}\\spad{y}) for \\spad{x} in \\spad{lx} for \\spad{y} in \\spad{ly}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) (|List| (|DoubleFloat|)) (|List| (|DoubleFloat|))) "\\spad{draw(lx,{}ly)} plots the curve constructed of points (\\spad{x},{}\\spad{y}) for \\spad{x} in \\spad{lx} for \\spad{y} in \\spad{ly}."))) 
NIL 
NIL 
(-249) 
((|constructor| (NIL "This package has no description")) (|units| (((|List| (|Float|)) (|List| (|DrawOption|)) (|List| (|Float|))) "\\spad{units(l,{}u)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{unit}. If the option does not exist the value,{} \\spad{u} is returned.")) (|coord| (((|Mapping| (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|))) (|List| (|DrawOption|)) (|Mapping| (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)))) "\\spad{coord(l,{}p)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{coord}. If the option does not exist the value,{} \\spad{p} is returned.")) (|tubeRadius| (((|Float|) (|List| (|DrawOption|)) (|Float|)) "\\spad{tubeRadius(l,{}n)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{tubeRadius}. If the option does not exist the value,{} \\spad{n} is returned.")) (|tubePoints| (((|PositiveInteger|) (|List| (|DrawOption|)) (|PositiveInteger|)) "\\spad{tubePoints(l,{}n)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{tubePoints}. If the option does not exist the value,{} \\spad{n} is returned.")) (|space| (((|ThreeSpace| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{space(l)} takes a list of draw options,{} \\spad{l},{} and checks to see if it contains the option \\spad{space}. If the the option doesn\\spad{'t} exist,{} then an empty space is returned.")) (|var2Steps| (((|PositiveInteger|) (|List| (|DrawOption|)) (|PositiveInteger|)) "\\spad{var2Steps(l,{}n)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{var2Steps}. If the option does not exist the value,{} \\spad{n} is returned.")) (|var1Steps| (((|PositiveInteger|) (|List| (|DrawOption|)) (|PositiveInteger|)) "\\spad{var1Steps(l,{}n)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{var1Steps}. If the option does not exist the value,{} \\spad{n} is returned.")) (|ranges| (((|List| (|Segment| (|Float|))) (|List| (|DrawOption|)) (|List| (|Segment| (|Float|)))) "\\spad{ranges(l,{}r)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{ranges}. If the option does not exist the value,{} \\spad{r} is returned.")) (|curveColorPalette| (((|Palette|) (|List| (|DrawOption|)) (|Palette|)) "\\spad{curveColorPalette(l,{}p)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{curveColorPalette}. If the option does not exist the value,{} \\spad{p} is returned.")) (|pointColorPalette| (((|Palette|) (|List| (|DrawOption|)) (|Palette|)) "\\spad{pointColorPalette(l,{}p)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{pointColorPalette}. If the option does not exist the value,{} \\spad{p} is returned.")) (|toScale| (((|Boolean|) (|List| (|DrawOption|)) (|Boolean|)) "\\spad{toScale(l,{}b)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{toScale}. If the option does not exist the value,{} \\spad{b} is returned.")) (|style| (((|String|) (|List| (|DrawOption|)) (|String|)) "\\spad{style(l,{}s)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{style}. If the option does not exist the value,{} \\spad{s} is returned.")) (|title| (((|String|) (|List| (|DrawOption|)) (|String|)) "\\spad{title(l,{}s)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{title}. If the option does not exist the value,{} \\spad{s} is returned.")) (|viewpoint| (((|Record| (|:| |theta| (|DoubleFloat|)) (|:| |phi| (|DoubleFloat|)) (|:| |scale| (|DoubleFloat|)) (|:| |scaleX| (|DoubleFloat|)) (|:| |scaleY| (|DoubleFloat|)) (|:| |scaleZ| (|DoubleFloat|)) (|:| |deltaX| (|DoubleFloat|)) (|:| |deltaY| (|DoubleFloat|))) (|List| (|DrawOption|)) (|Record| (|:| |theta| (|DoubleFloat|)) (|:| |phi| (|DoubleFloat|)) (|:| |scale| (|DoubleFloat|)) (|:| |scaleX| (|DoubleFloat|)) (|:| |scaleY| (|DoubleFloat|)) (|:| |scaleZ| (|DoubleFloat|)) (|:| |deltaX| (|DoubleFloat|)) (|:| |deltaY| (|DoubleFloat|)))) "\\spad{viewpoint(l,{}ls)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{viewpoint}. IF the option does not exist,{} the value \\spad{ls} is returned.")) (|clipBoolean| (((|Boolean|) (|List| (|DrawOption|)) (|Boolean|)) "\\spad{clipBoolean(l,{}b)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{clipBoolean}. If the option does not exist the value,{} \\spad{b} is returned.")) (|adaptive| (((|Boolean|) (|List| (|DrawOption|)) (|Boolean|)) "\\spad{adaptive(l,{}b)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{adaptive}. If the option does not exist the value,{} \\spad{b} is returned."))) 
NIL 
NIL 
(-250 S) 
((|constructor| (NIL "This package has no description")) (|option| (((|Union| |#1| "failed") (|List| (|DrawOption|)) (|Symbol|)) "\\spad{option(l,{}s)} determines whether the indicated drawing option,{} \\spad{s},{} is contained in the list of drawing options,{} \\spad{l},{} which is defined by the draw command."))) 
NIL 
NIL 
(-251) 
((|constructor| (NIL "DrawOption allows the user to specify defaults for the creation and rendering of plots.")) (|option?| (((|Boolean|) (|List| $) (|Symbol|)) "\\spad{option?()} is not to be used at the top level; option? internally returns \\spad{true} for drawing options which are indicated in a draw command,{} or \\spad{false} for those which are not.")) (|option| (((|Union| (|Any|) "failed") (|List| $) (|Symbol|)) "\\spad{option()} is not to be used at the top level; option determines internally which drawing options are indicated in a draw command.")) (|unit| (($ (|List| (|Float|))) "\\spad{unit(lf)} will mark off the units according to the indicated list \\spad{lf}. This option is expressed in the form \\spad{unit == [f1,{}f2]}.")) (|coord| (($ (|Mapping| (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)))) "\\spad{coord(p)} specifies a change of coordinates of point \\spad{p}. This option is expressed in the form \\spad{coord == p}.")) (|tubePoints| (($ (|PositiveInteger|)) "\\spad{tubePoints(n)} specifies the number of points,{} \\spad{n},{} defining the circle which creates the tube around a 3D curve,{} the default is 6. This option is expressed in the form \\spad{tubePoints == n}.")) (|var2Steps| (($ (|PositiveInteger|)) "\\spad{var2Steps(n)} indicates the number of subdivisions,{} \\spad{n},{} of the second range variable. This option is expressed in the form \\spad{var2Steps == n}.")) (|var1Steps| (($ (|PositiveInteger|)) "\\spad{var1Steps(n)} indicates the number of subdivisions,{} \\spad{n},{} of the first range variable. This option is expressed in the form \\spad{var1Steps == n}.")) (|space| (($ (|ThreeSpace| (|DoubleFloat|))) "\\spad{space specifies} the space into which we will draw. If none is given then a new space is created.")) (|ranges| (($ (|List| (|Segment| (|Float|)))) "\\spad{ranges(l)} provides a list of user-specified ranges \\spad{l}. This option is expressed in the form \\spad{ranges == l}.")) (|range| (($ (|List| (|Segment| (|Fraction| (|Integer|))))) "\\spad{range([i])} provides a user-specified range \\spad{i}. This option is expressed in the form \\spad{range == [i]}.") (($ (|List| (|Segment| (|Float|)))) "\\spad{range([l])} provides a user-specified range \\spad{l}. This option is expressed in the form \\spad{range == [l]}.")) (|tubeRadius| (($ (|Float|)) "\\spad{tubeRadius(r)} specifies a radius,{} \\spad{r},{} for a tube plot around a 3D curve; is expressed in the form \\spad{tubeRadius == 4}.")) (|colorFunction| (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) "\\spad{colorFunction(f(x,{}y,{}z))} specifies the color for three dimensional plots as a function of \\spad{x},{} \\spad{y},{} and \\spad{z} coordinates. This option is expressed in the form \\spad{colorFunction == f(x,{}y,{}z)}.") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) "\\spad{colorFunction(f(u,{}v))} specifies the color for three dimensional plots as a function based upon the two parametric variables. This option is expressed in the form \\spad{colorFunction == f(u,{}v)}.") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) "\\spad{colorFunction(f(z))} specifies the color based upon the \\spad{z}-component of three dimensional plots. This option is expressed in the form \\spad{colorFunction == f(z)}.")) (|curveColor| (($ (|Palette|)) "\\spad{curveColor(p)} specifies a color index for 2D graph curves from the spadcolors palette \\spad{p}. This option is expressed in the form \\spad{curveColor ==p}.") (($ (|Float|)) "\\spad{curveColor(v)} specifies a color,{} \\spad{v},{} for 2D graph curves. This option is expressed in the form \\spad{curveColor == v}.")) (|pointColor| (($ (|Palette|)) "\\spad{pointColor(p)} specifies a color index for 2D graph points from the spadcolors palette \\spad{p}. This option is expressed in the form \\spad{pointColor == p}.") (($ (|Float|)) "\\spad{pointColor(v)} specifies a color,{} \\spad{v},{} for 2D graph points. This option is expressed in the form \\spad{pointColor == v}.")) (|coordinates| (($ (|Mapping| (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)))) "\\spad{coordinates(p)} specifies a change of coordinate systems of point \\spad{p}. This option is expressed in the form \\spad{coordinates == p}.")) (|toScale| (($ (|Boolean|)) "\\spad{toScale(b)} specifies whether or not a plot is to be drawn to scale; if \\spad{b} is \\spad{true} it is drawn to scale,{} if \\spad{b} is \\spad{false} it is not. This option is expressed in the form \\spad{toScale == b}.")) (|style| (($ (|String|)) "\\spad{style(s)} specifies the drawing style in which the graph will be plotted by the indicated string \\spad{s}. This option is expressed in the form \\spad{style == s}.")) (|title| (($ (|String|)) "\\spad{title(s)} specifies a title for a plot by the indicated string \\spad{s}. This option is expressed in the form \\spad{title == s}.")) (|viewpoint| (($ (|Record| (|:| |theta| (|DoubleFloat|)) (|:| |phi| (|DoubleFloat|)) (|:| |scale| (|DoubleFloat|)) (|:| |scaleX| (|DoubleFloat|)) (|:| |scaleY| (|DoubleFloat|)) (|:| |scaleZ| (|DoubleFloat|)) (|:| |deltaX| (|DoubleFloat|)) (|:| |deltaY| (|DoubleFloat|)))) "\\spad{viewpoint(vp)} creates a viewpoint data structure corresponding to the list of values. The values are interpreted as [theta,{} phi,{} scale,{} scaleX,{} scaleY,{} scaleZ,{} deltaX,{} deltaY]. This option is expressed in the form \\spad{viewpoint == ls}.")) (|clip| (($ (|List| (|Segment| (|Float|)))) "\\spad{clip([l])} provides ranges for user-defined clipping as specified in the list \\spad{l}. This option is expressed in the form \\spad{clip == [l]}.") (($ (|Boolean|)) "\\spad{clip(b)} turns 2D clipping on if \\spad{b} is \\spad{true},{} or off if \\spad{b} is \\spad{false}. This option is expressed in the form \\spad{clip == b}.")) (|adaptive| (($ (|Boolean|)) "\\spad{adaptive(b)} turns adaptive 2D plotting on if \\spad{b} is \\spad{true},{} or off if \\spad{b} is \\spad{false}. This option is expressed in the form \\spad{adaptive == b}."))) 
NIL 
NIL 
(-252 R S V) 
((|constructor| (NIL "\\spadtype{DifferentialSparseMultivariatePolynomial} implements an ordinary differential polynomial ring by combining a domain belonging to the category \\spadtype{DifferentialVariableCategory} with the domain \\spadtype{SparseMultivariatePolynomial}."))) 
(((-4507 "*") |has| |#1| (-170)) (-4498 |has| |#1| (-550)) (-4503 |has| |#1| (-6 -4503)) (-4500 . T) (-4499 . T) (-4502 . T)) 
((|HasCategory| |#1| (QUOTE (-896))) (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-170))) (-3322 (|HasCategory| |#1| (QUOTE (-170))) (|HasCategory| |#1| (QUOTE (-550)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -873) (QUOTE (-375)))) (|HasCategory| |#3| (LIST (QUOTE -873) (QUOTE (-375))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -873) (QUOTE (-560)))) (|HasCategory| |#3| (LIST (QUOTE -873) (QUOTE (-560))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-375))))) (|HasCategory| |#3| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-375)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-560))))) (|HasCategory| |#3| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-560)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| |#3| (LIST (QUOTE -601) (QUOTE (-533))))) (|HasCategory| |#1| (QUOTE (-834))) (|HasCategory| |#1| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -43) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#1| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#1| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#1| (QUOTE (-221))) (|HasCategory| |#1| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#1| (QUOTE (-359))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -43) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#1| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560)))))) (|HasAttribute| |#1| (QUOTE -4503)) (|HasCategory| |#1| (QUOTE (-447))) (-3322 (|HasCategory| |#1| (QUOTE (-170))) (|HasCategory| |#1| (QUOTE (-447))) (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-896)))) (-3322 (|HasCategory| |#1| (QUOTE (-447))) (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-896)))) (-3322 (|HasCategory| |#1| (QUOTE (-447))) (|HasCategory| |#1| (QUOTE (-896)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-896)))) (-3322 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-896)))) (|HasCategory| |#1| (QUOTE (-146))))) 
(-253 S) 
((|constructor| (NIL "This category is part of the PAFF package")) (|tree| (($ (|List| |#1|)) "\\spad{tree(l)} creates a chain tree from the list \\spad{l}") (($ |#1|) "\\spad{tree(nd)} creates a tree with value \\spad{nd},{} and no children") (($ |#1| (|List| $)) "\\spad{tree(nd,{}ls)} creates a tree with value \\spad{nd},{} and children \\spad{ls}."))) 
((-4505 . T) (-4506 . T) (-3576 . T)) 
NIL 
(-254 S) 
((|constructor| (NIL "This category is part of the PAFF package")) (|fullOutput| (((|Boolean|)) "\\spad{fullOutput returns} the value of the flag set by fullOutput(\\spad{b}).") (((|Boolean|) (|Boolean|)) "\\spad{fullOutput(b)} sets a flag such that when \\spad{true},{} a coerce to OutputForm yields the full output of \\spad{tr},{} otherwise encode(\\spad{tr}) is output (see encode function). The default is \\spad{false}.")) (|fullOut| (((|OutputForm|) $) "\\spad{fullOut(tr)} yields a full output of \\spad{tr} (see function fullOutput).")) (|encode| (((|String|) $) "\\spad{encode(t)} returns a string indicating the \"shape\" of the tree"))) 
((-4505 . T) (-4506 . T)) 
((|HasCategory| |#1| (QUOTE (-1082))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082))))) 
(-255 K |symb| |PolyRing| E |ProjPt| PCS |Plc| DIVISOR |InfClsPoint| |DesTree| BLMET) 
((|constructor| (NIL "\\indented{1}{The following is all the categories,{} domains and package} used for the desingularisation be means of monoidal transformation (Blowing-up)")) (|genusTreeNeg| (((|Integer|) (|NonNegativeInteger|) (|List| |#10|)) "\\spad{genusTreeNeg(n,{}listOfTrees)} computes the \"genus\" of a curve that may be not absolutly irreducible,{} where \\spad{n} is the degree of a polynomial pol defining the curve and \\spad{listOfTrees} is all the desingularisation trees at all singular points on the curve defined by pol. A \"negative\" genus means that the curve is reducible \\spad{!!}.")) (|genusTree| (((|NonNegativeInteger|) (|NonNegativeInteger|) (|List| |#10|)) "\\spad{genusTree(n,{}listOfTrees)} computes the genus of a curve,{} where \\spad{n} is the degree of a polynomial pol defining the curve and \\spad{listOfTrees} is all the desingularisation trees at all singular points on the curve defined by pol.")) (|genusNeg| (((|Integer|) |#3|) "\\spad{genusNeg(pol)} computes the \"genus\" of a curve that may be not absolutly irreducible. A \"negative\" genus means that the curve is reducible \\spad{!!}.")) (|genus| (((|NonNegativeInteger|) |#3|) "\\spad{genus(pol)} computes the genus of the curve defined by \\spad{pol}.")) (|initializeParamOfPlaces| (((|Void|) |#10| (|List| |#3|)) "initParLocLeaves(\\spad{tr},{}listOfFnc) initialize the local parametrization at places corresponding to the leaves of \\spad{tr} according to the given list of functions in listOfFnc.") (((|Void|) |#10|) "initParLocLeaves(\\spad{tr}) initialize the local parametrization at places corresponding to the leaves of \\spad{tr}.")) (|initParLocLeaves| (((|Void|) |#10|) "\\spad{initParLocLeaves(tr)} initialize the local parametrization at simple points corresponding to the leaves of \\spad{tr}.")) (|fullParamInit| (((|Void|) |#10|) "\\spad{fullParamInit(tr)} initialize the local parametrization at all places (leaves of \\spad{tr}),{} computes the local exceptional divisor at each infinytly close points in the tree. This function is equivalent to the following called: initParLocLeaves(\\spad{tr}) initializeParamOfPlaces(\\spad{tr}) blowUpWithExcpDiv(\\spad{tr})")) (|desingTree| (((|List| |#10|) |#3|) "\\spad{desingTree(pol)} returns all the desingularisation trees of all singular points on the curve defined by \\spad{pol}.")) (|desingTreeAtPoint| ((|#10| |#5| |#3|) "\\spad{desingTreeAtPoint(pt,{}pol)} computes the desingularisation tree at the point \\spad{pt} on the curve defined by \\spad{pol}. This function recursively compute the tree.")) (|adjunctionDivisor| ((|#8| |#10|) "\\spad{adjunctionDivisor(tr)} compute the local adjunction divisor of a desingularisation tree \\spad{tr} of a singular point.")) (|divisorAtDesingTree| ((|#8| |#3| |#10|) "\\spad{divisorAtDesingTree(f,{}tr)} computes the local divisor of \\spad{f} at a desingularisation tree \\spad{tr} of a singular point."))) 
NIL 
NIL 
(-256 A S) 
((|constructor| (NIL "\\spadtype{DifferentialVariableCategory} constructs the set of derivatives of a given set of (ordinary) differential indeterminates. If \\spad{x},{}...,{}\\spad{y} is an ordered set of differential indeterminates,{} and the prime notation is used for differentiation,{} then the set of derivatives (including zero-th order) of the differential indeterminates is \\spad{x},{}\\spad{x'},{}\\spad{x''},{}...,{} \\spad{y},{}\\spad{y'},{}\\spad{y''},{}... (Note that in the interpreter,{} the \\spad{n}-th derivative of \\spad{y} is displayed as \\spad{y} with a subscript \\spad{n}.) This set is viewed as a set of algebraic indeterminates,{} totally ordered in a way compatible with differentiation and the given order on the differential indeterminates. Such a total order is called a ranking of the differential indeterminates. \\blankline A domain in this category is needed to construct a differential polynomial domain. Differential polynomials are ordered by a ranking on the derivatives,{} and by an order (extending the ranking) on on the set of differential monomials. One may thus associate a domain in this category with a ranking of the differential indeterminates,{} just as one associates a domain in the category \\spadtype{OrderedAbelianMonoidSup} with an ordering of the set of monomials in a set of algebraic indeterminates. The ranking is specified through the binary relation \\spadfun{<}. For example,{} one may define one derivative to be less than another by lexicographically comparing first the \\spadfun{order},{} then the given order of the differential indeterminates appearing in the derivatives. This is the default implementation. \\blankline The notion of weight generalizes that of degree. A polynomial domain may be made into a graded ring if a weight function is given on the set of indeterminates,{} Very often,{} a grading is the first step in ordering the set of monomials. For differential polynomial domains,{} this constructor provides a function \\spadfun{weight},{} which allows the assignment of a non-negative number to each derivative of a differential indeterminate. For example,{} one may define the weight of a derivative to be simply its \\spadfun{order} (this is the default assignment). This weight function can then be extended to the set of all differential polynomials,{} providing a graded ring structure.")) (|coerce| (($ |#2|) "\\spad{coerce(s)} returns \\spad{s},{} viewed as the zero-th order derivative of \\spad{s}.")) (|differentiate| (($ $ (|NonNegativeInteger|)) "\\spad{differentiate(v,{} n)} returns the \\spad{n}-th derivative of \\spad{v}.") (($ $) "\\spad{differentiate(v)} returns the derivative of \\spad{v}.")) (|weight| (((|NonNegativeInteger|) $) "\\spad{weight(v)} returns the weight of the derivative \\spad{v}.")) (|variable| ((|#2| $) "\\spad{variable(v)} returns \\spad{s} if \\spad{v} is any derivative of the differential indeterminate \\spad{s}.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(v)} returns \\spad{n} if \\spad{v} is the \\spad{n}-th derivative of any differential indeterminate.")) (|makeVariable| (($ |#2| (|NonNegativeInteger|)) "\\spad{makeVariable(s,{} n)} returns the \\spad{n}-th derivative of a differential indeterminate \\spad{s} as an algebraic indeterminate."))) 
NIL 
NIL 
(-257 S) 
((|constructor| (NIL "\\spadtype{DifferentialVariableCategory} constructs the set of derivatives of a given set of (ordinary) differential indeterminates. If \\spad{x},{}...,{}\\spad{y} is an ordered set of differential indeterminates,{} and the prime notation is used for differentiation,{} then the set of derivatives (including zero-th order) of the differential indeterminates is \\spad{x},{}\\spad{x'},{}\\spad{x''},{}...,{} \\spad{y},{}\\spad{y'},{}\\spad{y''},{}... (Note that in the interpreter,{} the \\spad{n}-th derivative of \\spad{y} is displayed as \\spad{y} with a subscript \\spad{n}.) This set is viewed as a set of algebraic indeterminates,{} totally ordered in a way compatible with differentiation and the given order on the differential indeterminates. Such a total order is called a ranking of the differential indeterminates. \\blankline A domain in this category is needed to construct a differential polynomial domain. Differential polynomials are ordered by a ranking on the derivatives,{} and by an order (extending the ranking) on on the set of differential monomials. One may thus associate a domain in this category with a ranking of the differential indeterminates,{} just as one associates a domain in the category \\spadtype{OrderedAbelianMonoidSup} with an ordering of the set of monomials in a set of algebraic indeterminates. The ranking is specified through the binary relation \\spadfun{<}. For example,{} one may define one derivative to be less than another by lexicographically comparing first the \\spadfun{order},{} then the given order of the differential indeterminates appearing in the derivatives. This is the default implementation. \\blankline The notion of weight generalizes that of degree. A polynomial domain may be made into a graded ring if a weight function is given on the set of indeterminates,{} Very often,{} a grading is the first step in ordering the set of monomials. For differential polynomial domains,{} this constructor provides a function \\spadfun{weight},{} which allows the assignment of a non-negative number to each derivative of a differential indeterminate. For example,{} one may define the weight of a derivative to be simply its \\spadfun{order} (this is the default assignment). This weight function can then be extended to the set of all differential polynomials,{} providing a graded ring structure.")) (|coerce| (($ |#1|) "\\spad{coerce(s)} returns \\spad{s},{} viewed as the zero-th order derivative of \\spad{s}.")) (|differentiate| (($ $ (|NonNegativeInteger|)) "\\spad{differentiate(v,{} n)} returns the \\spad{n}-th derivative of \\spad{v}.") (($ $) "\\spad{differentiate(v)} returns the derivative of \\spad{v}.")) (|weight| (((|NonNegativeInteger|) $) "\\spad{weight(v)} returns the weight of the derivative \\spad{v}.")) (|variable| ((|#1| $) "\\spad{variable(v)} returns \\spad{s} if \\spad{v} is any derivative of the differential indeterminate \\spad{s}.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(v)} returns \\spad{n} if \\spad{v} is the \\spad{n}-th derivative of any differential indeterminate.")) (|makeVariable| (($ |#1| (|NonNegativeInteger|)) "\\spad{makeVariable(s,{} n)} returns the \\spad{n}-th derivative of a differential indeterminate \\spad{s} as an algebraic indeterminate."))) 
NIL 
NIL 
(-258) 
((|constructor| (NIL "\\axiomType{e04AgentsPackage} is a package of numerical agents to be used to investigate attributes of an input function so as to decide the \\axiomFun{measure} of an appropriate numerical optimization routine.")) (|optAttributes| (((|List| (|String|)) (|Union| (|:| |noa| (|Record| (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |init| (|List| (|DoubleFloat|))) (|:| |lb| (|List| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |cf| (|List| (|Expression| (|DoubleFloat|)))) (|:| |ub| (|List| (|OrderedCompletion| (|DoubleFloat|)))))) (|:| |lsa| (|Record| (|:| |lfn| (|List| (|Expression| (|DoubleFloat|)))) (|:| |init| (|List| (|DoubleFloat|))))))) "\\spad{optAttributes(o)} is a function for supplying a list of attributes of an optimization problem.")) (|expenseOfEvaluation| (((|Float|) (|Record| (|:| |lfn| (|List| (|Expression| (|DoubleFloat|)))) (|:| |init| (|List| (|DoubleFloat|))))) "\\spad{expenseOfEvaluation(o)} returns the intensity value of the cost of evaluating the input set of functions. This is in terms of the number of ``operational units\\spad{''}. It returns a value in the range [0,{}1].")) (|changeNameToObjf| (((|Result|) (|Symbol|) (|Result|)) "\\spad{changeNameToObjf(s,{}r)} changes the name of item \\axiom{\\spad{s}} in \\axiom{\\spad{r}} to objf.")) (|varList| (((|List| (|Symbol|)) (|Expression| (|DoubleFloat|)) (|NonNegativeInteger|)) "\\spad{varList(e,{}n)} returns a list of \\axiom{\\spad{n}} indexed variables with name as in \\axiom{\\spad{e}}.")) (|variables| (((|List| (|Symbol|)) (|Record| (|:| |lfn| (|List| (|Expression| (|DoubleFloat|)))) (|:| |init| (|List| (|DoubleFloat|))))) "\\spad{variables(args)} returns the list of variables in \\axiom{\\spad{args}.\\spad{lfn}}")) (|quadratic?| (((|Boolean|) (|Expression| (|DoubleFloat|))) "\\spad{quadratic?(e)} tests if \\axiom{\\spad{e}} is a quadratic function.")) (|nonLinearPart| (((|List| (|Expression| (|DoubleFloat|))) (|List| (|Expression| (|DoubleFloat|)))) "\\spad{nonLinearPart(l)} returns the list of non-linear functions of \\spad{l}.")) (|linearPart| (((|List| (|Expression| (|DoubleFloat|))) (|List| (|Expression| (|DoubleFloat|)))) "\\spad{linearPart(l)} returns the list of linear functions of \\axiom{\\spad{l}}.")) (|linearMatrix| (((|Matrix| (|DoubleFloat|)) (|List| (|Expression| (|DoubleFloat|))) (|NonNegativeInteger|)) "\\spad{linearMatrix(l,{}n)} returns a matrix of coefficients of the linear functions in \\axiom{\\spad{l}}. If \\spad{l} is empty,{} the matrix has at least one row.")) (|linear?| (((|Boolean|) (|Expression| (|DoubleFloat|))) "\\spad{linear?(e)} tests if \\axiom{\\spad{e}} is a linear function.") (((|Boolean|) (|List| (|Expression| (|DoubleFloat|)))) "\\spad{linear?(l)} returns \\spad{true} if all the bounds \\spad{l} are either linear or simple.")) (|simpleBounds?| (((|Boolean|) (|List| (|Expression| (|DoubleFloat|)))) "\\spad{simpleBounds?(l)} returns \\spad{true} if the list of expressions \\spad{l} are simple.")) (|splitLinear| (((|Expression| (|DoubleFloat|)) (|Expression| (|DoubleFloat|))) "\\spad{splitLinear(f)} splits the linear part from an expression which it returns.")) (|sumOfSquares| (((|Union| (|Expression| (|DoubleFloat|)) "failed") (|Expression| (|DoubleFloat|))) "\\spad{sumOfSquares(f)} returns either an expression for which the square is the original function of \"failed\".")) (|sortConstraints| (((|Record| (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |init| (|List| (|DoubleFloat|))) (|:| |lb| (|List| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |cf| (|List| (|Expression| (|DoubleFloat|)))) (|:| |ub| (|List| (|OrderedCompletion| (|DoubleFloat|))))) (|Record| (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |init| (|List| (|DoubleFloat|))) (|:| |lb| (|List| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |cf| (|List| (|Expression| (|DoubleFloat|)))) (|:| |ub| (|List| (|OrderedCompletion| (|DoubleFloat|)))))) "\\spad{sortConstraints(args)} uses a simple bubblesort on the list of constraints using the degree of the expression on which to sort. Of course,{} it must match the bounds to the constraints.")) (|finiteBound| (((|List| (|DoubleFloat|)) (|List| (|OrderedCompletion| (|DoubleFloat|))) (|DoubleFloat|)) "\\spad{finiteBound(l,{}b)} repaces all instances of an infinite entry in \\axiom{\\spad{l}} by a finite entry \\axiom{\\spad{b}} or \\axiom{\\spad{-b}}."))) 
NIL 
NIL 
(-259) 
((|constructor| (NIL "\\axiomType{e04dgfAnnaType} is a domain of \\axiomType{NumericalOptimization} for the NAG routine E04DGF,{} a general optimization routine which can handle some singularities in the input function. The function \\axiomFun{measure} measures the usefulness of the routine E04DGF for the given problem. The function \\axiomFun{numericalOptimization} performs the optimization by using \\axiomType{NagOptimisationPackage}."))) 
NIL 
NIL 
(-260) 
((|constructor| (NIL "\\axiomType{e04fdfAnnaType} is a domain of \\axiomType{NumericalOptimization} for the NAG routine E04FDF,{} a general optimization routine which can handle some singularities in the input function. The function \\axiomFun{measure} measures the usefulness of the routine E04FDF for the given problem. The function \\axiomFun{numericalOptimization} performs the optimization by using \\axiomType{NagOptimisationPackage}."))) 
NIL 
NIL 
(-261) 
((|constructor| (NIL "\\axiomType{e04gcfAnnaType} is a domain of \\axiomType{NumericalOptimization} for the NAG routine E04GCF,{} a general optimization routine which can handle some singularities in the input function. The function \\axiomFun{measure} measures the usefulness of the routine E04GCF for the given problem. The function \\axiomFun{numericalOptimization} performs the optimization by using \\axiomType{NagOptimisationPackage}."))) 
NIL 
NIL 
(-262) 
((|constructor| (NIL "\\axiomType{e04jafAnnaType} is a domain of \\axiomType{NumericalOptimization} for the NAG routine E04JAF,{} a general optimization routine which can handle some singularities in the input function. The function \\axiomFun{measure} measures the usefulness of the routine E04JAF for the given problem. The function \\axiomFun{numericalOptimization} performs the optimization by using \\axiomType{NagOptimisationPackage}."))) 
NIL 
NIL 
(-263) 
((|constructor| (NIL "\\axiomType{e04mbfAnnaType} is a domain of \\axiomType{NumericalOptimization} for the NAG routine E04MBF,{} an optimization routine for Linear functions. The function \\axiomFun{measure} measures the usefulness of the routine E04MBF for the given problem. The function \\axiomFun{numericalOptimization} performs the optimization by using \\axiomType{NagOptimisationPackage}."))) 
NIL 
NIL 
(-264) 
((|constructor| (NIL "\\axiomType{e04nafAnnaType} is a domain of \\axiomType{NumericalOptimization} for the NAG routine E04NAF,{} an optimization routine for Quadratic functions. The function \\axiomFun{measure} measures the usefulness of the routine E04NAF for the given problem. The function \\axiomFun{numericalOptimization} performs the optimization by using \\axiomType{NagOptimisationPackage}."))) 
NIL 
NIL 
(-265) 
((|constructor| (NIL "\\axiomType{e04ucfAnnaType} is a domain of \\axiomType{NumericalOptimization} for the NAG routine E04UCF,{} a general optimization routine which can handle some singularities in the input function. The function \\axiomFun{measure} measures the usefulness of the routine E04UCF for the given problem. The function \\axiomFun{numericalOptimization} performs the optimization by using \\axiomType{NagOptimisationPackage}."))) 
NIL 
NIL 
(-266) 
((|constructor| (NIL "A domain used in the construction of the exterior algebra on a set \\spad{X} over a ring \\spad{R}. This domain represents the set of all ordered subsets of the set \\spad{X},{} assumed to be in correspondance with {1,{}2,{}3,{} ...}. The ordered subsets are themselves ordered lexicographically and are in bijective correspondance with an ordered basis of the exterior algebra. In this domain we are dealing strictly with the exponents of basis elements which can only be 0 or 1. \\blankline The multiplicative identity element of the exterior algebra corresponds to the empty subset of \\spad{X}. A coerce from List Integer to an ordered basis element is provided to allow the convenient input of expressions. Another exported function forgets the ordered structure and simply returns the list corresponding to an ordered subset.")) (|Nul| (($ (|NonNegativeInteger|)) "\\spad{Nul()} gives the basis element 1 for the algebra generated by \\spad{n} generators.")) (|exponents| (((|List| (|Integer|)) $) "\\spad{exponents(x)} converts a domain element into a list of zeros and ones corresponding to the exponents in the basis element that \\spad{x} represents.")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(x)} gives the numbers of 1\\spad{'s} in \\spad{x},{} \\spadignore{i.e.} the number of non-zero exponents in the basis element that \\spad{x} represents.")) (|coerce| (($ (|List| (|Integer|))) "\\spad{coerce(l)} converts a list of 0\\spad{'s} and 1\\spad{'s} into a basis element,{} where 1 (respectively 0) designates that the variable of the corresponding index of \\spad{l} is (respectively,{} is not) present. Error: if an element of \\spad{l} is not 0 or 1."))) 
NIL 
NIL 
(-267 R -2262) 
((|constructor| (NIL "Provides elementary functions over an integral domain.")) (|localReal?| (((|Boolean|) |#2|) "\\spad{localReal?(x)} should be local but conditional")) (|specialTrigs| (((|Union| |#2| "failed") |#2| (|List| (|Record| (|:| |func| |#2|) (|:| |pole| (|Boolean|))))) "\\spad{specialTrigs(x,{}l)} should be local but conditional")) (|iiacsch| ((|#2| |#2|) "\\spad{iiacsch(x)} should be local but conditional")) (|iiasech| ((|#2| |#2|) "\\spad{iiasech(x)} should be local but conditional")) (|iiacoth| ((|#2| |#2|) "\\spad{iiacoth(x)} should be local but conditional")) (|iiatanh| ((|#2| |#2|) "\\spad{iiatanh(x)} should be local but conditional")) (|iiacosh| ((|#2| |#2|) "\\spad{iiacosh(x)} should be local but conditional")) (|iiasinh| ((|#2| |#2|) "\\spad{iiasinh(x)} should be local but conditional")) (|iicsch| ((|#2| |#2|) "\\spad{iicsch(x)} should be local but conditional")) (|iisech| ((|#2| |#2|) "\\spad{iisech(x)} should be local but conditional")) (|iicoth| ((|#2| |#2|) "\\spad{iicoth(x)} should be local but conditional")) (|iitanh| ((|#2| |#2|) "\\spad{iitanh(x)} should be local but conditional")) (|iicosh| ((|#2| |#2|) "\\spad{iicosh(x)} should be local but conditional")) (|iisinh| ((|#2| |#2|) "\\spad{iisinh(x)} should be local but conditional")) (|iiacsc| ((|#2| |#2|) "\\spad{iiacsc(x)} should be local but conditional")) (|iiasec| ((|#2| |#2|) "\\spad{iiasec(x)} should be local but conditional")) (|iiacot| ((|#2| |#2|) "\\spad{iiacot(x)} should be local but conditional")) (|iiatan| ((|#2| |#2|) "\\spad{iiatan(x)} should be local but conditional")) (|iiacos| ((|#2| |#2|) "\\spad{iiacos(x)} should be local but conditional")) (|iiasin| ((|#2| |#2|) "\\spad{iiasin(x)} should be local but conditional")) (|iicsc| ((|#2| |#2|) "\\spad{iicsc(x)} should be local but conditional")) (|iisec| ((|#2| |#2|) "\\spad{iisec(x)} should be local but conditional")) (|iicot| ((|#2| |#2|) "\\spad{iicot(x)} should be local but conditional")) (|iitan| ((|#2| |#2|) "\\spad{iitan(x)} should be local but conditional")) (|iicos| ((|#2| |#2|) "\\spad{iicos(x)} should be local but conditional")) (|iisin| ((|#2| |#2|) "\\spad{iisin(x)} should be local but conditional")) (|iilog| ((|#2| |#2|) "\\spad{iilog(x)} should be local but conditional")) (|iiexp| ((|#2| |#2|) "\\spad{iiexp(x)} should be local but conditional")) (|iisqrt3| ((|#2|) "\\spad{iisqrt3()} should be local but conditional")) (|iisqrt2| ((|#2|) "\\spad{iisqrt2()} should be local but conditional")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(p)} returns an elementary operator with the same symbol as \\spad{p}")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(p)} returns \\spad{true} if operator \\spad{p} is elementary")) (|pi| ((|#2|) "\\spad{\\spad{pi}()} returns the \\spad{pi} operator")) (|acsch| ((|#2| |#2|) "\\spad{acsch(x)} applies the inverse hyperbolic cosecant operator to \\spad{x}")) (|asech| ((|#2| |#2|) "\\spad{asech(x)} applies the inverse hyperbolic secant operator to \\spad{x}")) (|acoth| ((|#2| |#2|) "\\spad{acoth(x)} applies the inverse hyperbolic cotangent operator to \\spad{x}")) (|atanh| ((|#2| |#2|) "\\spad{atanh(x)} applies the inverse hyperbolic tangent operator to \\spad{x}")) (|acosh| ((|#2| |#2|) "\\spad{acosh(x)} applies the inverse hyperbolic cosine operator to \\spad{x}")) (|asinh| ((|#2| |#2|) "\\spad{asinh(x)} applies the inverse hyperbolic sine operator to \\spad{x}")) (|csch| ((|#2| |#2|) "\\spad{csch(x)} applies the hyperbolic cosecant operator to \\spad{x}")) (|sech| ((|#2| |#2|) "\\spad{sech(x)} applies the hyperbolic secant operator to \\spad{x}")) (|coth| ((|#2| |#2|) "\\spad{coth(x)} applies the hyperbolic cotangent operator to \\spad{x}")) (|tanh| ((|#2| |#2|) "\\spad{tanh(x)} applies the hyperbolic tangent operator to \\spad{x}")) (|cosh| ((|#2| |#2|) "\\spad{cosh(x)} applies the hyperbolic cosine operator to \\spad{x}")) (|sinh| ((|#2| |#2|) "\\spad{sinh(x)} applies the hyperbolic sine operator to \\spad{x}")) (|acsc| ((|#2| |#2|) "\\spad{acsc(x)} applies the inverse cosecant operator to \\spad{x}")) (|asec| ((|#2| |#2|) "\\spad{asec(x)} applies the inverse secant operator to \\spad{x}")) (|acot| ((|#2| |#2|) "\\spad{acot(x)} applies the inverse cotangent operator to \\spad{x}")) (|atan| ((|#2| |#2|) "\\spad{atan(x)} applies the inverse tangent operator to \\spad{x}")) (|acos| ((|#2| |#2|) "\\spad{acos(x)} applies the inverse cosine operator to \\spad{x}")) (|asin| ((|#2| |#2|) "\\spad{asin(x)} applies the inverse sine operator to \\spad{x}")) (|csc| ((|#2| |#2|) "\\spad{csc(x)} applies the cosecant operator to \\spad{x}")) (|sec| ((|#2| |#2|) "\\spad{sec(x)} applies the secant operator to \\spad{x}")) (|cot| ((|#2| |#2|) "\\spad{cot(x)} applies the cotangent operator to \\spad{x}")) (|tan| ((|#2| |#2|) "\\spad{tan(x)} applies the tangent operator to \\spad{x}")) (|cos| ((|#2| |#2|) "\\spad{cos(x)} applies the cosine operator to \\spad{x}")) (|sin| ((|#2| |#2|) "\\spad{sin(x)} applies the sine operator to \\spad{x}")) (|log| ((|#2| |#2|) "\\spad{log(x)} applies the logarithm operator to \\spad{x}")) (|exp| ((|#2| |#2|) "\\spad{exp(x)} applies the exponential operator to \\spad{x}"))) 
NIL 
NIL 
(-268 R -2262) 
((|constructor| (NIL "ElementaryFunctionStructurePackage provides functions to test the algebraic independence of various elementary functions,{} using the Risch structure theorem (real and complex versions). It also provides transformations on elementary functions which are not considered simplifications.")) (|tanQ| ((|#2| (|Fraction| (|Integer|)) |#2|) "\\spad{tanQ(q,{}a)} is a local function with a conditional implementation.")) (|rootNormalize| ((|#2| |#2| (|Kernel| |#2|)) "\\spad{rootNormalize(f,{} k)} returns \\spad{f} rewriting either \\spad{k} which must be an \\spad{n}th-root in terms of radicals already in \\spad{f},{} or some radicals in \\spad{f} in terms of \\spad{k}.")) (|validExponential| (((|Union| |#2| "failed") (|List| (|Kernel| |#2|)) |#2| (|Symbol|)) "\\spad{validExponential([k1,{}...,{}kn],{}f,{}x)} returns \\spad{g} if \\spad{exp(f)=g} and \\spad{g} involves only \\spad{k1...kn},{} and \"failed\" otherwise.")) (|realElementary| ((|#2| |#2| (|Symbol|)) "\\spad{realElementary(f,{}x)} rewrites the kernels of \\spad{f} involving \\spad{x} in terms of the 4 fundamental real transcendental elementary functions: \\spad{log,{} exp,{} tan,{} atan}.") ((|#2| |#2|) "\\spad{realElementary(f)} rewrites \\spad{f} in terms of the 4 fundamental real transcendental elementary functions: \\spad{log,{} exp,{} tan,{} atan}.")) (|rischNormalize| (((|Record| (|:| |func| |#2|) (|:| |kers| (|List| (|Kernel| |#2|))) (|:| |vals| (|List| |#2|))) |#2| (|Symbol|)) "\\spad{rischNormalize(f,{} x)} returns \\spad{[g,{} [k1,{}...,{}kn],{} [h1,{}...,{}hn]]} such that \\spad{g = normalize(f,{} x)} and each \\spad{\\spad{ki}} was rewritten as \\spad{\\spad{hi}} during the normalization.")) (|normalize| ((|#2| |#2| (|Symbol|)) "\\spad{normalize(f,{} x)} rewrites \\spad{f} using the least possible number of real algebraically independent kernels involving \\spad{x}.") ((|#2| |#2|) "\\spad{normalize(f)} rewrites \\spad{f} using the least possible number of real algebraically independent kernels."))) 
NIL 
NIL 
(-269 |Coef| UTS ULS) 
((|constructor| (NIL "This package provides elementary functions on any Laurent series domain over a field which was constructed from a Taylor series domain. These functions are implemented by calling the corresponding functions on the Taylor series domain. We also provide 'partial functions' which compute transcendental functions of Laurent series when possible and return \"failed\" when this is not possible.")) (|acsch| ((|#3| |#3|) "\\spad{acsch(z)} returns the inverse hyperbolic cosecant of Laurent series \\spad{z}.")) (|asech| ((|#3| |#3|) "\\spad{asech(z)} returns the inverse hyperbolic secant of Laurent series \\spad{z}.")) (|acoth| ((|#3| |#3|) "\\spad{acoth(z)} returns the inverse hyperbolic cotangent of Laurent series \\spad{z}.")) (|atanh| ((|#3| |#3|) "\\spad{atanh(z)} returns the inverse hyperbolic tangent of Laurent series \\spad{z}.")) (|acosh| ((|#3| |#3|) "\\spad{acosh(z)} returns the inverse hyperbolic cosine of Laurent series \\spad{z}.")) (|asinh| ((|#3| |#3|) "\\spad{asinh(z)} returns the inverse hyperbolic sine of Laurent series \\spad{z}.")) (|csch| ((|#3| |#3|) "\\spad{csch(z)} returns the hyperbolic cosecant of Laurent series \\spad{z}.")) (|sech| ((|#3| |#3|) "\\spad{sech(z)} returns the hyperbolic secant of Laurent series \\spad{z}.")) (|coth| ((|#3| |#3|) "\\spad{coth(z)} returns the hyperbolic cotangent of Laurent series \\spad{z}.")) (|tanh| ((|#3| |#3|) "\\spad{tanh(z)} returns the hyperbolic tangent of Laurent series \\spad{z}.")) (|cosh| ((|#3| |#3|) "\\spad{cosh(z)} returns the hyperbolic cosine of Laurent series \\spad{z}.")) (|sinh| ((|#3| |#3|) "\\spad{sinh(z)} returns the hyperbolic sine of Laurent series \\spad{z}.")) (|acsc| ((|#3| |#3|) "\\spad{acsc(z)} returns the arc-cosecant of Laurent series \\spad{z}.")) (|asec| ((|#3| |#3|) "\\spad{asec(z)} returns the arc-secant of Laurent series \\spad{z}.")) (|acot| ((|#3| |#3|) "\\spad{acot(z)} returns the arc-cotangent of Laurent series \\spad{z}.")) (|atan| ((|#3| |#3|) "\\spad{atan(z)} returns the arc-tangent of Laurent series \\spad{z}.")) (|acos| ((|#3| |#3|) "\\spad{acos(z)} returns the arc-cosine of Laurent series \\spad{z}.")) (|asin| ((|#3| |#3|) "\\spad{asin(z)} returns the arc-sine of Laurent series \\spad{z}.")) (|csc| ((|#3| |#3|) "\\spad{csc(z)} returns the cosecant of Laurent series \\spad{z}.")) (|sec| ((|#3| |#3|) "\\spad{sec(z)} returns the secant of Laurent series \\spad{z}.")) (|cot| ((|#3| |#3|) "\\spad{cot(z)} returns the cotangent of Laurent series \\spad{z}.")) (|tan| ((|#3| |#3|) "\\spad{tan(z)} returns the tangent of Laurent series \\spad{z}.")) (|cos| ((|#3| |#3|) "\\spad{cos(z)} returns the cosine of Laurent series \\spad{z}.")) (|sin| ((|#3| |#3|) "\\spad{sin(z)} returns the sine of Laurent series \\spad{z}.")) (|log| ((|#3| |#3|) "\\spad{log(z)} returns the logarithm of Laurent series \\spad{z}.")) (|exp| ((|#3| |#3|) "\\spad{exp(z)} returns the exponential of Laurent series \\spad{z}.")) (** ((|#3| |#3| (|Fraction| (|Integer|))) "\\spad{s ** r} raises a Laurent series \\spad{s} to a rational power \\spad{r}"))) 
NIL 
((|HasCategory| |#1| (QUOTE (-359)))) 
(-270 |Coef| ULS UPXS EFULS) 
((|constructor| (NIL "This package provides elementary functions on any Laurent series domain over a field which was constructed from a Taylor series domain. These functions are implemented by calling the corresponding functions on the Taylor series domain. We also provide 'partial functions' which compute transcendental functions of Laurent series when possible and return \"failed\" when this is not possible.")) (|acsch| ((|#3| |#3|) "\\spad{acsch(z)} returns the inverse hyperbolic cosecant of a Puiseux series \\spad{z}.")) (|asech| ((|#3| |#3|) "\\spad{asech(z)} returns the inverse hyperbolic secant of a Puiseux series \\spad{z}.")) (|acoth| ((|#3| |#3|) "\\spad{acoth(z)} returns the inverse hyperbolic cotangent of a Puiseux series \\spad{z}.")) (|atanh| ((|#3| |#3|) "\\spad{atanh(z)} returns the inverse hyperbolic tangent of a Puiseux series \\spad{z}.")) (|acosh| ((|#3| |#3|) "\\spad{acosh(z)} returns the inverse hyperbolic cosine of a Puiseux series \\spad{z}.")) (|asinh| ((|#3| |#3|) "\\spad{asinh(z)} returns the inverse hyperbolic sine of a Puiseux series \\spad{z}.")) (|csch| ((|#3| |#3|) "\\spad{csch(z)} returns the hyperbolic cosecant of a Puiseux series \\spad{z}.")) (|sech| ((|#3| |#3|) "\\spad{sech(z)} returns the hyperbolic secant of a Puiseux series \\spad{z}.")) (|coth| ((|#3| |#3|) "\\spad{coth(z)} returns the hyperbolic cotangent of a Puiseux series \\spad{z}.")) (|tanh| ((|#3| |#3|) "\\spad{tanh(z)} returns the hyperbolic tangent of a Puiseux series \\spad{z}.")) (|cosh| ((|#3| |#3|) "\\spad{cosh(z)} returns the hyperbolic cosine of a Puiseux series \\spad{z}.")) (|sinh| ((|#3| |#3|) "\\spad{sinh(z)} returns the hyperbolic sine of a Puiseux series \\spad{z}.")) (|acsc| ((|#3| |#3|) "\\spad{acsc(z)} returns the arc-cosecant of a Puiseux series \\spad{z}.")) (|asec| ((|#3| |#3|) "\\spad{asec(z)} returns the arc-secant of a Puiseux series \\spad{z}.")) (|acot| ((|#3| |#3|) "\\spad{acot(z)} returns the arc-cotangent of a Puiseux series \\spad{z}.")) (|atan| ((|#3| |#3|) "\\spad{atan(z)} returns the arc-tangent of a Puiseux series \\spad{z}.")) (|acos| ((|#3| |#3|) "\\spad{acos(z)} returns the arc-cosine of a Puiseux series \\spad{z}.")) (|asin| ((|#3| |#3|) "\\spad{asin(z)} returns the arc-sine of a Puiseux series \\spad{z}.")) (|csc| ((|#3| |#3|) "\\spad{csc(z)} returns the cosecant of a Puiseux series \\spad{z}.")) (|sec| ((|#3| |#3|) "\\spad{sec(z)} returns the secant of a Puiseux series \\spad{z}.")) (|cot| ((|#3| |#3|) "\\spad{cot(z)} returns the cotangent of a Puiseux series \\spad{z}.")) (|tan| ((|#3| |#3|) "\\spad{tan(z)} returns the tangent of a Puiseux series \\spad{z}.")) (|cos| ((|#3| |#3|) "\\spad{cos(z)} returns the cosine of a Puiseux series \\spad{z}.")) (|sin| ((|#3| |#3|) "\\spad{sin(z)} returns the sine of a Puiseux series \\spad{z}.")) (|log| ((|#3| |#3|) "\\spad{log(z)} returns the logarithm of a Puiseux series \\spad{z}.")) (|exp| ((|#3| |#3|) "\\spad{exp(z)} returns the exponential of a Puiseux series \\spad{z}.")) (** ((|#3| |#3| (|Fraction| (|Integer|))) "\\spad{z ** r} raises a Puiseaux series \\spad{z} to a rational power \\spad{r}"))) 
NIL 
((|HasCategory| |#1| (QUOTE (-359)))) 
(-271 A S) 
((|constructor| (NIL "An extensible aggregate is one which allows insertion and deletion of entries. These aggregates are models of lists and streams which are represented by linked structures so as to make insertion,{} deletion,{} and concatenation efficient. However,{} access to elements of these extensible aggregates is generally slow since access is made from the end. See \\spadtype{FlexibleArray} for an exception.")) (|removeDuplicates!| (($ $) "\\spad{removeDuplicates!(u)} destructively removes duplicates from \\spad{u}.")) (|select!| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{select!(p,{}u)} destructively changes \\spad{u} by keeping only values \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})}.")) (|merge!| (($ $ $) "\\spad{merge!(u,{}v)} destructively merges \\spad{u} and \\spad{v} in ascending order.") (($ (|Mapping| (|Boolean|) |#2| |#2|) $ $) "\\spad{merge!(p,{}u,{}v)} destructively merges \\spad{u} and \\spad{v} using predicate \\spad{p}.")) (|insert!| (($ $ $ (|Integer|)) "\\spad{insert!(v,{}u,{}i)} destructively inserts aggregate \\spad{v} into \\spad{u} at position \\spad{i}.") (($ |#2| $ (|Integer|)) "\\spad{insert!(x,{}u,{}i)} destructively inserts \\spad{x} into \\spad{u} at position \\spad{i}.")) (|remove!| (($ |#2| $) "\\spad{remove!(x,{}u)} destructively removes all values \\spad{x} from \\spad{u}.") (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{remove!(p,{}u)} destructively removes all elements \\spad{x} of \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}.")) (|delete!| (($ $ (|UniversalSegment| (|Integer|))) "\\spad{delete!(u,{}i..j)} destructively deletes elements \\spad{u}.\\spad{i} through \\spad{u}.\\spad{j}.") (($ $ (|Integer|)) "\\indented{1}{delete!(\\spad{u},{}\\spad{i}) destructively deletes the \\axiom{\\spad{i}}th element of \\spad{u}.} \\blankline \\spad{E} Data:=Record(age:Integer,{}gender:String) \\spad{E} a1:AssociationList(String,{}Data):=table() \\spad{E} \\spad{a1}.\"tim\":=[55,{}\"male\"]\\$Data \\spad{E} delete!(\\spad{a1},{}1)")) (|concat!| (($ $ $) "\\spad{concat!(u,{}v)} destructively appends \\spad{v} to the end of \\spad{u}. \\spad{v} is unchanged") (($ $ |#2|) "\\spad{concat!(u,{}x)} destructively adds element \\spad{x} to the end of \\spad{u}."))) 
NIL 
((|HasCategory| |#2| (QUOTE (-834))) (|HasCategory| |#2| (QUOTE (-1082)))) 
(-272 S) 
((|constructor| (NIL "An extensible aggregate is one which allows insertion and deletion of entries. These aggregates are models of lists and streams which are represented by linked structures so as to make insertion,{} deletion,{} and concatenation efficient. However,{} access to elements of these extensible aggregates is generally slow since access is made from the end. See \\spadtype{FlexibleArray} for an exception.")) (|removeDuplicates!| (($ $) "\\spad{removeDuplicates!(u)} destructively removes duplicates from \\spad{u}.")) (|select!| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{select!(p,{}u)} destructively changes \\spad{u} by keeping only values \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})}.")) (|merge!| (($ $ $) "\\spad{merge!(u,{}v)} destructively merges \\spad{u} and \\spad{v} in ascending order.") (($ (|Mapping| (|Boolean|) |#1| |#1|) $ $) "\\spad{merge!(p,{}u,{}v)} destructively merges \\spad{u} and \\spad{v} using predicate \\spad{p}.")) (|insert!| (($ $ $ (|Integer|)) "\\spad{insert!(v,{}u,{}i)} destructively inserts aggregate \\spad{v} into \\spad{u} at position \\spad{i}.") (($ |#1| $ (|Integer|)) "\\spad{insert!(x,{}u,{}i)} destructively inserts \\spad{x} into \\spad{u} at position \\spad{i}.")) (|remove!| (($ |#1| $) "\\spad{remove!(x,{}u)} destructively removes all values \\spad{x} from \\spad{u}.") (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{remove!(p,{}u)} destructively removes all elements \\spad{x} of \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}.")) (|delete!| (($ $ (|UniversalSegment| (|Integer|))) "\\spad{delete!(u,{}i..j)} destructively deletes elements \\spad{u}.\\spad{i} through \\spad{u}.\\spad{j}.") (($ $ (|Integer|)) "\\indented{1}{delete!(\\spad{u},{}\\spad{i}) destructively deletes the \\axiom{\\spad{i}}th element of \\spad{u}.} \\blankline \\spad{E} Data:=Record(age:Integer,{}gender:String) \\spad{E} a1:AssociationList(String,{}Data):=table() \\spad{E} \\spad{a1}.\"tim\":=[55,{}\"male\"]\\$Data \\spad{E} delete!(\\spad{a1},{}1)")) (|concat!| (($ $ $) "\\spad{concat!(u,{}v)} destructively appends \\spad{v} to the end of \\spad{u}. \\spad{v} is unchanged") (($ $ |#1|) "\\spad{concat!(u,{}x)} destructively adds element \\spad{x} to the end of \\spad{u}."))) 
((-4506 . T) (-3576 . T)) 
NIL 
(-273 S) 
((|constructor| (NIL "Category for the elementary functions.")) (** (($ $ $) "\\spad{x**y} returns \\spad{x} to the power \\spad{y}.")) (|exp| (($ $) "\\spad{exp(x)} returns \\%\\spad{e} to the power \\spad{x}.")) (|log| (($ $) "\\spad{log(x)} returns the natural logarithm of \\spad{x}."))) 
NIL 
NIL 
(-274) 
((|constructor| (NIL "Category for the elementary functions.")) (** (($ $ $) "\\spad{x**y} returns \\spad{x} to the power \\spad{y}.")) (|exp| (($ $) "\\spad{exp(x)} returns \\%\\spad{e} to the power \\spad{x}.")) (|log| (($ $) "\\spad{log(x)} returns the natural logarithm of \\spad{x}."))) 
NIL 
NIL 
(-275 |Coef| UTS) 
((|constructor| (NIL "The elliptic functions \\spad{sn},{} \\spad{sc} and \\spad{dn} are expanded as Taylor series.")) (|sncndn| (((|List| (|Stream| |#1|)) (|Stream| |#1|) |#1|) "\\spad{sncndn(s,{}c)} is used internally.")) (|dn| ((|#2| |#2| |#1|) "\\spad{dn(x,{}k)} expands the elliptic function \\spad{dn} as a Taylor \\indented{1}{series.}")) (|cn| ((|#2| |#2| |#1|) "\\spad{cn(x,{}k)} expands the elliptic function \\spad{cn} as a Taylor \\indented{1}{series.}")) (|sn| ((|#2| |#2| |#1|) "\\spad{sn(x,{}k)} expands the elliptic function \\spad{sn} as a Taylor \\indented{1}{series.}"))) 
NIL 
NIL 
(-276 S |Index|) 
((|constructor| (NIL "An eltable over domains \\spad{D} and \\spad{I} is a structure which can be viewed as a function from \\spad{D} to \\spad{I}. Examples of eltable structures range from data structures,{} \\spadignore{e.g.} those of type \\spadtype{List},{} to algebraic structures like \\spadtype{Polynomial}.")) (|elt| ((|#2| $ |#1|) "\\spad{elt(u,{}i)} (also written: \\spad{u} . \\spad{i}) returns the element of \\spad{u} indexed by \\spad{i}. Error: if \\spad{i} is not an index of \\spad{u}."))) 
NIL 
NIL 
(-277 S |Dom| |Im|) 
((|constructor| (NIL "An eltable aggregate is one which can be viewed as a function. For example,{} the list \\axiom{[1,{}7,{}4]} can applied to 0,{}1,{} and 2 respectively will return the integers 1,{}7,{} and 4; thus this list may be viewed as mapping 0 to 1,{} 1 to 7 and 2 to 4. In general,{} an aggregate can map members of a domain Dom to an image domain Im.")) (|qsetelt!| ((|#3| $ |#2| |#3|) "\\spad{qsetelt!(u,{}x,{}y)} sets the image of \\axiom{\\spad{x}} to be \\axiom{\\spad{y}} under \\axiom{\\spad{u}},{} without checking that \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}}. If such a check is required use the function \\axiom{setelt}.")) (|setelt| ((|#3| $ |#2| |#3|) "\\spad{setelt(u,{}x,{}y)} sets the image of \\spad{x} to be \\spad{y} under \\spad{u},{} assuming \\spad{x} is in the domain of \\spad{u}. Error: if \\spad{x} is not in the domain of \\spad{u}.")) (|qelt| ((|#3| $ |#2|) "\\spad{qelt(u,{} x)} applies \\axiom{\\spad{u}} to \\axiom{\\spad{x}} without checking whether \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}}. If \\axiom{\\spad{x}} is not in the domain of \\axiom{\\spad{u}} a memory-access violation may occur. If a check on whether \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}} is required,{} use the function \\axiom{elt}.")) (|elt| ((|#3| $ |#2| |#3|) "\\spad{elt(u,{} x,{} y)} applies \\spad{u} to \\spad{x} if \\spad{x} is in the domain of \\spad{u},{} and returns \\spad{y} otherwise. For example,{} if \\spad{u} is a polynomial in \\axiom{\\spad{x}} over the rationals,{} \\axiom{elt(\\spad{u},{}\\spad{n},{}0)} may define the coefficient of \\axiom{\\spad{x}} to the power \\spad{n},{} returning 0 when \\spad{n} is out of range."))) 
NIL 
((|HasAttribute| |#1| (QUOTE -4506))) 
(-278 |Dom| |Im|) 
((|constructor| (NIL "An eltable aggregate is one which can be viewed as a function. For example,{} the list \\axiom{[1,{}7,{}4]} can applied to 0,{}1,{} and 2 respectively will return the integers 1,{}7,{} and 4; thus this list may be viewed as mapping 0 to 1,{} 1 to 7 and 2 to 4. In general,{} an aggregate can map members of a domain Dom to an image domain Im.")) (|qsetelt!| ((|#2| $ |#1| |#2|) "\\spad{qsetelt!(u,{}x,{}y)} sets the image of \\axiom{\\spad{x}} to be \\axiom{\\spad{y}} under \\axiom{\\spad{u}},{} without checking that \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}}. If such a check is required use the function \\axiom{setelt}.")) (|setelt| ((|#2| $ |#1| |#2|) "\\spad{setelt(u,{}x,{}y)} sets the image of \\spad{x} to be \\spad{y} under \\spad{u},{} assuming \\spad{x} is in the domain of \\spad{u}. Error: if \\spad{x} is not in the domain of \\spad{u}.")) (|qelt| ((|#2| $ |#1|) "\\spad{qelt(u,{} x)} applies \\axiom{\\spad{u}} to \\axiom{\\spad{x}} without checking whether \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}}. If \\axiom{\\spad{x}} is not in the domain of \\axiom{\\spad{u}} a memory-access violation may occur. If a check on whether \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}} is required,{} use the function \\axiom{elt}.")) (|elt| ((|#2| $ |#1| |#2|) "\\spad{elt(u,{} x,{} y)} applies \\spad{u} to \\spad{x} if \\spad{x} is in the domain of \\spad{u},{} and returns \\spad{y} otherwise. For example,{} if \\spad{u} is a polynomial in \\axiom{\\spad{x}} over the rationals,{} \\axiom{elt(\\spad{u},{}\\spad{n},{}0)} may define the coefficient of \\axiom{\\spad{x}} to the power \\spad{n},{} returning 0 when \\spad{n} is out of range."))) 
NIL 
NIL 
(-279 S R |Mod| -4331 -4208 |exactQuo|) 
((|constructor| (NIL "These domains are used for the factorization and gcds of univariate polynomials over the integers in order to work modulo different primes. See \\spadtype{ModularRing},{} \\spadtype{ModularField}")) (|elt| ((|#2| $ |#2|) "\\spad{elt(x,{}r)} or \\spad{x}.\\spad{r} is not documented")) (|inv| (($ $) "\\spad{inv(x)} is not documented")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} is not documented")) (|exQuo| (((|Union| $ "failed") $ $) "\\spad{exQuo(x,{}y)} is not documented")) (|reduce| (($ |#2| |#3|) "\\spad{reduce(r,{}m)} is not documented")) (|coerce| ((|#2| $) "\\spad{coerce(x)} is not documented")) (|modulus| ((|#3| $) "\\spad{modulus(x)} is not documented"))) 
((-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-280) 
((|constructor| (NIL "Entire Rings (non-commutative Integral Domains),{} \\spadignore{i.e.} a ring not necessarily commutative which has no zero divisors. \\blankline Axioms\\spad{\\br} \\tab{5}\\spad{ab=0 => a=0 or b=0} \\spad{--} known as noZeroDivisors\\spad{\\br} \\tab{5}\\spad{not(1=0)}")) (|noZeroDivisors| ((|attribute|) "if a product is zero then one of the factors must be zero."))) 
((-4498 . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-281 R) 
((|constructor| (NIL "This is a package for the exact computation of eigenvalues and eigenvectors. This package can be made to work for matrices with coefficients which are rational functions over a ring where we can factor polynomials. Rational eigenvalues are always explicitly computed while the non-rational ones are expressed in terms of their minimal polynomial.")) (|eigenvectors| (((|List| (|Record| (|:| |eigval| (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|)))) (|:| |eigmult| (|NonNegativeInteger|)) (|:| |eigvec| (|List| (|Matrix| (|Fraction| (|Polynomial| |#1|))))))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{eigenvectors(m)} returns the eigenvalues and eigenvectors for the matrix \\spad{m}. The rational eigenvalues and the correspondent eigenvectors are explicitely computed,{} while the non rational ones are given via their minimal polynomial and the corresponding eigenvectors are expressed in terms of a \"generic\" root of such a polynomial.")) (|generalizedEigenvectors| (((|List| (|Record| (|:| |eigval| (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|)))) (|:| |geneigvec| (|List| (|Matrix| (|Fraction| (|Polynomial| |#1|))))))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{generalizedEigenvectors(m)} returns the generalized eigenvectors of the matrix \\spad{m}.")) (|generalizedEigenvector| (((|List| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|Record| (|:| |eigval| (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|)))) (|:| |eigmult| (|NonNegativeInteger|)) (|:| |eigvec| (|List| (|Matrix| (|Fraction| (|Polynomial| |#1|)))))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{generalizedEigenvector(eigen,{}m)} returns the generalized eigenvectors of the matrix relative to the eigenvalue \\spad{eigen},{} as returned by the function eigenvectors.") (((|List| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|))) (|Matrix| (|Fraction| (|Polynomial| |#1|))) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{generalizedEigenvector(alpha,{}m,{}k,{}g)} returns the generalized eigenvectors of the matrix relative to the eigenvalue \\spad{alpha}. The integers \\spad{k} and \\spad{g} are respectively the algebraic and the geometric multiplicity of tye eigenvalue \\spad{alpha}. \\spad{alpha} can be either rational or not. In the seconda case apha is the minimal polynomial of the eigenvalue.")) (|eigenvector| (((|List| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{eigenvector(eigval,{}m)} returns the eigenvectors belonging to the eigenvalue \\spad{eigval} for the matrix \\spad{m}.")) (|eigenvalues| (((|List| (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|)))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{eigenvalues(m)} returns the eigenvalues of the matrix \\spad{m} which are expressible as rational functions over the rational numbers.")) (|characteristicPolynomial| (((|Polynomial| |#1|) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{characteristicPolynomial(m)} returns the characteristicPolynomial of the matrix \\spad{m} using a new generated symbol symbol as the main variable.") (((|Polynomial| |#1|) (|Matrix| (|Fraction| (|Polynomial| |#1|))) (|Symbol|)) "\\spad{characteristicPolynomial(m,{}var)} returns the characteristicPolynomial of the matrix \\spad{m} using the symbol \\spad{var} as the main variable."))) 
NIL 
NIL 
(-282 S R) 
((|constructor| (NIL "This package provides operations for mapping the sides of equations.")) (|map| (((|Equation| |#2|) (|Mapping| |#2| |#1|) (|Equation| |#1|)) "\\spad{map(f,{}eq)} returns an equation where \\spad{f} is applied to the sides of \\spad{eq}"))) 
NIL 
NIL 
(-283 S) 
((|constructor| (NIL "Equations as mathematical objects. All properties of the basis domain,{} \\spadignore{e.g.} being an abelian group are carried over the equation domain,{} by performing the structural operations on the left and on the right hand side.")) (|subst| (($ $ $) "\\spad{subst(eq1,{}eq2)} substitutes \\spad{eq2} into both sides of \\spad{eq1} the \\spad{lhs} of \\spad{eq2} should be a kernel")) (|inv| (($ $) "\\spad{inv(x)} returns the multiplicative inverse of \\spad{x}.")) (/ (($ $ $) "\\spad{e1/e2} produces a new equation by dividing the left and right hand sides of equations \\spad{e1} and \\spad{e2}.")) (|factorAndSplit| (((|List| $) $) "\\spad{factorAndSplit(eq)} make the right hand side 0 and factors the new left hand side. Each factor is equated to 0 and put into the resulting list without repetitions.")) (|rightOne| (((|Union| $ "failed") $) "\\spad{rightOne(eq)} divides by the right hand side.") (((|Union| $ "failed") $) "\\spad{rightOne(eq)} divides by the right hand side,{} if possible.")) (|leftOne| (((|Union| $ "failed") $) "\\spad{leftOne(eq)} divides by the left hand side.") (((|Union| $ "failed") $) "\\spad{leftOne(eq)} divides by the left hand side,{} if possible.")) (* (($ $ |#1|) "\\spad{eqn*x} produces a new equation by multiplying both sides of equation eqn by \\spad{x}.") (($ |#1| $) "\\spad{x*eqn} produces a new equation by multiplying both sides of equation eqn by \\spad{x}.")) (- (($ $ |#1|) "\\spad{eqn-x} produces a new equation by subtracting \\spad{x} from both sides of equation eqn.") (($ |#1| $) "\\spad{x-eqn} produces a new equation by subtracting both sides of equation eqn from \\spad{x}.")) (|rightZero| (($ $) "\\spad{rightZero(eq)} subtracts the right hand side.")) (|leftZero| (($ $) "\\spad{leftZero(eq)} subtracts the left hand side.")) (+ (($ $ |#1|) "\\spad{eqn+x} produces a new equation by adding \\spad{x} to both sides of equation eqn.") (($ |#1| $) "\\spad{x+eqn} produces a new equation by adding \\spad{x} to both sides of equation eqn.")) (|eval| (($ $ (|List| $)) "\\spad{eval(eqn,{} [x1=v1,{} ... xn=vn])} replaces \\spad{xi} by \\spad{vi} in equation \\spad{eqn}.") (($ $ $) "\\spad{eval(eqn,{} x=f)} replaces \\spad{x} by \\spad{f} in equation \\spad{eqn}.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,{}eqn)} constructs a new equation by applying \\spad{f} to both sides of \\spad{eqn}.")) (|rhs| ((|#1| $) "\\spad{rhs(eqn)} returns the right hand side of equation \\spad{eqn}.")) (|lhs| ((|#1| $) "\\spad{lhs(eqn)} returns the left hand side of equation \\spad{eqn}.")) (|swap| (($ $) "\\spad{swap(eq)} interchanges left and right hand side of equation \\spad{eq}.")) (|equation| (($ |#1| |#1|) "\\spad{equation(a,{}b)} creates an equation.")) (= (($ |#1| |#1|) "\\spad{a=b} creates an equation."))) 
((-4502 -3322 (|has| |#1| (-1039)) (|has| |#1| (-471))) (-4499 |has| |#1| (-1039)) (-4500 |has| |#1| (-1039))) 
((|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-1082))) (|HasCategory| |#1| (QUOTE (-1039))) (|HasCategory| |#1| (LIST (QUOTE -887) (QUOTE (-1153)))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#1| (QUOTE (-1039)))) (|HasCategory| |#1| (QUOTE (-471))) (|HasCategory| |#1| (LIST (QUOTE -515) (QUOTE (-1153)) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082)))) (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-291))) (-3322 (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-471)))) (-3322 (|HasCategory| |#1| (QUOTE (-471))) (|HasCategory| |#1| (QUOTE (-1039)))) (|HasCategory| |#1| (QUOTE (-170))) (-3322 (|HasCategory| |#1| (QUOTE (-170))) (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-1039)))) (-3322 (|HasCategory| |#1| (QUOTE (-170))) (|HasCategory| |#1| (QUOTE (-359)))) (|HasCategory| |#1| (QUOTE (-708))) (-3322 (|HasCategory| |#1| (QUOTE (-471))) (|HasCategory| |#1| (QUOTE (-708)))) (|HasCategory| |#1| (QUOTE (-1094))) (-3322 (|HasCategory| |#1| (QUOTE (-471))) (|HasCategory| |#1| (QUOTE (-708))) (|HasCategory| |#1| (QUOTE (-1094)))) (|HasCategory| |#1| (QUOTE (-21))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-170))) (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-1039)))) (-3322 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-708)))) (|HasCategory| |#1| (QUOTE (-25))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-170))) (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-1039)))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-170))) (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-471))) (|HasCategory| |#1| (QUOTE (-708))) (|HasCategory| |#1| (QUOTE (-1039))) (|HasCategory| |#1| (QUOTE (-1094))) (|HasCategory| |#1| (QUOTE (-1082))))) 
(-284 |Key| |Entry|) 
((|constructor| (NIL "This domain provides tables where the keys are compared using \\spadfun{eq?}. Thus keys are considered equal only if they are the same instance of a structure."))) 
((-4505 . T) (-4506 . T)) 
((|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (QUOTE (-1082))) (-12 (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (LIST (QUOTE -298) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2286) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3071) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (QUOTE (-1082)))) (|HasCategory| |#1| (QUOTE (-834))) (|HasCategory| |#2| (QUOTE (-1082))) (-3322 (|HasCategory| (-2 (|:| -2286 |#1|) (|:| -3071 |#2|)) (QUOTE (-1082))) (|HasCategory| |#2| (QUOTE (-1082)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-1082))))) 
(-285) 
((|constructor| (NIL "ErrorFunctions implements error functions callable from the system interpreter. Typically,{} these functions would be called in user functions. The simple forms of the functions take one argument which is either a string (an error message) or a list of strings which all together make up a message. The list can contain formatting codes (see below). The more sophisticated versions takes two arguments where the first argument is the name of the function from which the error was invoked and the second argument is either a string or a list of strings,{} as above. When you use the one argument version in an interpreter function,{} the system will automatically insert the name of the function as the new first argument. Thus in the user interpreter function\\spad{\\br} \\tab{5}\\spad{f x == if x < 0 then error \"negative argument\" else x}\\spad{\\br} the call to error will actually be of the form\\spad{\\br} \\tab{5}\\spad{error(\"f\",{}\"negative argument\")}\\spad{\\br} because the interpreter will have created a new first argument. \\blankline Formatting codes: error messages may contain the following formatting codes (they should either start or end a string or else have blanks around them):\\spad{\\br} \\spad{\\%l}\\tab{6}start a new line\\spad{\\br} \\spad{\\%b}\\tab{6}start printing in a bold font (where available)\\spad{\\br} \\spad{\\%d}\\tab{6}stop printing in a bold font (where available)\\spad{\\br} \\spad{\\%ceon}\\tab{3}start centering message lines\\spad{\\br} \\spad{\\%ceoff}\\tab{2}stop centering message lines\\spad{\\br} \\spad{\\%rjon}\\tab{3}start displaying lines \"ragged left\"\\spad{\\br} \\spad{\\%rjoff}\\tab{2}stop displaying lines \"ragged left\"\\spad{\\br} \\spad{\\%i}\\tab{6}indent following lines 3 additional spaces\\spad{\\br} \\spad{\\%u}\\tab{6}unindent following lines 3 additional spaces\\spad{\\br} \\spad{\\%xN}\\tab{5}insert \\spad{N} blanks (eg,{} \\spad{\\%x10} inserts 10 blanks) \\blankline")) (|error| (((|Exit|) (|String|) (|List| (|String|))) "\\spad{error(nam,{}lmsg)} displays error messages \\spad{lmsg} preceded by a message containing the name \\spad{nam} of the function in which the error is contained.") (((|Exit|) (|String|) (|String|)) "\\spad{error(nam,{}msg)} displays error message \\spad{msg} preceded by a message containing the name \\spad{nam} of the function in which the error is contained.") (((|Exit|) (|List| (|String|))) "\\spad{error(lmsg)} displays error message \\spad{lmsg} and terminates.") (((|Exit|) (|String|)) "\\spad{error(msg)} displays error message \\spad{msg} and terminates."))) 
NIL 
NIL 
(-286 -2262 S) 
((|constructor| (NIL "This package allows a map from any expression space into any object to be lifted to a kernel over the expression set,{} using a given property of the operator of the kernel.")) (|map| ((|#2| (|Mapping| |#2| |#1|) (|String|) (|Kernel| |#1|)) "\\spad{map(f,{} p,{} k)} uses the property \\spad{p} of the operator of \\spad{k},{} in order to lift \\spad{f} and apply it to \\spad{k}."))) 
NIL 
NIL 
(-287 E -2262) 
((|constructor| (NIL "This package allows a mapping \\spad{E} \\spad{->} \\spad{F} to be lifted to a kernel over \\spad{E}; This lifting can fail if the operator of the kernel cannot be applied in \\spad{F}; Do not use this package with \\spad{E} = \\spad{F},{} since this may drop some properties of the operators.")) (|map| ((|#2| (|Mapping| |#2| |#1|) (|Kernel| |#1|)) "\\spad{map(f,{} k)} returns \\spad{g = op(f(a1),{}...,{}f(an))} where \\spad{k = op(a1,{}...,{}an)}."))) 
NIL 
NIL 
(-288 A B) 
((|constructor| (NIL "\\spad{ExpertSystemContinuityPackage1} exports a function to check range inclusion")) (|in?| (((|Boolean|) (|DoubleFloat|)) "\\spad{in?(p)} tests whether point \\spad{p} is internal to the range [\\spad{A..B}]"))) 
NIL 
NIL 
(-289) 
((|constructor| (NIL "ExpertSystemContinuityPackage is a package of functions for the use of domains belonging to the category \\axiomType{NumericalIntegration}.")) (|sdf2lst| (((|List| (|String|)) (|Stream| (|DoubleFloat|))) "\\spad{sdf2lst(ln)} coerces a Stream of \\axiomType{DoubleFloat} to \\axiomType{List}(\\axiomType{String})")) (|ldf2lst| (((|List| (|String|)) (|List| (|DoubleFloat|))) "\\spad{ldf2lst(ln)} coerces a List of \\axiomType{DoubleFloat} to \\axiomType{List}(\\axiomType{String})")) (|df2st| (((|String|) (|DoubleFloat|)) "\\spad{df2st(n)} coerces a \\axiomType{DoubleFloat} to \\axiomType{String}")) (|polynomialZeros| (((|List| (|DoubleFloat|)) (|Polynomial| (|Fraction| (|Integer|))) (|Symbol|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{polynomialZeros(fn,{}var,{}range)} calculates the real zeros of the polynomial which are contained in the given interval. It returns a list of points (\\axiomType{Doublefloat}) for which the univariate polynomial \\spad{fn} is zero.")) (|singularitiesOf| (((|Stream| (|DoubleFloat|)) (|Vector| (|Expression| (|DoubleFloat|))) (|List| (|Symbol|)) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{singularitiesOf(v,{}vars,{}range)} returns a list of points (\\axiomType{Doublefloat}) at which a NAG fortran version of \\spad{v} will most likely produce an error. This includes those points which evaluate to 0/0.") (((|Stream| (|DoubleFloat|)) (|Expression| (|DoubleFloat|)) (|List| (|Symbol|)) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{singularitiesOf(e,{}vars,{}range)} returns a list of points (\\axiomType{Doublefloat}) at which a NAG fortran version of \\spad{e} will most likely produce an error. This includes those points which evaluate to 0/0.")) (|zerosOf| (((|Stream| (|DoubleFloat|)) (|Expression| (|DoubleFloat|)) (|List| (|Symbol|)) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{zerosOf(e,{}vars,{}range)} returns a list of points (\\axiomType{Doublefloat}) at which a NAG fortran version of \\spad{e} will most likely produce an error.")) (|problemPoints| (((|List| (|DoubleFloat|)) (|Expression| (|DoubleFloat|)) (|Symbol|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{problemPoints(f,{}var,{}range)} returns a list of possible problem points by looking at the zeros of the denominator of the function \\spad{f} if it can be retracted to \\axiomType{Polynomial(DoubleFloat)}.")) (|functionIsFracPolynomial?| (((|Boolean|) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{functionIsFracPolynomial?(args)} tests whether the function can be retracted to \\axiomType{Fraction(Polynomial(DoubleFloat))}")) (|gethi| (((|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{gethi(u)} gets the \\axiomType{DoubleFloat} equivalent of the second endpoint of the range \\axiom{\\spad{u}}")) (|getlo| (((|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{getlo(u)} gets the \\axiomType{DoubleFloat} equivalent of the first endpoint of the range \\axiom{\\spad{u}}"))) 
NIL 
NIL 
(-290 S) 
((|constructor| (NIL "An expression space is a set which is closed under certain operators.")) (|odd?| (((|Boolean|) $) "\\spad{odd? x} is \\spad{true} if \\spad{x} is an odd integer.")) (|even?| (((|Boolean|) $) "\\spad{even? x} is \\spad{true} if \\spad{x} is an even integer.")) (|definingPolynomial| (($ $) "\\spad{definingPolynomial(x)} returns an expression \\spad{p} such that \\spad{p(x) = 0}.")) (|minPoly| (((|SparseUnivariatePolynomial| $) (|Kernel| $)) "\\spad{minPoly(k)} returns \\spad{p} such that \\spad{p(k) = 0}.")) (|eval| (($ $ (|BasicOperator|) (|Mapping| $ $)) "\\spad{eval(x,{} s,{} f)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|BasicOperator|) (|Mapping| $ (|List| $))) "\\spad{eval(x,{} s,{} f)} replaces every \\spad{s(a1,{}..,{}am)} in \\spad{x} by \\spad{f(a1,{}..,{}am)} for any \\spad{a1},{}...,{}\\spad{am}.") (($ $ (|List| (|BasicOperator|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x,{} [s1,{}...,{}sm],{} [f1,{}...,{}fm])} replaces every \\spad{\\spad{si}(a1,{}...,{}an)} in \\spad{x} by \\spad{\\spad{fi}(a1,{}...,{}an)} for any \\spad{a1},{}...,{}\\spad{an}.") (($ $ (|List| (|BasicOperator|)) (|List| (|Mapping| $ $))) "\\spad{eval(x,{} [s1,{}...,{}sm],{} [f1,{}...,{}fm])} replaces every \\spad{\\spad{si}(a)} in \\spad{x} by \\spad{\\spad{fi}(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|Mapping| $ $)) "\\spad{eval(x,{} s,{} f)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|Mapping| $ (|List| $))) "\\spad{eval(x,{} s,{} f)} replaces every \\spad{s(a1,{}..,{}am)} in \\spad{x} by \\spad{f(a1,{}..,{}am)} for any \\spad{a1},{}...,{}\\spad{am}.") (($ $ (|List| (|Symbol|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x,{} [s1,{}...,{}sm],{} [f1,{}...,{}fm])} replaces every \\spad{\\spad{si}(a1,{}...,{}an)} in \\spad{x} by \\spad{\\spad{fi}(a1,{}...,{}an)} for any \\spad{a1},{}...,{}\\spad{an}.") (($ $ (|List| (|Symbol|)) (|List| (|Mapping| $ $))) "\\spad{eval(x,{} [s1,{}...,{}sm],{} [f1,{}...,{}fm])} replaces every \\spad{\\spad{si}(a)} in \\spad{x} by \\spad{\\spad{fi}(a)} for any \\spad{a}.")) (|freeOf?| (((|Boolean|) $ (|Symbol|)) "\\spad{freeOf?(x,{} s)} tests if \\spad{x} does not contain any operator whose name is \\spad{s}.") (((|Boolean|) $ $) "\\spad{freeOf?(x,{} y)} tests if \\spad{x} does not contain any occurrence of \\spad{y},{} where \\spad{y} is a single kernel.")) (|map| (($ (|Mapping| $ $) (|Kernel| $)) "\\spad{map(f,{} k)} returns \\spad{op(f(x1),{}...,{}f(xn))} where \\spad{k = op(x1,{}...,{}xn)}.")) (|kernel| (($ (|BasicOperator|) (|List| $)) "\\spad{kernel(op,{} [f1,{}...,{}fn])} constructs \\spad{op(f1,{}...,{}fn)} without evaluating it.") (($ (|BasicOperator|) $) "\\spad{kernel(op,{} x)} constructs \\spad{op}(\\spad{x}) without evaluating it.")) (|is?| (((|Boolean|) $ (|Symbol|)) "\\spad{is?(x,{} s)} tests if \\spad{x} is a kernel and is the name of its operator is \\spad{s}.") (((|Boolean|) $ (|BasicOperator|)) "\\spad{is?(x,{} op)} tests if \\spad{x} is a kernel and is its operator is op.")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} tests if \\% accepts \\spad{op} as applicable to its elements.")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\%.")) (|operators| (((|List| (|BasicOperator|)) $) "\\spad{operators(f)} returns all the basic operators appearing in \\spad{f},{} no matter what their levels are.")) (|tower| (((|List| (|Kernel| $)) $) "\\spad{tower(f)} returns all the kernels appearing in \\spad{f},{} no matter what their levels are.")) (|kernels| (((|List| (|Kernel| $)) $) "\\spad{kernels(f)} returns the list of all the top-level kernels appearing in \\spad{f},{} but not the ones appearing in the arguments of the top-level kernels.")) (|mainKernel| (((|Union| (|Kernel| $) "failed") $) "\\spad{mainKernel(f)} returns a kernel of \\spad{f} with maximum nesting level,{} or if \\spad{f} has no kernels (\\spadignore{i.e.} \\spad{f} is a constant).")) (|height| (((|NonNegativeInteger|) $) "\\spad{height(f)} returns the highest nesting level appearing in \\spad{f}. Constants have height 0. Symbols have height 1. For any operator op and expressions \\spad{f1},{}...,{}\\spad{fn},{} \\spad{op(f1,{}...,{}fn)} has height equal to \\spad{1 + max(height(f1),{}...,{}height(fn))}.")) (|distribute| (($ $ $) "\\spad{distribute(f,{} g)} expands all the kernels in \\spad{f} that contain \\spad{g} in their arguments and that are formally enclosed by a \\spadfunFrom{box}{ExpressionSpace} or a \\spadfunFrom{paren}{ExpressionSpace} expression.") (($ $) "\\spad{distribute(f)} expands all the kernels in \\spad{f} that are formally enclosed by a \\spadfunFrom{box}{ExpressionSpace} or \\spadfunFrom{paren}{ExpressionSpace} expression.")) (|paren| (($ (|List| $)) "\\spad{paren([f1,{}...,{}fn])} returns \\spad{(f1,{}...,{}fn)}. This prevents the \\spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \\spad{atan(paren [x,{} 2])} returns the formal kernel \\spad{atan((x,{} 2))}.") (($ $) "\\spad{paren(f)} returns (\\spad{f}). This prevents \\spad{f} from being evaluated when operators are applied to it. For example,{} \\spad{log(1)} returns 0,{} but \\spad{log(paren 1)} returns the formal kernel log((1)).")) (|box| (($ (|List| $)) "\\spad{box([f1,{}...,{}fn])} returns \\spad{(f1,{}...,{}fn)} with a 'box' around them that prevents the \\spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \\spad{atan(box [x,{} 2])} returns the formal kernel \\spad{atan(x,{} 2)}.") (($ $) "\\spad{box(f)} returns \\spad{f} with a 'box' around it that prevents \\spad{f} from being evaluated when operators are applied to it. For example,{} \\spad{log(1)} returns 0,{} but \\spad{log(box 1)} returns the formal kernel log(1).")) (|subst| (($ $ (|List| (|Kernel| $)) (|List| $)) "\\spad{subst(f,{} [k1...,{}kn],{} [g1,{}...,{}gn])} replaces the kernels \\spad{k1},{}...,{}\\spad{kn} by \\spad{g1},{}...,{}\\spad{gn} formally in \\spad{f}.") (($ $ (|List| (|Equation| $))) "\\spad{subst(f,{} [k1 = g1,{}...,{}kn = gn])} replaces the kernels \\spad{k1},{}...,{}\\spad{kn} by \\spad{g1},{}...,{}\\spad{gn} formally in \\spad{f}.") (($ $ (|Equation| $)) "\\spad{subst(f,{} k = g)} replaces the kernel \\spad{k} by \\spad{g} formally in \\spad{f}.")) (|elt| (($ (|BasicOperator|) (|List| $)) "\\spad{elt(op,{}[x1,{}...,{}xn])} or \\spad{op}([\\spad{x1},{}...,{}\\spad{xn}]) applies the \\spad{n}-ary operator \\spad{op} to \\spad{x1},{}...,{}\\spad{xn}.") (($ (|BasicOperator|) $ $ $ $) "\\spad{elt(op,{}x,{}y,{}z,{}t)} or \\spad{op}(\\spad{x},{} \\spad{y},{} \\spad{z},{} \\spad{t}) applies the 4-ary operator \\spad{op} to \\spad{x},{} \\spad{y},{} \\spad{z} and \\spad{t}.") (($ (|BasicOperator|) $ $ $) "\\spad{elt(op,{}x,{}y,{}z)} or \\spad{op}(\\spad{x},{} \\spad{y},{} \\spad{z}) applies the ternary operator \\spad{op} to \\spad{x},{} \\spad{y} and \\spad{z}.") (($ (|BasicOperator|) $ $) "\\spad{elt(op,{}x,{}y)} or \\spad{op}(\\spad{x},{} \\spad{y}) applies the binary operator \\spad{op} to \\spad{x} and \\spad{y}.") (($ (|BasicOperator|) $) "\\spad{elt(op,{}x)} or \\spad{op}(\\spad{x}) applies the unary operator \\spad{op} to \\spad{x}."))) 
NIL 
((|HasCategory| |#1| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-1039)))) 
(-291) 
((|constructor| (NIL "An expression space is a set which is closed under certain operators.")) (|odd?| (((|Boolean|) $) "\\spad{odd? x} is \\spad{true} if \\spad{x} is an odd integer.")) (|even?| (((|Boolean|) $) "\\spad{even? x} is \\spad{true} if \\spad{x} is an even integer.")) (|definingPolynomial| (($ $) "\\spad{definingPolynomial(x)} returns an expression \\spad{p} such that \\spad{p(x) = 0}.")) (|minPoly| (((|SparseUnivariatePolynomial| $) (|Kernel| $)) "\\spad{minPoly(k)} returns \\spad{p} such that \\spad{p(k) = 0}.")) (|eval| (($ $ (|BasicOperator|) (|Mapping| $ $)) "\\spad{eval(x,{} s,{} f)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|BasicOperator|) (|Mapping| $ (|List| $))) "\\spad{eval(x,{} s,{} f)} replaces every \\spad{s(a1,{}..,{}am)} in \\spad{x} by \\spad{f(a1,{}..,{}am)} for any \\spad{a1},{}...,{}\\spad{am}.") (($ $ (|List| (|BasicOperator|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x,{} [s1,{}...,{}sm],{} [f1,{}...,{}fm])} replaces every \\spad{\\spad{si}(a1,{}...,{}an)} in \\spad{x} by \\spad{\\spad{fi}(a1,{}...,{}an)} for any \\spad{a1},{}...,{}\\spad{an}.") (($ $ (|List| (|BasicOperator|)) (|List| (|Mapping| $ $))) "\\spad{eval(x,{} [s1,{}...,{}sm],{} [f1,{}...,{}fm])} replaces every \\spad{\\spad{si}(a)} in \\spad{x} by \\spad{\\spad{fi}(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|Mapping| $ $)) "\\spad{eval(x,{} s,{} f)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|Mapping| $ (|List| $))) "\\spad{eval(x,{} s,{} f)} replaces every \\spad{s(a1,{}..,{}am)} in \\spad{x} by \\spad{f(a1,{}..,{}am)} for any \\spad{a1},{}...,{}\\spad{am}.") (($ $ (|List| (|Symbol|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x,{} [s1,{}...,{}sm],{} [f1,{}...,{}fm])} replaces every \\spad{\\spad{si}(a1,{}...,{}an)} in \\spad{x} by \\spad{\\spad{fi}(a1,{}...,{}an)} for any \\spad{a1},{}...,{}\\spad{an}.") (($ $ (|List| (|Symbol|)) (|List| (|Mapping| $ $))) "\\spad{eval(x,{} [s1,{}...,{}sm],{} [f1,{}...,{}fm])} replaces every \\spad{\\spad{si}(a)} in \\spad{x} by \\spad{\\spad{fi}(a)} for any \\spad{a}.")) (|freeOf?| (((|Boolean|) $ (|Symbol|)) "\\spad{freeOf?(x,{} s)} tests if \\spad{x} does not contain any operator whose name is \\spad{s}.") (((|Boolean|) $ $) "\\spad{freeOf?(x,{} y)} tests if \\spad{x} does not contain any occurrence of \\spad{y},{} where \\spad{y} is a single kernel.")) (|map| (($ (|Mapping| $ $) (|Kernel| $)) "\\spad{map(f,{} k)} returns \\spad{op(f(x1),{}...,{}f(xn))} where \\spad{k = op(x1,{}...,{}xn)}.")) (|kernel| (($ (|BasicOperator|) (|List| $)) "\\spad{kernel(op,{} [f1,{}...,{}fn])} constructs \\spad{op(f1,{}...,{}fn)} without evaluating it.") (($ (|BasicOperator|) $) "\\spad{kernel(op,{} x)} constructs \\spad{op}(\\spad{x}) without evaluating it.")) (|is?| (((|Boolean|) $ (|Symbol|)) "\\spad{is?(x,{} s)} tests if \\spad{x} is a kernel and is the name of its operator is \\spad{s}.") (((|Boolean|) $ (|BasicOperator|)) "\\spad{is?(x,{} op)} tests if \\spad{x} is a kernel and is its operator is op.")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} tests if \\% accepts \\spad{op} as applicable to its elements.")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\%.")) (|operators| (((|List| (|BasicOperator|)) $) "\\spad{operators(f)} returns all the basic operators appearing in \\spad{f},{} no matter what their levels are.")) (|tower| (((|List| (|Kernel| $)) $) "\\spad{tower(f)} returns all the kernels appearing in \\spad{f},{} no matter what their levels are.")) (|kernels| (((|List| (|Kernel| $)) $) "\\spad{kernels(f)} returns the list of all the top-level kernels appearing in \\spad{f},{} but not the ones appearing in the arguments of the top-level kernels.")) (|mainKernel| (((|Union| (|Kernel| $) "failed") $) "\\spad{mainKernel(f)} returns a kernel of \\spad{f} with maximum nesting level,{} or if \\spad{f} has no kernels (\\spadignore{i.e.} \\spad{f} is a constant).")) (|height| (((|NonNegativeInteger|) $) "\\spad{height(f)} returns the highest nesting level appearing in \\spad{f}. Constants have height 0. Symbols have height 1. For any operator op and expressions \\spad{f1},{}...,{}\\spad{fn},{} \\spad{op(f1,{}...,{}fn)} has height equal to \\spad{1 + max(height(f1),{}...,{}height(fn))}.")) (|distribute| (($ $ $) "\\spad{distribute(f,{} g)} expands all the kernels in \\spad{f} that contain \\spad{g} in their arguments and that are formally enclosed by a \\spadfunFrom{box}{ExpressionSpace} or a \\spadfunFrom{paren}{ExpressionSpace} expression.") (($ $) "\\spad{distribute(f)} expands all the kernels in \\spad{f} that are formally enclosed by a \\spadfunFrom{box}{ExpressionSpace} or \\spadfunFrom{paren}{ExpressionSpace} expression.")) (|paren| (($ (|List| $)) "\\spad{paren([f1,{}...,{}fn])} returns \\spad{(f1,{}...,{}fn)}. This prevents the \\spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \\spad{atan(paren [x,{} 2])} returns the formal kernel \\spad{atan((x,{} 2))}.") (($ $) "\\spad{paren(f)} returns (\\spad{f}). This prevents \\spad{f} from being evaluated when operators are applied to it. For example,{} \\spad{log(1)} returns 0,{} but \\spad{log(paren 1)} returns the formal kernel log((1)).")) (|box| (($ (|List| $)) "\\spad{box([f1,{}...,{}fn])} returns \\spad{(f1,{}...,{}fn)} with a 'box' around them that prevents the \\spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \\spad{atan(box [x,{} 2])} returns the formal kernel \\spad{atan(x,{} 2)}.") (($ $) "\\spad{box(f)} returns \\spad{f} with a 'box' around it that prevents \\spad{f} from being evaluated when operators are applied to it. For example,{} \\spad{log(1)} returns 0,{} but \\spad{log(box 1)} returns the formal kernel log(1).")) (|subst| (($ $ (|List| (|Kernel| $)) (|List| $)) "\\spad{subst(f,{} [k1...,{}kn],{} [g1,{}...,{}gn])} replaces the kernels \\spad{k1},{}...,{}\\spad{kn} by \\spad{g1},{}...,{}\\spad{gn} formally in \\spad{f}.") (($ $ (|List| (|Equation| $))) "\\spad{subst(f,{} [k1 = g1,{}...,{}kn = gn])} replaces the kernels \\spad{k1},{}...,{}\\spad{kn} by \\spad{g1},{}...,{}\\spad{gn} formally in \\spad{f}.") (($ $ (|Equation| $)) "\\spad{subst(f,{} k = g)} replaces the kernel \\spad{k} by \\spad{g} formally in \\spad{f}.")) (|elt| (($ (|BasicOperator|) (|List| $)) "\\spad{elt(op,{}[x1,{}...,{}xn])} or \\spad{op}([\\spad{x1},{}...,{}\\spad{xn}]) applies the \\spad{n}-ary operator \\spad{op} to \\spad{x1},{}...,{}\\spad{xn}.") (($ (|BasicOperator|) $ $ $ $) "\\spad{elt(op,{}x,{}y,{}z,{}t)} or \\spad{op}(\\spad{x},{} \\spad{y},{} \\spad{z},{} \\spad{t}) applies the 4-ary operator \\spad{op} to \\spad{x},{} \\spad{y},{} \\spad{z} and \\spad{t}.") (($ (|BasicOperator|) $ $ $) "\\spad{elt(op,{}x,{}y,{}z)} or \\spad{op}(\\spad{x},{} \\spad{y},{} \\spad{z}) applies the ternary operator \\spad{op} to \\spad{x},{} \\spad{y} and \\spad{z}.") (($ (|BasicOperator|) $ $) "\\spad{elt(op,{}x,{}y)} or \\spad{op}(\\spad{x},{} \\spad{y}) applies the binary operator \\spad{op} to \\spad{x} and \\spad{y}.") (($ (|BasicOperator|) $) "\\spad{elt(op,{}x)} or \\spad{op}(\\spad{x}) applies the unary operator \\spad{op} to \\spad{x}."))) 
NIL 
NIL 
(-292 R1) 
((|constructor| (NIL "\\axiom{\\spad{ExpertSystemToolsPackage1}} contains some useful functions for use by the computational agents of Ordinary Differential Equation solvers.")) (|neglist| (((|List| |#1|) (|List| |#1|)) "\\spad{neglist(l)} returns only the negative elements of the list \\spad{l}"))) 
NIL 
NIL 
(-293 R1 R2) 
((|constructor| (NIL "\\axiom{\\spad{ExpertSystemToolsPackage2}} contains some useful functions for use by the computational agents of Ordinary Differential Equation solvers.")) (|map| (((|Matrix| |#2|) (|Mapping| |#2| |#1|) (|Matrix| |#1|)) "\\spad{map(f,{}m)} applies a mapping \\spad{f:R1} \\spad{->} \\spad{R2} onto a matrix \\spad{m} in \\spad{R1} returning a matrix in \\spad{R2}"))) 
NIL 
NIL 
(-294) 
((|constructor| (NIL "\\axiom{ExpertSystemToolsPackage} contains some useful functions for use by the computational agents of numerical solvers.")) (|mat| (((|Matrix| (|DoubleFloat|)) (|List| (|DoubleFloat|)) (|NonNegativeInteger|)) "\\spad{mat(a,{}n)} constructs a one-dimensional matrix of a.")) (|fi2df| (((|DoubleFloat|) (|Fraction| (|Integer|))) "\\spad{fi2df(f)} coerces a \\axiomType{Fraction Integer} to \\axiomType{DoubleFloat}")) (|df2ef| (((|Expression| (|Float|)) (|DoubleFloat|)) "\\spad{df2ef(a)} coerces a \\axiomType{DoubleFloat} to \\axiomType{Expression Float}")) (|pdf2df| (((|DoubleFloat|) (|Polynomial| (|DoubleFloat|))) "\\spad{pdf2df(p)} coerces a \\axiomType{Polynomial DoubleFloat} to \\axiomType{DoubleFloat}. It is an error if \\axiom{\\spad{p}} is not retractable to DoubleFloat.")) (|pdf2ef| (((|Expression| (|Float|)) (|Polynomial| (|DoubleFloat|))) "\\spad{pdf2ef(p)} coerces a \\axiomType{Polynomial DoubleFloat} to \\axiomType{Expression Float}")) (|iflist2Result| (((|Result|) (|Record| (|:| |stiffness| (|Float|)) (|:| |stability| (|Float|)) (|:| |expense| (|Float|)) (|:| |accuracy| (|Float|)) (|:| |intermediateResults| (|Float|)))) "\\spad{iflist2Result(m)} converts a attributes record into a \\axiomType{Result}")) (|att2Result| (((|Result|) (|Record| (|:| |endPointContinuity| (|Union| (|:| |continuous| "Continuous at the end points") (|:| |lowerSingular| "There is a singularity at the lower end point") (|:| |upperSingular| "There is a singularity at the upper end point") (|:| |bothSingular| "There are singularities at both end points") (|:| |notEvaluated| "End point continuity not yet evaluated"))) (|:| |singularitiesStream| (|Union| (|:| |str| (|Stream| (|DoubleFloat|))) (|:| |notEvaluated| "Internal singularities not yet evaluated"))) (|:| |range| (|Union| (|:| |finite| "The range is finite") (|:| |lowerInfinite| "The bottom of range is infinite") (|:| |upperInfinite| "The top of range is infinite") (|:| |bothInfinite| "Both top and bottom points are infinite") (|:| |notEvaluated| "Range not yet evaluated"))))) "\\spad{att2Result(m)} converts a attributes record into a \\axiomType{Result}")) (|measure2Result| (((|Result|) (|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|))) (|:| |extra| (|Result|)))) "\\spad{measure2Result(m)} converts a measure record into a \\axiomType{Result}") (((|Result|) (|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|))))) "\\spad{measure2Result(m)} converts a measure record into a \\axiomType{Result}")) (|outputMeasure| (((|String|) (|Float|)) "\\spad{outputMeasure(n)} rounds \\spad{n} to 3 decimal places and outputs it as a string")) (|concat| (((|Result|) (|List| (|Result|))) "\\spad{concat(l)} concatenates a list of aggregates of type \\axiomType{Result}") (((|Result|) (|Result|) (|Result|)) "\\spad{concat(a,{}b)} adds two aggregates of type \\axiomType{Result}.")) (|gethi| (((|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{gethi(u)} gets the \\axiomType{DoubleFloat} equivalent of the second endpoint of the range \\spad{u}")) (|getlo| (((|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{getlo(u)} gets the \\axiomType{DoubleFloat} equivalent of the first endpoint of the range \\spad{u}")) (|sdf2lst| (((|List| (|String|)) (|Stream| (|DoubleFloat|))) "\\spad{sdf2lst(ln)} coerces a \\axiomType{Stream DoubleFloat} to \\axiomType{String}")) (|ldf2lst| (((|List| (|String|)) (|List| (|DoubleFloat|))) "\\spad{ldf2lst(ln)} coerces a \\axiomType{List DoubleFloat} to \\axiomType{List String}")) (|f2st| (((|String|) (|Float|)) "\\spad{f2st(n)} coerces a \\axiomType{Float} to \\axiomType{String}")) (|df2st| (((|String|) (|DoubleFloat|)) "\\spad{df2st(n)} coerces a \\axiomType{DoubleFloat} to \\axiomType{String}")) (|in?| (((|Boolean|) (|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{in?(p,{}range)} tests whether point \\spad{p} is internal to the \\spad{range} \\spad{range}")) (|vedf2vef| (((|Vector| (|Expression| (|Float|))) (|Vector| (|Expression| (|DoubleFloat|)))) "\\spad{vedf2vef(v)} maps \\axiomType{Vector Expression DoubleFloat} to \\axiomType{Vector Expression Float}")) (|edf2ef| (((|Expression| (|Float|)) (|Expression| (|DoubleFloat|))) "\\spad{edf2ef(e)} maps \\axiomType{Expression DoubleFloat} to \\axiomType{Expression Float}")) (|ldf2vmf| (((|Vector| (|MachineFloat|)) (|List| (|DoubleFloat|))) "\\spad{ldf2vmf(l)} coerces a \\axiomType{List DoubleFloat} to \\axiomType{List MachineFloat}")) (|df2mf| (((|MachineFloat|) (|DoubleFloat|)) "\\spad{df2mf(n)} coerces a \\axiomType{DoubleFloat} to \\axiomType{MachineFloat}")) (|dflist| (((|List| (|DoubleFloat|)) (|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))))) "\\spad{dflist(l)} returns a list of \\axiomType{DoubleFloat} equivalents of list \\spad{l}")) (|dfRange| (((|Segment| (|OrderedCompletion| (|DoubleFloat|))) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{dfRange(r)} converts a range including \\inputbitmap{\\htbmdir{}/plusminus.bitmap} \\infty to \\axiomType{DoubleFloat} equavalents.")) (|edf2efi| (((|Expression| (|Fraction| (|Integer|))) (|Expression| (|DoubleFloat|))) "\\spad{edf2efi(e)} coerces \\axiomType{Expression DoubleFloat} into \\axiomType{Expression Fraction Integer}")) (|numberOfOperations| (((|Record| (|:| |additions| (|Integer|)) (|:| |multiplications| (|Integer|)) (|:| |exponentiations| (|Integer|)) (|:| |functionCalls| (|Integer|))) (|Vector| (|Expression| (|DoubleFloat|)))) "\\spad{numberOfOperations(ode)} counts additions,{} multiplications,{} exponentiations and function calls in the input set of expressions.")) (|expenseOfEvaluation| (((|Float|) (|Vector| (|Expression| (|DoubleFloat|)))) "\\spad{expenseOfEvaluation(o)} gives an approximation of the cost of evaluating a list of expressions in terms of the number of basic operations. < 0.3 inexpensive ; 0.5 neutral ; > 0.7 very expensive 400 `operation units' \\spad{->} 0.75 200 `operation units' \\spad{->} 0.5 83 `operation units' \\spad{->} 0.25 \\spad{**} = 4 units ,{} function calls = 10 units.")) (|isQuotient| (((|Union| (|Expression| (|DoubleFloat|)) "failed") (|Expression| (|DoubleFloat|))) "\\spad{isQuotient(expr)} returns the quotient part of the input expression or \\spad{\"failed\"} if the expression is not of that form.")) (|edf2df| (((|DoubleFloat|) (|Expression| (|DoubleFloat|))) "\\spad{edf2df(n)} maps \\axiomType{Expression DoubleFloat} to \\axiomType{DoubleFloat} It is an error if \\spad{n} is not coercible to DoubleFloat")) (|edf2fi| (((|Fraction| (|Integer|)) (|Expression| (|DoubleFloat|))) "\\spad{edf2fi(n)} maps \\axiomType{Expression DoubleFloat} to \\axiomType{Fraction Integer} It is an error if \\spad{n} is not coercible to Fraction Integer")) (|df2fi| (((|Fraction| (|Integer|)) (|DoubleFloat|)) "\\spad{df2fi(n)} is a function to convert a \\axiomType{DoubleFloat} to a \\axiomType{Fraction Integer}")) (|convert| (((|List| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|List| (|Segment| (|OrderedCompletion| (|Float|))))) "\\spad{convert(l)} is a function to convert a \\axiomType{Segment OrderedCompletion Float} to a \\axiomType{Segment OrderedCompletion DoubleFloat}")) (|socf2socdf| (((|Segment| (|OrderedCompletion| (|DoubleFloat|))) (|Segment| (|OrderedCompletion| (|Float|)))) "\\spad{socf2socdf(a)} is a function to convert a \\axiomType{Segment OrderedCompletion Float} to a \\axiomType{Segment OrderedCompletion DoubleFloat}")) (|ocf2ocdf| (((|OrderedCompletion| (|DoubleFloat|)) (|OrderedCompletion| (|Float|))) "\\spad{ocf2ocdf(a)} is a function to convert an \\axiomType{OrderedCompletion Float} to an \\axiomType{OrderedCompletion DoubleFloat}")) (|ef2edf| (((|Expression| (|DoubleFloat|)) (|Expression| (|Float|))) "\\spad{ef2edf(f)} is a function to convert an \\axiomType{Expression Float} to an \\axiomType{Expression DoubleFloat}")) (|f2df| (((|DoubleFloat|) (|Float|)) "\\spad{f2df(f)} is a function to convert a \\axiomType{Float} to a \\axiomType{DoubleFloat}"))) 
NIL 
NIL 
(-295 S) 
((|constructor| (NIL "A constructive euclidean domain,{} \\spadignore{i.e.} one can divide producing a quotient and a remainder where the remainder is either zero or is smaller (\\spadfun{euclideanSize}) than the divisor. \\blankline Conditional attributes\\spad{\\br} \\tab{5}multiplicativeValuation\\tab{5}Size(a*b)=Size(a)*Size(\\spad{b})\\spad{\\br} \\tab{5}additiveValuation\\tab{11}Size(a*b)=Size(a)+Size(\\spad{b})")) (|multiEuclidean| (((|Union| (|List| $) "failed") (|List| $) $) "\\spad{multiEuclidean([f1,{}...,{}fn],{}z)} returns a list of coefficients \\spad{[a1,{} ...,{} an]} such that \\spad{ z / prod \\spad{fi} = sum aj/fj}. If no such list of coefficients exists,{} \"failed\" is returned.")) (|extendedEuclidean| (((|Union| (|Record| (|:| |coef1| $) (|:| |coef2| $)) "failed") $ $ $) "\\spad{extendedEuclidean(x,{}y,{}z)} either returns a record rec where \\spad{rec.coef1*x+rec.coef2*y=z} or returns \"failed\" if \\spad{z} cannot be expressed as a linear combination of \\spad{x} and \\spad{y}.") (((|Record| (|:| |coef1| $) (|:| |coef2| $) (|:| |generator| $)) $ $) "\\spad{extendedEuclidean(x,{}y)} returns a record rec where \\spad{rec.coef1*x+rec.coef2*y = rec.generator} and rec.generator is a \\spad{gcd} of \\spad{x} and \\spad{y}. The \\spad{gcd} is unique only up to associates if \\spadatt{canonicalUnitNormal} is not asserted. \\spadfun{principalIdeal} provides a version of this operation which accepts an arbitrary length list of arguments.")) (|rem| (($ $ $) "\\spad{x rem y} is the same as \\spad{divide(x,{}y).remainder}. See \\spadfunFrom{divide}{EuclideanDomain}.")) (|quo| (($ $ $) "\\spad{x quo y} is the same as \\spad{divide(x,{}y).quotient}. See \\spadfunFrom{divide}{EuclideanDomain}.")) (|divide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{divide(x,{}y)} divides \\spad{x} by \\spad{y} producing a record containing a \\spad{quotient} and \\spad{remainder},{} where the remainder is smaller (see \\spadfunFrom{sizeLess?}{EuclideanDomain}) than the divisor \\spad{y}.")) (|euclideanSize| (((|NonNegativeInteger|) $) "\\spad{euclideanSize(x)} returns the euclidean size of the element \\spad{x}. Error: if \\spad{x} is zero.")) (|sizeLess?| (((|Boolean|) $ $) "\\spad{sizeLess?(x,{}y)} tests whether \\spad{x} is strictly smaller than \\spad{y} with respect to the \\spadfunFrom{euclideanSize}{EuclideanDomain}."))) 
NIL 
NIL 
(-296) 
((|constructor| (NIL "A constructive euclidean domain,{} \\spadignore{i.e.} one can divide producing a quotient and a remainder where the remainder is either zero or is smaller (\\spadfun{euclideanSize}) than the divisor. \\blankline Conditional attributes\\spad{\\br} \\tab{5}multiplicativeValuation\\tab{5}Size(a*b)=Size(a)*Size(\\spad{b})\\spad{\\br} \\tab{5}additiveValuation\\tab{11}Size(a*b)=Size(a)+Size(\\spad{b})")) (|multiEuclidean| (((|Union| (|List| $) "failed") (|List| $) $) "\\spad{multiEuclidean([f1,{}...,{}fn],{}z)} returns a list of coefficients \\spad{[a1,{} ...,{} an]} such that \\spad{ z / prod \\spad{fi} = sum aj/fj}. If no such list of coefficients exists,{} \"failed\" is returned.")) (|extendedEuclidean| (((|Union| (|Record| (|:| |coef1| $) (|:| |coef2| $)) "failed") $ $ $) "\\spad{extendedEuclidean(x,{}y,{}z)} either returns a record rec where \\spad{rec.coef1*x+rec.coef2*y=z} or returns \"failed\" if \\spad{z} cannot be expressed as a linear combination of \\spad{x} and \\spad{y}.") (((|Record| (|:| |coef1| $) (|:| |coef2| $) (|:| |generator| $)) $ $) "\\spad{extendedEuclidean(x,{}y)} returns a record rec where \\spad{rec.coef1*x+rec.coef2*y = rec.generator} and rec.generator is a \\spad{gcd} of \\spad{x} and \\spad{y}. The \\spad{gcd} is unique only up to associates if \\spadatt{canonicalUnitNormal} is not asserted. \\spadfun{principalIdeal} provides a version of this operation which accepts an arbitrary length list of arguments.")) (|rem| (($ $ $) "\\spad{x rem y} is the same as \\spad{divide(x,{}y).remainder}. See \\spadfunFrom{divide}{EuclideanDomain}.")) (|quo| (($ $ $) "\\spad{x quo y} is the same as \\spad{divide(x,{}y).quotient}. See \\spadfunFrom{divide}{EuclideanDomain}.")) (|divide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{divide(x,{}y)} divides \\spad{x} by \\spad{y} producing a record containing a \\spad{quotient} and \\spad{remainder},{} where the remainder is smaller (see \\spadfunFrom{sizeLess?}{EuclideanDomain}) than the divisor \\spad{y}.")) (|euclideanSize| (((|NonNegativeInteger|) $) "\\spad{euclideanSize(x)} returns the euclidean size of the element \\spad{x}. Error: if \\spad{x} is zero.")) (|sizeLess?| (((|Boolean|) $ $) "\\spad{sizeLess?(x,{}y)} tests whether \\spad{x} is strictly smaller than \\spad{y} with respect to the \\spadfunFrom{euclideanSize}{EuclideanDomain}."))) 
((-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-297 S R) 
((|constructor| (NIL "This category provides \\spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\\spad{''} substitutions.")) (|eval| (($ $ (|List| (|Equation| |#2|))) "\\spad{eval(f,{} [x1 = v1,{}...,{}xn = vn])} replaces \\spad{xi} by \\spad{vi} in \\spad{f}.") (($ $ (|Equation| |#2|)) "\\spad{eval(f,{}x = v)} replaces \\spad{x} by \\spad{v} in \\spad{f}."))) 
NIL 
NIL 
(-298 R) 
((|constructor| (NIL "This category provides \\spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\\spad{''} substitutions.")) (|eval| (($ $ (|List| (|Equation| |#1|))) "\\spad{eval(f,{} [x1 = v1,{}...,{}xn = vn])} replaces \\spad{xi} by \\spad{vi} in \\spad{f}.") (($ $ (|Equation| |#1|)) "\\spad{eval(f,{}x = v)} replaces \\spad{x} by \\spad{v} in \\spad{f}."))) 
NIL 
NIL 
(-299 -2262) 
((|constructor| (NIL "This package is to be used in conjuction with the CycleIndicators package. It provides an evaluation function for SymmetricPolynomials.")) (|eval| ((|#1| (|Mapping| |#1| (|Integer|)) (|SymmetricPolynomial| (|Fraction| (|Integer|)))) "\\spad{eval(f,{}s)} evaluates the cycle index \\spad{s} by applying \\indented{1}{the function \\spad{f} to each integer in a monomial partition,{}} \\indented{1}{forms their product and sums the results over all monomials.}"))) 
NIL 
NIL 
(-300) 
((|constructor| (NIL "A function which does not return directly to its caller should have Exit as its return type. \\blankline Note that It is convenient to have a formal \\spad{coerce} into each type from type Exit. This allows,{} for example,{} errors to be raised in one half of a type-balanced \\spad{if}."))) 
NIL 
NIL 
(-301 R FE |var| |cen|) 
((|constructor| (NIL "UnivariatePuiseuxSeriesWithExponentialSingularity is a domain used to represent essential singularities of functions. Objects in this domain are quotients of sums,{} where each term in the sum is a univariate Puiseux series times the exponential of a univariate Puiseux series.")) (|coerce| (($ (|UnivariatePuiseuxSeries| |#2| |#3| |#4|)) "\\spad{coerce(f)} converts a \\spadtype{UnivariatePuiseuxSeries} to an \\spadtype{ExponentialExpansion}.")) (|limitPlus| (((|Union| (|OrderedCompletion| |#2|) "failed") $) "\\spad{limitPlus(f(var))} returns \\spad{limit(var -> a+,{}f(var))}."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-896))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (LIST (QUOTE -1029) (QUOTE (-1153)))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-146))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-148))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-1013))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-807))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-1128))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (LIST (QUOTE -873) (QUOTE (-560)))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (LIST (QUOTE -873) (QUOTE (-375)))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-375))))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-560))))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-221))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (LIST (QUOTE -515) (QUOTE (-1153)) (LIST (QUOTE -1221) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (LIST (QUOTE -298) (LIST (QUOTE -1221) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (LIST (QUOTE -276) (LIST (QUOTE -1221) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)) (LIST (QUOTE -1221) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-296))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-542))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-834))) (-3322 (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-807))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-834)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-896)))) (-3322 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-896)))) (|HasCategory| (-1221 |#1| |#2| |#3| |#4|) (QUOTE (-146))))) 
(-302 R S) 
((|constructor| (NIL "Lifting of maps to Expressions.")) (|map| (((|Expression| |#2|) (|Mapping| |#2| |#1|) (|Expression| |#1|)) "\\spad{map(f,{} e)} applies \\spad{f} to all the constants appearing in \\spad{e}."))) 
NIL 
NIL 
(-303 R FE) 
((|constructor| (NIL "This package provides functions to convert functional expressions to power series.")) (|series| (((|Any|) |#2| (|Equation| |#2|) (|Fraction| (|Integer|))) "\\spad{series(f,{}x = a,{}n)} expands the expression \\spad{f} as a series in powers of (\\spad{x} - a); terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2| (|Equation| |#2|)) "\\spad{series(f,{}x = a)} expands the expression \\spad{f} as a series in powers of (\\spad{x} - a).") (((|Any|) |#2| (|Fraction| (|Integer|))) "\\spad{series(f,{}n)} returns a series expansion of the expression \\spad{f}. Note that \\spad{f} should have only one variable; the series will be expanded in powers of that variable and terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2|) "\\spad{series(f)} returns a series expansion of the expression \\spad{f}. Note that \\spad{f} should have only one variable; the series will be expanded in powers of that variable.") (((|Any|) (|Symbol|)) "\\spad{series(x)} returns \\spad{x} viewed as a series.")) (|puiseux| (((|Any|) |#2| (|Equation| |#2|) (|Fraction| (|Integer|))) "\\spad{puiseux(f,{}x = a,{}n)} expands the expression \\spad{f} as a Puiseux series in powers of \\spad{(x - a)}; terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2| (|Equation| |#2|)) "\\spad{puiseux(f,{}x = a)} expands the expression \\spad{f} as a Puiseux series in powers of \\spad{(x - a)}.") (((|Any|) |#2| (|Fraction| (|Integer|))) "\\spad{puiseux(f,{}n)} returns a Puiseux expansion of the expression \\spad{f}. Note that \\spad{f} should have only one variable; the series will be expanded in powers of that variable and terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2|) "\\spad{puiseux(f)} returns a Puiseux expansion of the expression \\spad{f}. Note that \\spad{f} should have only one variable; the series will be expanded in powers of that variable.") (((|Any|) (|Symbol|)) "\\spad{puiseux(x)} returns \\spad{x} viewed as a Puiseux series.")) (|laurent| (((|Any|) |#2| (|Equation| |#2|) (|Integer|)) "\\spad{laurent(f,{}x = a,{}n)} expands the expression \\spad{f} as a Laurent series in powers of \\spad{(x - a)}; terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2| (|Equation| |#2|)) "\\spad{laurent(f,{}x = a)} expands the expression \\spad{f} as a Laurent series in powers of \\spad{(x - a)}.") (((|Any|) |#2| (|Integer|)) "\\spad{laurent(f,{}n)} returns a Laurent expansion of the expression \\spad{f}. Note that \\spad{f} should have only one variable; the series will be expanded in powers of that variable and terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2|) "\\spad{laurent(f)} returns a Laurent expansion of the expression \\spad{f}. Note that \\spad{f} should have only one variable; the series will be expanded in powers of that variable.") (((|Any|) (|Symbol|)) "\\spad{laurent(x)} returns \\spad{x} viewed as a Laurent series.")) (|taylor| (((|Any|) |#2| (|Equation| |#2|) (|NonNegativeInteger|)) "\\spad{taylor(f,{}x = a)} expands the expression \\spad{f} as a Taylor series in powers of \\spad{(x - a)}; terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2| (|Equation| |#2|)) "\\spad{taylor(f,{}x = a)} expands the expression \\spad{f} as a Taylor series in powers of \\spad{(x - a)}.") (((|Any|) |#2| (|NonNegativeInteger|)) "\\spad{taylor(f,{}n)} returns a Taylor expansion of the expression \\spad{f}. Note that \\spad{f} should have only one variable; the series will be expanded in powers of that variable and terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2|) "\\spad{taylor(f)} returns a Taylor expansion of the expression \\spad{f}. Note that \\spad{f} should have only one variable; the series will be expanded in powers of that variable.") (((|Any|) (|Symbol|)) "\\spad{taylor(x)} returns \\spad{x} viewed as a Taylor series."))) 
NIL 
NIL 
(-304 R) 
((|constructor| (NIL "Top-level mathematical expressions involving symbolic functions.")) (|squareFreePolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{squareFreePolynomial(p)} is not documented")) (|factorPolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{factorPolynomial(p)} is not documented")) (|simplifyPower| (($ $ (|Integer|)) "simplifyPower?(\\spad{f},{}\\spad{n}) is not documented")) (|number?| (((|Boolean|) $) "\\spad{number?(f)} tests if \\spad{f} is rational")) (|reduce| (($ $) "\\spad{reduce(f)} simplifies all the unreduced algebraic quantities present in \\spad{f} by applying their defining relations."))) 
((-4502 -3322 (-1367 (|has| |#1| (-1039)) (|has| |#1| (-622 (-560)))) (-12 (|has| |#1| (-550)) (-3322 (-1367 (|has| |#1| (-1039)) (|has| |#1| (-622 (-560)))) (|has| |#1| (-1039)) (|has| |#1| (-471)))) (|has| |#1| (-1039)) (|has| |#1| (-471))) (-4500 |has| |#1| (-170)) (-4499 |has| |#1| (-170)) ((-4507 "*") |has| |#1| (-550)) (-4498 |has| |#1| (-550)) (-4503 |has| |#1| (-550)) (-4497 |has| |#1| (-550))) 
((|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-170))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-1039))) (-3322 (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-1039)))) (|HasCategory| |#1| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-471))) (|HasCategory| |#1| (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| |#1| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#1| (LIST (QUOTE -873) (QUOTE (-560)))) (|HasCategory| |#1| (LIST (QUOTE -873) (QUOTE (-375)))) (|HasCategory| |#1| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-375))))) (|HasCategory| |#1| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-560))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-550)))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-170))) (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-1039)))) (-12 (|HasCategory| |#1| (QUOTE (-447))) (|HasCategory| |#1| (QUOTE (-550)))) (-3322 (|HasCategory| |#1| (QUOTE (-471))) (|HasCategory| |#1| (QUOTE (-550)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-1039)))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-1039)))) (-3322 (|HasCategory| |#1| (QUOTE (-471))) (|HasCategory| |#1| (QUOTE (-1039)))) (|HasCategory| |#1| (QUOTE (-21))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-170))) (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-1039)))) (-3322 (-12 (|HasCategory| |#1| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-1039)))) (|HasCategory| |#1| (QUOTE (-21)))) (|HasCategory| |#1| (QUOTE (-25))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-170))) (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-1039)))) (-3322 (-12 (|HasCategory| |#1| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-1039)))) (|HasCategory| |#1| (QUOTE (-25)))) (|HasCategory| |#1| (QUOTE (-1094))) (-3322 (|HasCategory| |#1| (QUOTE (-471))) (|HasCategory| |#1| (QUOTE (-1094)))) (-3322 (-12 (|HasCategory| |#1| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-1039)))) (|HasCategory| |#1| (QUOTE (-1094)))) (-3322 (-12 (|HasCategory| |#1| (LIST (QUOTE -622) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-1039)))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-1094)))) (|HasCategory| |#1| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-550))))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#1| (QUOTE (-550)))) (-3322 (-12 (|HasCategory| |#1| (LIST (QUOTE -1029) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#1| (QUOTE (-550)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#1| (QUOTE (-550))))) (|HasCategory| $ (QUOTE (-1039))) (|HasCategory| $ (LIST (QUOTE -1029) (QUOTE (-560))))) 
(-305 R -2262) 
((|constructor| (NIL "Taylor series solutions of explicit ODE\\spad{'s}.")) (|seriesSolve| (((|Any|) |#2| (|BasicOperator|) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve(eq,{} y,{} x = a,{} [b0,{}...,{}bn])} is equivalent to \\spad{seriesSolve(eq = 0,{} y,{} x = a,{} [b0,{}...,{}b(n-1)])}.") (((|Any|) |#2| (|BasicOperator|) (|Equation| |#2|) (|Equation| |#2|)) "\\spad{seriesSolve(eq,{} y,{} x = a,{} y a = b)} is equivalent to \\spad{seriesSolve(eq=0,{} y,{} x=a,{} y a = b)}.") (((|Any|) |#2| (|BasicOperator|) (|Equation| |#2|) |#2|) "\\spad{seriesSolve(eq,{} y,{} x = a,{} b)} is equivalent to \\spad{seriesSolve(eq = 0,{} y,{} x = a,{} y a = b)}.") (((|Any|) (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) |#2|) "\\spad{seriesSolve(eq,{}y,{} x=a,{} b)} is equivalent to \\spad{seriesSolve(eq,{} y,{} x=a,{} y a = b)}.") (((|Any|) (|List| |#2|) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| (|Equation| |#2|))) "seriesSolve([\\spad{eq1},{}...,{}eqn],{} [\\spad{y1},{}...,{}\\spad{yn}],{} \\spad{x} = a,{}[\\spad{y1} a = \\spad{b1},{}...,{} \\spad{yn} a = \\spad{bn}]) is equivalent to \\spad{seriesSolve([eq1=0,{}...,{}eqn=0],{} [y1,{}...,{}yn],{} x = a,{} [y1 a = b1,{}...,{} yn a = bn])}.") (((|Any|) (|List| |#2|) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve([eq1,{}...,{}eqn],{} [y1,{}...,{}yn],{} x=a,{} [b1,{}...,{}bn])} is equivalent to \\spad{seriesSolve([eq1=0,{}...,{}eqn=0],{} [y1,{}...,{}yn],{} x=a,{} [b1,{}...,{}bn])}.") (((|Any|) (|List| (|Equation| |#2|)) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve([eq1,{}...,{}eqn],{} [y1,{}...,{}yn],{} x=a,{} [b1,{}...,{}bn])} is equivalent to \\spad{seriesSolve([eq1,{}...,{}eqn],{} [y1,{}...,{}yn],{} x = a,{} [y1 a = b1,{}...,{} yn a = bn])}.") (((|Any|) (|List| (|Equation| |#2|)) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| (|Equation| |#2|))) "\\spad{seriesSolve([eq1,{}...,{}eqn],{}[y1,{}...,{}yn],{}x = a,{}[y1 a = b1,{}...,{}yn a = bn])} returns a taylor series solution of \\spad{[eq1,{}...,{}eqn]} around \\spad{x = a} with initial conditions \\spad{\\spad{yi}(a) = \\spad{bi}}. Note that eqi must be of the form \\spad{\\spad{fi}(x,{} y1 x,{} y2 x,{}...,{} yn x) y1'(x) + \\spad{gi}(x,{} y1 x,{} y2 x,{}...,{} yn x) = h(x,{} y1 x,{} y2 x,{}...,{} yn x)}.") (((|Any|) (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve(eq,{}y,{}x=a,{}[b0,{}...,{}b(n-1)])} returns a Taylor series solution of \\spad{eq} around \\spad{x = a} with initial conditions \\spad{y(a) = b0},{} \\spad{y'(a) = b1},{} \\spad{y''(a) = b2},{} ...,{}\\spad{y(n-1)(a) = b(n-1)} \\spad{eq} must be of the form \\spad{f(x,{} y x,{} y'(x),{}...,{} y(n-1)(x)) y(n)(x) + g(x,{}y x,{}y'(x),{}...,{}y(n-1)(x)) = h(x,{}y x,{} y'(x),{}...,{} y(n-1)(x))}.") (((|Any|) (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) (|Equation| |#2|)) "\\spad{seriesSolve(eq,{}y,{}x=a,{} y a = b)} returns a Taylor series solution of \\spad{eq} around \\spad{x} = a with initial condition \\spad{y(a) = b}. Note that \\spad{eq} must be of the form \\spad{f(x,{} y x) y'(x) + g(x,{} y x) = h(x,{} y x)}."))) 
NIL 
NIL 
(-306 R -2262 UTSF UTSSUPF) 
((|constructor| (NIL "This package has no description"))) 
NIL 
NIL 
(-307) 
((|constructor| (NIL "Package for constructing tubes around 3-dimensional parametric curves.")) (|tubePlot| (((|TubePlot| (|Plot3D|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|) (|String|)) "\\spad{tubePlot(f,{}g,{}h,{}colorFcn,{}a..b,{}r,{}n,{}s)} puts a tube of radius \\spad{r} with \\spad{n} points on each circle about the curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} for \\spad{t} in \\spad{[a,{}b]}. If \\spad{s} = \"closed\",{} the tube is considered to be closed; if \\spad{s} = \"open\",{} the tube is considered to be open.") (((|TubePlot| (|Plot3D|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|)) "\\spad{tubePlot(f,{}g,{}h,{}colorFcn,{}a..b,{}r,{}n)} puts a tube of radius \\spad{r} with \\spad{n} points on each circle about the curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} for \\spad{t} in \\spad{[a,{}b]}. The tube is considered to be open.") (((|TubePlot| (|Plot3D|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Integer|) (|String|)) "\\spad{tubePlot(f,{}g,{}h,{}colorFcn,{}a..b,{}r,{}n,{}s)} puts a tube of radius \\spad{r(t)} with \\spad{n} points on each circle about the curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} for \\spad{t} in \\spad{[a,{}b]}. If \\spad{s} = \"closed\",{} the tube is considered to be closed; if \\spad{s} = \"open\",{} the tube is considered to be open.") (((|TubePlot| (|Plot3D|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Integer|)) "\\spad{tubePlot(f,{}g,{}h,{}colorFcn,{}a..b,{}r,{}n)} puts a tube of radius \\spad{r}(\\spad{t}) with \\spad{n} points on each circle about the curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} for \\spad{t} in \\spad{[a,{}b]}. The tube is considered to be open.")) (|constantToUnaryFunction| (((|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|DoubleFloat|)) "\\spad{constantToUnaryFunction(s)} is a local function which takes the value of \\spad{s},{} which may be a function of a constant,{} and returns a function which always returns the value \\spadtype{DoubleFloat} \\spad{s}."))) 
NIL 
NIL 
(-308 FE |var| |cen|) 
((|constructor| (NIL "ExponentialOfUnivariatePuiseuxSeries is a domain used to represent essential singularities of functions. An object in this domain is a function of the form \\spad{exp(f(x))},{} where \\spad{f(x)} is a Puiseux series with no terms of non-negative degree. Objects are ordered according to order of singularity,{} with functions which tend more rapidly to zero or infinity considered to be larger. Thus,{} if \\spad{order(f(x)) < order(g(x))},{} \\spadignore{i.e.} the first non-zero term of \\spad{f(x)} has lower degree than the first non-zero term of \\spad{g(x)},{} then \\spad{exp(f(x)) > exp(g(x))}. If \\spad{order(f(x)) = order(g(x))},{} then the ordering is essentially random. This domain is used in computing limits involving functions with essential singularities.")) (|exponentialOrder| (((|Fraction| (|Integer|)) $) "\\spad{exponentialOrder(exp(c * x **(-n) + ...))} returns \\spad{-n}. exponentialOrder(0) returns \\spad{0}.")) (|exponent| (((|UnivariatePuiseuxSeries| |#1| |#2| |#3|) $) "\\spad{exponent(exp(f(x)))} returns \\spad{f(x)}")) (|exponential| (($ (|UnivariatePuiseuxSeries| |#1| |#2| |#3|)) "\\spad{exponential(f(x))} returns \\spad{exp(f(x))}. Note: the function does NOT check that \\spad{f(x)} has no non-negative terms."))) 
(((-4507 "*") |has| |#1| (-170)) (-4498 |has| |#1| (-550)) (-4503 |has| |#1| (-359)) (-4497 |has| |#1| (-359)) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| |#1| (LIST (QUOTE -43) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-170))) (-3322 (|HasCategory| |#1| (QUOTE (-170))) (|HasCategory| |#1| (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -403) (QUOTE (-560))) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -403) (QUOTE (-560))) (|devaluate| |#1|))))) (|HasCategory| (-403 (-560)) (QUOTE (-1094))) (|HasCategory| |#1| (QUOTE (-359))) (-3322 (|HasCategory| |#1| (QUOTE (-170))) (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-550)))) (-3322 (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-550)))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -403) (QUOTE (-560)))))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -403) (QUOTE (-560)))))) (|HasSignature| |#1| (LIST (QUOTE -1625) (LIST (|devaluate| |#1|) (QUOTE (-1153)))))) (-3322 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-560)))) (|HasCategory| |#1| (LIST (QUOTE -43) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasCategory| |#1| (QUOTE (-951))) (|HasCategory| |#1| (QUOTE (-1173)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -43) (LIST (QUOTE -403) (QUOTE (-560))))) (|HasSignature| |#1| (LIST (QUOTE -2283) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1153))))) (|HasSignature| |#1| (LIST (QUOTE -3412) (LIST (LIST (QUOTE -626) (QUOTE (-1153))) (|devaluate| |#1|))))))) 
(-309 K) 
((|constructor| (NIL "Part of the Package for Algebraic Function Fields in one variable PAFF"))) 
NIL 
NIL 
(-310 M) 
((|constructor| (NIL "computes various functions on factored arguments.")) (|log| (((|List| (|Record| (|:| |coef| (|NonNegativeInteger|)) (|:| |logand| |#1|))) (|Factored| |#1|)) "\\spad{log(f)} returns \\spad{[(a1,{}b1),{}...,{}(am,{}bm)]} such that the logarithm of \\spad{f} is equal to \\spad{a1*log(b1) + ... + am*log(bm)}.")) (|nthRoot| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#1|) (|:| |radicand| (|List| |#1|))) (|Factored| |#1|) (|NonNegativeInteger|)) "\\spad{nthRoot(f,{} n)} returns \\spad{(p,{} r,{} [r1,{}...,{}rm])} such that the \\spad{n}th-root of \\spad{f} is equal to \\spad{r * \\spad{p}th-root(r1 * ... * rm)},{} where \\spad{r1},{}...,{}\\spad{rm} are distinct factors of \\spad{f},{} each of which has an exponent smaller than \\spad{p} in \\spad{f}."))) 
NIL 
NIL 
(-311 K) 
((|constructor| (NIL "Part of the Package for Algebraic Function Fields in one variable PAFF"))) 
NIL 
NIL 
(-312 E OV R P) 
((|constructor| (NIL "This package provides utilities used by the factorizers which operate on polynomials represented as univariate polynomials with multivariate coefficients.")) (|ran| ((|#3| (|Integer|)) "\\spad{ran(k)} computes a random integer between \\spad{-k} and \\spad{k} as a member of \\spad{R}.")) (|normalDeriv| (((|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|) (|Integer|)) "\\spad{normalDeriv(poly,{}i)} computes the \\spad{i}th derivative of \\spad{poly} divided by i!.")) (|raisePolynomial| (((|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#3|)) "\\spad{raisePolynomial(rpoly)} converts \\spad{rpoly} from a univariate polynomial over \\spad{r} to be a univariate polynomial with polynomial coefficients.")) (|lowerPolynomial| (((|SparseUnivariatePolynomial| |#3|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{lowerPolynomial(upoly)} converts \\spad{upoly} to be a univariate polynomial over \\spad{R}. An error if the coefficients contain variables.")) (|variables| (((|List| |#2|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{variables(upoly)} returns the list of variables for the coefficients of \\spad{upoly}.")) (|degree| (((|List| (|NonNegativeInteger|)) (|SparseUnivariatePolynomial| |#4|) (|List| |#2|)) "\\spad{degree(upoly,{} lvar)} returns a list containing the maximum degree for each variable in lvar.")) (|completeEval| (((|SparseUnivariatePolynomial| |#3|) (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|List| |#3|)) "\\spad{completeEval(upoly,{} lvar,{} lval)} evaluates the polynomial \\spad{upoly} with each variable in \\spad{lvar} replaced by the corresponding value in lval. Substitutions are done for all variables in \\spad{upoly} producing a univariate polynomial over \\spad{R}."))) 
NIL 
NIL 
(-313 S) 
((|constructor| (NIL "The free abelian group on a set \\spad{S} is the monoid of finite sums of the form \\spad{reduce(+,{}[\\spad{ni} * \\spad{si}])} where the \\spad{si}\\spad{'s} are in \\spad{S},{} and the \\spad{ni}\\spad{'s} are integers. The operation is commutative."))) 
((-4500 . T) (-4499 . T)) 
((|HasCategory| |#1| (QUOTE (-834))) (|HasCategory| (-560) (QUOTE (-779)))) 
(-314 S E) 
((|constructor| (NIL "A free abelian monoid on a set \\spad{S} is the monoid of finite sums of the form \\spad{reduce(+,{}[\\spad{ni} * \\spad{si}])} where the \\spad{si}\\spad{'s} are in \\spad{S},{} and the \\spad{ni}\\spad{'s} are in a given abelian monoid. The operation is commutative.")) (|highCommonTerms| (($ $ $) "\\spad{highCommonTerms(e1 a1 + ... + en an,{} f1 b1 + ... + fm bm)} returns \\spad{reduce(+,{}[max(\\spad{ei},{} \\spad{fi}) \\spad{ci}])} where \\spad{ci} ranges in the intersection of \\spad{{a1,{}...,{}an}} and \\spad{{b1,{}...,{}bm}}.")) (|mapGen| (($ (|Mapping| |#1| |#1|) $) "\\spad{mapGen(f,{} e1 a1 +...+ en an)} returns \\spad{e1 f(a1) +...+ en f(an)}.")) (|mapCoef| (($ (|Mapping| |#2| |#2|) $) "\\spad{mapCoef(f,{} e1 a1 +...+ en an)} returns \\spad{f(e1) a1 +...+ f(en) an}.")) (|coefficient| ((|#2| |#1| $) "\\spad{coefficient(s,{} e1 a1 + ... + en an)} returns \\spad{ei} such that \\spad{ai} = \\spad{s},{} or 0 if \\spad{s} is not one of the \\spad{ai}\\spad{'s}.")) (|nthFactor| ((|#1| $ (|Integer|)) "\\spad{nthFactor(x,{} n)} returns the factor of the n^th term of \\spad{x}.")) (|nthCoef| ((|#2| $ (|Integer|)) "\\spad{nthCoef(x,{} n)} returns the coefficient of the n^th term of \\spad{x}.")) (|terms| (((|List| (|Record| (|:| |gen| |#1|) (|:| |exp| |#2|))) $) "\\spad{terms(e1 a1 + ... + en an)} returns \\spad{[[a1,{} e1],{}...,{}[an,{} en]]}.")) (|size| (((|NonNegativeInteger|) $) "\\indented{1}{size(\\spad{x}) returns the number of terms in \\spad{x}.} \\indented{1}{mapGen(\\spad{f},{} \\spad{a1}\\spad{\\^}\\spad{e1} ... an\\spad{\\^}en) returns} \\spad{f(a1)\\^e1 ... f(an)\\^en}.")) (* (($ |#2| |#1|) "\\spad{e * s} returns \\spad{e} times \\spad{s}.")) (+ (($ |#1| $) "\\spad{s + x} returns the sum of \\spad{s} and \\spad{x}."))) 
NIL 
NIL 
(-315 S) 
((|constructor| (NIL "The free abelian monoid on a set \\spad{S} is the monoid of finite sums of the form \\spad{reduce(+,{}[\\spad{ni} * \\spad{si}])} where the \\spad{si}\\spad{'s} are in \\spad{S},{} and the \\spad{ni}\\spad{'s} are non-negative integers. The operation is commutative."))) 
NIL 
((|HasCategory| (-755) (QUOTE (-779)))) 
(-316 E R1 A1 R2 A2) 
((|constructor| (NIL "This package provides a mapping function for \\spadtype{FiniteAbelianMonoidRing} The packages defined in this file provide fast fraction free rational interpolation algorithms. (see \\spad{FAMR2},{} FFFG,{} FFFGF,{} NEWTON)")) (|map| ((|#5| (|Mapping| |#4| |#2|) |#3|) "\\spad{map}(\\spad{f},{} a) applies the map \\spad{f} to each coefficient in a. It is assumed that \\spad{f} maps 0 to 0"))) 
NIL 
NIL 
(-317 S R E) 
((|constructor| (NIL "This category is similar to AbelianMonoidRing,{} except that the sum is assumed to be finite. It is a useful model for polynomials,{} but is somewhat more general.")) (|primitivePart| (($ $) "\\spad{primitivePart(p)} returns the unit normalized form of polynomial \\spad{p} divided by the content of \\spad{p}.")) (|content| ((|#2| $) "\\spad{content(p)} gives the \\spad{gcd} of the coefficients of polynomial \\spad{p}.")) (|exquo| (((|Union| $ "failed") $ |#2|) "\\spad{exquo(p,{}r)} returns the exact quotient of polynomial \\spad{p} by \\spad{r},{} or \"failed\" if none exists.")) (|binomThmExpt| (($ $ $ (|NonNegativeInteger|)) "\\spad{binomThmExpt(p,{}q,{}n)} returns \\spad{(x+y)^n} by means of the binomial theorem trick.")) (|pomopo!| (($ $ |#2| |#3| $) "\\spad{pomopo!(p1,{}r,{}e,{}p2)} returns \\spad{p1 + monomial(e,{}r) * p2} and may use \\spad{p1} as workspace. The constaant \\spad{r} is assumed to be nonzero.")) (|mapExponents| (($ (|Mapping| |#3| |#3|) $) "\\spad{mapExponents(fn,{}u)} maps function \\spad{fn} onto the exponents of the non-zero monomials of polynomial \\spad{u}.")) (|minimumDegree| ((|#3| $) "\\spad{minimumDegree(p)} gives the least exponent of a non-zero term of polynomial \\spad{p}. Error: if applied to 0.")) (|numberOfMonomials| (((|NonNegativeInteger|) $) "\\spad{numberOfMonomials(p)} gives the number of non-zero monomials in polynomial \\spad{p}.")) (|coefficients| (((|List| |#2|) $) "\\spad{coefficients(p)} gives the list of non-zero coefficients of polynomial \\spad{p}.")) (|ground| ((|#2| $) "\\spad{ground(p)} retracts polynomial \\spad{p} to the coefficient ring.")) (|ground?| (((|Boolean|) $) "\\spad{ground?(p)} tests if polynomial \\spad{p} is a member of the coefficient ring."))) 
NIL 
((|HasCategory| |#2| (QUOTE (-447))) (|HasCategory| |#2| (QUOTE (-550))) (|HasCategory| |#2| (QUOTE (-170)))) 
(-318 R E) 
((|constructor| (NIL "This category is similar to AbelianMonoidRing,{} except that the sum is assumed to be finite. It is a useful model for polynomials,{} but is somewhat more general.")) (|primitivePart| (($ $) "\\spad{primitivePart(p)} returns the unit normalized form of polynomial \\spad{p} divided by the content of \\spad{p}.")) (|content| ((|#1| $) "\\spad{content(p)} gives the \\spad{gcd} of the coefficients of polynomial \\spad{p}.")) (|exquo| (((|Union| $ "failed") $ |#1|) "\\spad{exquo(p,{}r)} returns the exact quotient of polynomial \\spad{p} by \\spad{r},{} or \"failed\" if none exists.")) (|binomThmExpt| (($ $ $ (|NonNegativeInteger|)) "\\spad{binomThmExpt(p,{}q,{}n)} returns \\spad{(x+y)^n} by means of the binomial theorem trick.")) (|pomopo!| (($ $ |#1| |#2| $) "\\spad{pomopo!(p1,{}r,{}e,{}p2)} returns \\spad{p1 + monomial(e,{}r) * p2} and may use \\spad{p1} as workspace. The constaant \\spad{r} is assumed to be nonzero.")) (|mapExponents| (($ (|Mapping| |#2| |#2|) $) "\\spad{mapExponents(fn,{}u)} maps function \\spad{fn} onto the exponents of the non-zero monomials of polynomial \\spad{u}.")) (|minimumDegree| ((|#2| $) "\\spad{minimumDegree(p)} gives the least exponent of a non-zero term of polynomial \\spad{p}. Error: if applied to 0.")) (|numberOfMonomials| (((|NonNegativeInteger|) $) "\\spad{numberOfMonomials(p)} gives the number of non-zero monomials in polynomial \\spad{p}.")) (|coefficients| (((|List| |#1|) $) "\\spad{coefficients(p)} gives the list of non-zero coefficients of polynomial \\spad{p}.")) (|ground| ((|#1| $) "\\spad{ground(p)} retracts polynomial \\spad{p} to the coefficient ring.")) (|ground?| (((|Boolean|) $) "\\spad{ground?(p)} tests if polynomial \\spad{p} is a member of the coefficient ring."))) 
(((-4507 "*") |has| |#1| (-170)) (-4498 |has| |#1| (-550)) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-319 S) 
((|constructor| (NIL "A FlexibleArray is the notion of an array intended to allow for growth at the end only. Hence the following efficient operations \\spad{append(x,{}a)} meaning append item \\spad{x} at the end of the array \\spad{a} \\spad{delete(a,{}n)} meaning delete the last item from the array \\spad{a} Flexible arrays support the other operations inherited from \\spadtype{ExtensibleLinearAggregate}. However,{} these are not efficient. Flexible arrays combine the \\spad{O(1)} access time property of arrays with growing and shrinking at the end in \\spad{O(1)} (average) time. This is done by using an ordinary array which may have zero or more empty slots at the end. When the array becomes full it is copied into a new larger (50\\% larger) array. Conversely,{} when the array becomes less than 1/2 full,{} it is copied into a smaller array. Flexible arrays provide for an efficient implementation of many data structures in particular heaps,{} stacks and sets."))) 
((-4506 . T) (-4505 . T)) 
((|HasCategory| |#1| (QUOTE (-1082))) (|HasCategory| |#1| (LIST (QUOTE -601) (QUOTE (-533)))) (|HasCategory| |#1| (QUOTE (-834))) (-3322 (|HasCategory| |#1| (QUOTE (-834))) (|HasCategory| |#1| (QUOTE (-1082)))) (|HasCategory| (-560) (QUOTE (-834))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082)))) (-3322 (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-834)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-1082)))))) 
(-320 S -2262) 
((|constructor| (NIL "FiniteAlgebraicExtensionField \\spad{F} is the category of fields which are finite algebraic extensions of the field \\spad{F}. If \\spad{F} is finite then any finite algebraic extension of \\spad{F} is finite,{} too. Let \\spad{K} be a finite algebraic extension of the finite field \\spad{F}. The exponentiation of elements of \\spad{K} defines a \\spad{Z}-module structure on the multiplicative group of \\spad{K}. The additive group of \\spad{K} becomes a module over the ring of polynomials over \\spad{F} via the operation \\spadfun{linearAssociatedExp}(a:K,{}f:SparseUnivariatePolynomial \\spad{F}) which is linear over \\spad{F},{} \\spadignore{i.e.} for elements a from \\spad{K},{} \\spad{c},{}\\spad{d} from \\spad{F} and \\spad{f},{}\\spad{g} univariate polynomials over \\spad{F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals \\spad{c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus \\spad{d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from \\spad{F}[\\spad{X}]: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\spad{\\$}SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is a**(q**k) where q=size()\\spad{\\$}\\spad{F}. The operations order and discreteLog associated with the multiplicative exponentiation have additive analogues associated to the operation \\spadfun{linearAssociatedExp}. These are the functions \\spadfun{linearAssociatedOrder} and \\spadfun{linearAssociatedLog},{} respectively.")) (|linearAssociatedLog| (((|Union| (|SparseUnivariatePolynomial| |#2|) "failed") $ $) "\\spad{linearAssociatedLog(b,{}a)} returns a polynomial \\spad{g},{} such that the \\spadfun{linearAssociatedExp}(\\spad{b},{}\\spad{g}) equals a. If there is no such polynomial \\spad{g},{} then \\spadfun{linearAssociatedLog} fails.") (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{linearAssociatedLog(a)} returns a polynomial \\spad{g},{} such that \\spadfun{linearAssociatedExp}(normalElement(),{}\\spad{g}) equals a.")) (|linearAssociatedOrder| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{linearAssociatedOrder(a)} retruns the monic polynomial \\spad{g} of least degree,{} such that \\spadfun{linearAssociatedExp}(a,{}\\spad{g}) is 0.")) (|linearAssociatedExp| (($ $ (|SparseUnivariatePolynomial| |#2|)) "\\spad{linearAssociatedExp(a,{}f)} is linear over \\spad{F},{} \\spadignore{i.e.} for elements a from \\spad{\\$},{} \\spad{c},{}\\spad{d} form \\spad{F} and \\spad{f},{}\\spad{g} univariate polynomials over \\spad{F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals \\spad{c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus \\spad{d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from \\spad{F}[\\spad{X}]: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\spad{\\$}SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is a**(q**k),{} where q=size()\\spad{\\$}\\spad{F}.")) (|generator| (($) "\\spad{generator()} returns a root of the defining polynomial. This element generates the field as an algebra over the ground field.")) (|normal?| (((|Boolean|) $) "\\spad{normal?(a)} tests whether the element \\spad{a} is normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i),{} 0 <= i <= extensionDegree()-1} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Implementation according to Lidl/Niederreiter: Theorem 2.39.")) (|normalElement| (($) "\\spad{normalElement()} returns a element,{} normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i),{} 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. At the first call,{} the element is computed by \\spadfunFrom{createNormalElement}{FiniteAlgebraicExtensionField} then cached in a global variable. On subsequent calls,{} the element is retrieved by referencing the global variable.")) (|createNormalElement| (($) "\\spad{createNormalElement()} computes a normal element over the ground field \\spad{F},{} that is,{} \\spad{a**(q**i),{} 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Reference: Such an element exists Lidl/Niederreiter: Theorem 2.35.")) (|trace| (($ $ (|PositiveInteger|)) "\\spad{trace(a,{}d)} computes the trace of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size \\spad{q}. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note that \\spad{trace(a,{}d)=reduce(+,{}[a**(q**(d*i)) for i in 0..n/d])}.") ((|#2| $) "\\spad{trace(a)} computes the trace of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|norm| (($ $ (|PositiveInteger|)) "\\spad{norm(a,{}d)} computes the norm of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note that norm(a,{}\\spad{d}) = reduce(*,{}[a**(\\spad{q**}(d*i)) for \\spad{i} in 0..\\spad{n/d}])") ((|#2| $) "\\spad{norm(a)} computes the norm of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|degree| (((|PositiveInteger|) $) "\\spad{degree(a)} returns the degree of the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|extensionDegree| (((|PositiveInteger|)) "\\spad{extensionDegree()} returns the degree of field extension.")) (|definingPolynomial| (((|SparseUnivariatePolynomial| |#2|)) "\\spad{definingPolynomial()} returns the polynomial used to define the field extension.")) (|minimalPolynomial| (((|SparseUnivariatePolynomial| $) $ (|PositiveInteger|)) "\\spad{minimalPolynomial(x,{}n)} computes the minimal polynomial of \\spad{x} over the field of extension degree \\spad{n} over the ground field \\spad{F}.") (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{minimalPolynomial(a)} returns the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|represents| (($ (|Vector| |#2|)) "\\spad{represents([a1,{}..,{}an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed basis.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $)) "\\spad{coordinates([v1,{}...,{}vm])} returns the coordinates of the \\spad{vi}\\spad{'s} with to the fixed basis. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#2|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{F}-vectorspace basis.")) (|basis| (((|Vector| $) (|PositiveInteger|)) "\\spad{basis(n)} returns a fixed basis of a subfield of \\spad{\\$} as \\spad{F}-vectorspace.") (((|Vector| $)) "\\spad{basis()} returns a fixed basis of \\spad{\\$} as \\spad{F}-vectorspace."))) 
NIL 
((|HasCategory| |#2| (QUOTE (-364)))) 
(-321 -2262) 
((|constructor| (NIL "FiniteAlgebraicExtensionField \\spad{F} is the category of fields which are finite algebraic extensions of the field \\spad{F}. If \\spad{F} is finite then any finite algebraic extension of \\spad{F} is finite,{} too. Let \\spad{K} be a finite algebraic extension of the finite field \\spad{F}. The exponentiation of elements of \\spad{K} defines a \\spad{Z}-module structure on the multiplicative group of \\spad{K}. The additive group of \\spad{K} becomes a module over the ring of polynomials over \\spad{F} via the operation \\spadfun{linearAssociatedExp}(a:K,{}f:SparseUnivariatePolynomial \\spad{F}) which is linear over \\spad{F},{} \\spadignore{i.e.} for elements a from \\spad{K},{} \\spad{c},{}\\spad{d} from \\spad{F} and \\spad{f},{}\\spad{g} univariate polynomials over \\spad{F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals \\spad{c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus \\spad{d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from \\spad{F}[\\spad{X}]: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\spad{\\$}SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is a**(q**k) where q=size()\\spad{\\$}\\spad{F}. The operations order and discreteLog associated with the multiplicative exponentiation have additive analogues associated to the operation \\spadfun{linearAssociatedExp}. These are the functions \\spadfun{linearAssociatedOrder} and \\spadfun{linearAssociatedLog},{} respectively.")) (|linearAssociatedLog| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") $ $) "\\spad{linearAssociatedLog(b,{}a)} returns a polynomial \\spad{g},{} such that the \\spadfun{linearAssociatedExp}(\\spad{b},{}\\spad{g}) equals a. If there is no such polynomial \\spad{g},{} then \\spadfun{linearAssociatedLog} fails.") (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{linearAssociatedLog(a)} returns a polynomial \\spad{g},{} such that \\spadfun{linearAssociatedExp}(normalElement(),{}\\spad{g}) equals a.")) (|linearAssociatedOrder| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{linearAssociatedOrder(a)} retruns the monic polynomial \\spad{g} of least degree,{} such that \\spadfun{linearAssociatedExp}(a,{}\\spad{g}) is 0.")) (|linearAssociatedExp| (($ $ (|SparseUnivariatePolynomial| |#1|)) "\\spad{linearAssociatedExp(a,{}f)} is linear over \\spad{F},{} \\spadignore{i.e.} for elements a from \\spad{\\$},{} \\spad{c},{}\\spad{d} form \\spad{F} and \\spad{f},{}\\spad{g} univariate polynomials over \\spad{F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals \\spad{c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus \\spad{d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from \\spad{F}[\\spad{X}]: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\spad{\\$}SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is a**(q**k),{} where q=size()\\spad{\\$}\\spad{F}.")) (|generator| (($) "\\spad{generator()} returns a root of the defining polynomial. This element generates the field as an algebra over the ground field.")) (|normal?| (((|Boolean|) $) "\\spad{normal?(a)} tests whether the element \\spad{a} is normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i),{} 0 <= i <= extensionDegree()-1} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Implementation according to Lidl/Niederreiter: Theorem 2.39.")) (|normalElement| (($) "\\spad{normalElement()} returns a element,{} normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i),{} 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. At the first call,{} the element is computed by \\spadfunFrom{createNormalElement}{FiniteAlgebraicExtensionField} then cached in a global variable. On subsequent calls,{} the element is retrieved by referencing the global variable.")) (|createNormalElement| (($) "\\spad{createNormalElement()} computes a normal element over the ground field \\spad{F},{} that is,{} \\spad{a**(q**i),{} 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Reference: Such an element exists Lidl/Niederreiter: Theorem 2.35.")) (|trace| (($ $ (|PositiveInteger|)) "\\spad{trace(a,{}d)} computes the trace of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size \\spad{q}. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note that \\spad{trace(a,{}d)=reduce(+,{}[a**(q**(d*i)) for i in 0..n/d])}.") ((|#1| $) "\\spad{trace(a)} computes the trace of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|norm| (($ $ (|PositiveInteger|)) "\\spad{norm(a,{}d)} computes the norm of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note that norm(a,{}\\spad{d}) = reduce(*,{}[a**(\\spad{q**}(d*i)) for \\spad{i} in 0..\\spad{n/d}])") ((|#1| $) "\\spad{norm(a)} computes the norm of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|degree| (((|PositiveInteger|) $) "\\spad{degree(a)} returns the degree of the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|extensionDegree| (((|PositiveInteger|)) "\\spad{extensionDegree()} returns the degree of field extension.")) (|definingPolynomial| (((|SparseUnivariatePolynomial| |#1|)) "\\spad{definingPolynomial()} returns the polynomial used to define the field extension.")) (|minimalPolynomial| (((|SparseUnivariatePolynomial| $) $ (|PositiveInteger|)) "\\spad{minimalPolynomial(x,{}n)} computes the minimal polynomial of \\spad{x} over the field of extension degree \\spad{n} over the ground field \\spad{F}.") (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{minimalPolynomial(a)} returns the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|represents| (($ (|Vector| |#1|)) "\\spad{represents([a1,{}..,{}an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed basis.")) (|coordinates| (((|Matrix| |#1|) (|Vector| $)) "\\spad{coordinates([v1,{}...,{}vm])} returns the coordinates of the \\spad{vi}\\spad{'s} with to the fixed basis. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#1|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{F}-vectorspace basis.")) (|basis| (((|Vector| $) (|PositiveInteger|)) "\\spad{basis(n)} returns a fixed basis of a subfield of \\spad{\\$} as \\spad{F}-vectorspace.") (((|Vector| $)) "\\spad{basis()} returns a fixed basis of \\spad{\\$} as \\spad{F}-vectorspace."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-322) 
((|constructor| (NIL "This domain builds representations of program code segments for use with the FortranProgram domain.")) (|setLabelValue| (((|SingleInteger|) (|SingleInteger|)) "\\spad{setLabelValue(i)} resets the counter which produces labels to \\spad{i}")) (|getCode| (((|SExpression|) $) "\\spad{getCode(f)} returns a Lisp list of strings representing \\spad{f} in Fortran notation. This is used by the FortranProgram domain.")) (|printCode| (((|Void|) $) "\\spad{printCode(f)} prints out \\spad{f} in FORTRAN notation.")) (|code| (((|Union| (|:| |nullBranch| "null") (|:| |assignmentBranch| (|Record| (|:| |var| (|Symbol|)) (|:| |arrayIndex| (|List| (|Polynomial| (|Integer|)))) (|:| |rand| (|Record| (|:| |ints2Floats?| (|Boolean|)) (|:| |expr| (|OutputForm|)))))) (|:| |arrayAssignmentBranch| (|Record| (|:| |var| (|Symbol|)) (|:| |rand| (|OutputForm|)) (|:| |ints2Floats?| (|Boolean|)))) (|:| |conditionalBranch| (|Record| (|:| |switch| (|Switch|)) (|:| |thenClause| $) (|:| |elseClause| $))) (|:| |returnBranch| (|Record| (|:| |empty?| (|Boolean|)) (|:| |value| (|Record| (|:| |ints2Floats?| (|Boolean|)) (|:| |expr| (|OutputForm|)))))) (|:| |blockBranch| (|List| $)) (|:| |commentBranch| (|List| (|String|))) (|:| |callBranch| (|String|)) (|:| |forBranch| (|Record| (|:| |range| (|SegmentBinding| (|Polynomial| (|Integer|)))) (|:| |span| (|Polynomial| (|Integer|))) (|:| |body| $))) (|:| |labelBranch| (|SingleInteger|)) (|:| |loopBranch| (|Record| (|:| |switch| (|Switch|)) (|:| |body| $))) (|:| |commonBranch| (|Record| (|:| |name| (|Symbol|)) (|:| |contents| (|List| (|Symbol|))))) (|:| |printBranch| (|List| (|OutputForm|)))) $) "\\spad{code(f)} returns the internal representation of the object represented by \\spad{f}.")) (|operation| (((|Union| (|:| |Null| "null") (|:| |Assignment| "assignment") (|:| |Conditional| "conditional") (|:| |Return| "return") (|:| |Block| "block") (|:| |Comment| "comment") (|:| |Call| "call") (|:| |For| "for") (|:| |While| "while") (|:| |Repeat| "repeat") (|:| |Goto| "goto") (|:| |Continue| "continue") (|:| |ArrayAssignment| "arrayAssignment") (|:| |Save| "save") (|:| |Stop| "stop") (|:| |Common| "common") (|:| |Print| "print")) $) "\\spad{operation(f)} returns the name of the operation represented by \\spad{f}.")) (|common| (($ (|Symbol|) (|List| (|Symbol|))) "\\spad{common(name,{}contents)} creates a representation a named common block.")) (|printStatement| (($ (|List| (|OutputForm|))) "\\spad{printStatement(l)} creates a representation of a PRINT statement.")) (|save| (($) "\\spad{save()} creates a representation of a SAVE statement.")) (|stop| (($) "\\spad{stop()} creates a representation of a STOP statement.")) (|block| (($ (|List| $)) "\\spad{block(l)} creates a representation of the statements in \\spad{l} as a block.")) (|assign| (($ (|Symbol|) (|List| (|Polynomial| (|Integer|))) (|Expression| (|Complex| (|Float|)))) "\\spad{assign(x,{}l,{}y)} creates a representation of the assignment of \\spad{y} to the \\spad{l}\\spad{'}th element of array \\spad{x} (\\spad{l} is a list of indices).") (($ (|Symbol|) (|List| (|Polynomial| (|Integer|))) (|Expression| (|Float|))) "\\spad{assign(x,{}l,{}y)} creates a representation of the assignment of \\spad{y} to the \\spad{l}\\spad{'}th element of array \\spad{x} (\\spad{l} is a list of indices).") (($ (|Symbol|) (|List| (|Polynomial| (|Integer|))) (|Expression| (|Integer|))) "\\spad{assign(x,{}l,{}y)} creates a representation of the assignment of \\spad{y} to the \\spad{l}\\spad{'}th element of array \\spad{x} (\\spad{l} is a list of indices).") (($ (|Symbol|) (|Vector| (|Expression| (|Complex| (|Float|))))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Vector| (|Expression| (|Float|)))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Vector| (|Expression| (|Integer|)))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|Expression| (|Complex| (|Float|))))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|Expression| (|Float|)))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|Expression| (|Integer|)))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Expression| (|Complex| (|Float|)))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Expression| (|Float|))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Expression| (|Integer|))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|List| (|Polynomial| (|Integer|))) (|Expression| (|MachineComplex|))) "\\spad{assign(x,{}l,{}y)} creates a representation of the assignment of \\spad{y} to the \\spad{l}\\spad{'}th element of array \\spad{x} (\\spad{l} is a list of indices).") (($ (|Symbol|) (|List| (|Polynomial| (|Integer|))) (|Expression| (|MachineFloat|))) "\\spad{assign(x,{}l,{}y)} creates a representation of the assignment of \\spad{y} to the \\spad{l}\\spad{'}th element of array \\spad{x} (\\spad{l} is a list of indices).") (($ (|Symbol|) (|List| (|Polynomial| (|Integer|))) (|Expression| (|MachineInteger|))) "\\spad{assign(x,{}l,{}y)} creates a representation of the assignment of \\spad{y} to the \\spad{l}\\spad{'}th element of array \\spad{x} (\\spad{l} is a list of indices).") (($ (|Symbol|) (|Vector| (|Expression| (|MachineComplex|)))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Vector| (|Expression| (|MachineFloat|)))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Vector| (|Expression| (|MachineInteger|)))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|Expression| (|MachineComplex|)))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|Expression| (|MachineFloat|)))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|Expression| (|MachineInteger|)))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Vector| (|MachineComplex|))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Vector| (|MachineFloat|))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Vector| (|MachineInteger|))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|MachineComplex|))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|MachineFloat|))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|MachineInteger|))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Expression| (|MachineComplex|))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Expression| (|MachineFloat|))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Expression| (|MachineInteger|))) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|String|)) "\\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.")) (|cond| (($ (|Switch|) $ $) "\\spad{cond(s,{}e,{}f)} creates a representation of the FORTRAN expression IF (\\spad{s}) THEN \\spad{e} ELSE \\spad{f}.") (($ (|Switch|) $) "\\spad{cond(s,{}e)} creates a representation of the FORTRAN expression IF (\\spad{s}) THEN \\spad{e}.")) (|returns| (($ (|Expression| (|Complex| (|Float|)))) "\\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.") (($ (|Expression| (|Integer|))) "\\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.") (($ (|Expression| (|Float|))) "\\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.") (($ (|Expression| (|MachineComplex|))) "\\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.") (($ (|Expression| (|MachineInteger|))) "\\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.") (($ (|Expression| (|MachineFloat|))) "\\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.") (($) "\\spad{returns()} creates a representation of a FORTRAN RETURN statement.")) (|call| (($ (|String|)) "\\spad{call(s)} creates a representation of a FORTRAN CALL statement")) (|comment| (($ (|List| (|String|))) "\\spad{comment(s)} creates a representation of the Strings \\spad{s} as a multi-line FORTRAN comment.") (($ (|String|)) "\\spad{comment(s)} creates a representation of the String \\spad{s} as a single FORTRAN comment.")) (|continue| (($ (|SingleInteger|)) "\\spad{continue(l)} creates a representation of a FORTRAN CONTINUE labelled with \\spad{l}")) (|goto| (($ (|SingleInteger|)) "\\spad{goto(l)} creates a representation of a FORTRAN GOTO statement")) (|repeatUntilLoop| (($ (|Switch|) $) "\\spad{repeatUntilLoop(s,{}c)} creates a repeat ... until loop in FORTRAN.")) (|whileLoop| (($ (|Switch|) $) "\\spad{whileLoop(s,{}c)} creates a while loop in FORTRAN.")) (|forLoop| (($ (|SegmentBinding| (|Polynomial| (|Integer|))) (|Polynomial| (|Integer|)) $) "\\spad{forLoop(i=1..10,{}n,{}c)} creates a representation of a FORTRAN DO loop with \\spad{i} ranging over the values 1 to 10 by \\spad{n}.") (($ (|SegmentBinding| (|Polynomial| (|Integer|))) $) "\\spad{forLoop(i=1..10,{}c)} creates a representation of a FORTRAN DO loop with \\spad{i} ranging over the values 1 to 10.")) (|coerce| (((|OutputForm|) $) "\\spad{coerce(f)} returns an object of type OutputForm."))) 
NIL 
NIL 
(-323 E) 
((|constructor| (NIL "This domain creates kernels for use in Fourier series")) (|argument| ((|#1| $) "\\spad{argument(x)} returns the argument of a given sin/cos expressions")) (|sin?| (((|Boolean|) $) "\\spad{sin?(x)} returns \\spad{true} if term is a sin,{} otherwise \\spad{false}")) (|cos| (($ |#1|) "\\spad{cos(x)} makes a cos kernel for use in Fourier series")) (|sin| (($ |#1|) "\\spad{sin(x)} makes a sin kernel for use in Fourier series"))) 
NIL 
NIL 
(-324) 
((|constructor| (NIL "\\spadtype{FortranCodePackage1} provides some utilities for producing useful objects in FortranCode domain. The Package may be used with the FortranCode domain and its \\spad{printCode} or possibly via an outputAsFortran. (The package provides items of use in connection with ASPs in the AXIOM-NAG link and,{} where appropriate,{} naming accords with that in IRENA.) The easy-to-use functions use Fortran loop variables \\spad{I1},{} \\spad{I2},{} and it is users' responsibility to check that this is sensible. The advanced functions use SegmentBinding to allow users control over Fortran loop variable names.")) (|identitySquareMatrix| (((|FortranCode|) (|Symbol|) (|Polynomial| (|Integer|))) "\\spad{identitySquareMatrix(s,{}p)} \\undocumented{}")) (|zeroSquareMatrix| (((|FortranCode|) (|Symbol|) (|Polynomial| (|Integer|))) "\\spad{zeroSquareMatrix(s,{}p)} \\undocumented{}")) (|zeroMatrix| (((|FortranCode|) (|Symbol|) (|SegmentBinding| (|Polynomial| (|Integer|))) (|SegmentBinding| (|Polynomial| (|Integer|)))) "\\spad{zeroMatrix(s,{}b,{}d)} in this version gives the user control over names of Fortran variables used in loops.") (((|FortranCode|) (|Symbol|) (|Polynomial| (|Integer|)) (|Polynomial| (|Integer|))) "\\spad{zeroMatrix(s,{}p,{}q)} uses loop variables in the Fortran,{} \\spad{I1} and \\spad{I2}")) (|zeroVector| (((|FortranCode|) (|Symbol|) (|Polynomial| (|Integer|))) "\\spad{zeroVector(s,{}p)} \\undocumented{}"))) 
NIL 
NIL 
(-325 R1 UP1 UPUP1 F1 R2 UP2 UPUP2 F2) 
((|constructor| (NIL "Lift a map to finite divisors.")) (|map| (((|FiniteDivisor| |#5| |#6| |#7| |#8|) (|Mapping| |#5| |#1|) (|FiniteDivisor| |#1| |#2| |#3| |#4|)) "\\spad{map(f,{}d)} \\undocumented{}"))) 
NIL 
NIL 
(-326 S -2262 UP UPUP R) 
((|constructor| (NIL "This category describes finite rational divisors on a curve,{} that is finite formal sums SUM(\\spad{n} * \\spad{P}) where the \\spad{n}\\spad{'s} are integers and the \\spad{P}\\spad{'s} are finite rational points on the curve.")) (|generator| (((|Union| |#5| "failed") $) "\\spad{generator(d)} returns \\spad{f} if \\spad{(f) = d},{} \"failed\" if \\spad{d} is not principal.")) (|principal?| (((|Boolean|) $) "\\spad{principal?(D)} tests if the argument is the divisor of a function.")) (|reduce| (($ $) "\\spad{reduce(D)} converts \\spad{D} to some reduced form (the reduced forms can be differents in different implementations).")) (|decompose| (((|Record| (|:| |id| (|FractionalIdeal| |#3| (|Fraction| |#3|) |#4| |#5|)) (|:| |principalPart| |#5|)) $) "\\spad{decompose(d)} returns \\spad{[id,{} f]} where \\spad{d = (id) + div(f)}.")) (|divisor| (($ |#5| |#3| |#3| |#3| |#2|) "\\spad{divisor(h,{} d,{} d',{} g,{} r)} returns the sum of all the finite points where \\spad{h/d} has residue \\spad{r}. \\spad{h} must be integral. \\spad{d} must be squarefree. \\spad{d'} is some derivative of \\spad{d} (not necessarily dd/dx). \\spad{g = gcd(d,{}discriminant)} contains the ramified zeros of \\spad{d}") (($ |#2| |#2| (|Integer|)) "\\spad{divisor(a,{} b,{} n)} makes the divisor \\spad{nP} where \\spad{P:} \\spad{(x = a,{} y = b)}. \\spad{P} is allowed to be singular if \\spad{n} is a multiple of the rank.") (($ |#2| |#2|) "\\spad{divisor(a,{} b)} makes the divisor \\spad{P:} \\spad{(x = a,{} y = b)}. Error: if \\spad{P} is singular.") (($ |#5|) "\\spad{divisor(g)} returns the divisor of the function \\spad{g}.") (($ (|FractionalIdeal| |#3| (|Fraction| |#3|) |#4| |#5|)) "\\spad{divisor(I)} makes a divisor \\spad{D} from an ideal \\spad{I}.")) (|ideal| (((|FractionalIdeal| |#3| (|Fraction| |#3|) |#4| |#5|) $) "\\spad{ideal(D)} returns the ideal corresponding to a divisor \\spad{D}."))) 
NIL 
NIL 
(-327 -2262 UP UPUP R) 
((|constructor| (NIL "This category describes finite rational divisors on a curve,{} that is finite formal sums SUM(\\spad{n} * \\spad{P}) where the \\spad{n}\\spad{'s} are integers and the \\spad{P}\\spad{'s} are finite rational points on the curve.")) (|generator| (((|Union| |#4| "failed") $) "\\spad{generator(d)} returns \\spad{f} if \\spad{(f) = d},{} \"failed\" if \\spad{d} is not principal.")) (|principal?| (((|Boolean|) $) "\\spad{principal?(D)} tests if the argument is the divisor of a function.")) (|reduce| (($ $) "\\spad{reduce(D)} converts \\spad{D} to some reduced form (the reduced forms can be differents in different implementations).")) (|decompose| (((|Record| (|:| |id| (|FractionalIdeal| |#2| (|Fraction| |#2|) |#3| |#4|)) (|:| |principalPart| |#4|)) $) "\\spad{decompose(d)} returns \\spad{[id,{} f]} where \\spad{d = (id) + div(f)}.")) (|divisor| (($ |#4| |#2| |#2| |#2| |#1|) "\\spad{divisor(h,{} d,{} d',{} g,{} r)} returns the sum of all the finite points where \\spad{h/d} has residue \\spad{r}. \\spad{h} must be integral. \\spad{d} must be squarefree. \\spad{d'} is some derivative of \\spad{d} (not necessarily dd/dx). \\spad{g = gcd(d,{}discriminant)} contains the ramified zeros of \\spad{d}") (($ |#1| |#1| (|Integer|)) "\\spad{divisor(a,{} b,{} n)} makes the divisor \\spad{nP} where \\spad{P:} \\spad{(x = a,{} y = b)}. \\spad{P} is allowed to be singular if \\spad{n} is a multiple of the rank.") (($ |#1| |#1|) "\\spad{divisor(a,{} b)} makes the divisor \\spad{P:} \\spad{(x = a,{} y = b)}. Error: if \\spad{P} is singular.") (($ |#4|) "\\spad{divisor(g)} returns the divisor of the function \\spad{g}.") (($ (|FractionalIdeal| |#2| (|Fraction| |#2|) |#3| |#4|)) "\\spad{divisor(I)} makes a divisor \\spad{D} from an ideal \\spad{I}.")) (|ideal| (((|FractionalIdeal| |#2| (|Fraction| |#2|) |#3| |#4|) $) "\\spad{ideal(D)} returns the ideal corresponding to a divisor \\spad{D}."))) 
NIL 
NIL 
(-328 -2262 UP UPUP R) 
((|constructor| (NIL "This domains implements finite rational divisors on a curve,{} that is finite formal sums SUM(\\spad{n} * \\spad{P}) where the \\spad{n}\\spad{'s} are integers and the \\spad{P}\\spad{'s} are finite rational points on the curve.")) (|lSpaceBasis| (((|Vector| |#4|) $) "\\spad{lSpaceBasis(d)} returns a basis for \\spad{L(d) = {f | (f) >= -d}} as a module over \\spad{K[x]}.")) (|finiteBasis| (((|Vector| |#4|) $) "\\spad{finiteBasis(d)} returns a basis for \\spad{d} as a module over \\spad{K}[\\spad{x}]."))) 
NIL 
NIL 
(-329 S R) 
((|constructor| (NIL "This category provides a selection of evaluation operations depending on what the argument type \\spad{R} provides.")) (|map| (($ (|Mapping| |#2| |#2|) $) "\\spad{map(f,{} ex)} evaluates ex,{} applying \\spad{f} to values of type \\spad{R} in ex."))) 
NIL 
((|HasCategory| |#2| (LIST (QUOTE -515) (QUOTE (-1153)) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -298) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -276) (|devaluate| |#2|) (|devaluate| |#2|)))) 
(-330 R) 
((|constructor| (NIL "This category provides a selection of evaluation operations depending on what the argument type \\spad{R} provides.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,{} ex)} evaluates ex,{} applying \\spad{f} to values of type \\spad{R} in ex."))) 
NIL 
NIL 
(-331 |basicSymbols| |subscriptedSymbols| R) 
((|constructor| (NIL "A domain of expressions involving functions which can be translated into standard Fortran-77,{} with some extra extensions from the NAG Fortran Library.")) (|useNagFunctions| (((|Boolean|) (|Boolean|)) "\\spad{useNagFunctions(v)} sets the flag which controls whether NAG functions \\indented{1}{are being used for mathematical and machine constants.\\space{2}The previous} \\indented{1}{value is returned.}") (((|Boolean|)) "\\spad{useNagFunctions()} indicates whether NAG functions are being used \\indented{1}{for mathematical and machine constants.}")) (|variables| (((|List| (|Symbol|)) $) "\\spad{variables(e)} return a list of all the variables in \\spad{e}.")) (|pi| (($) "\\spad{\\spad{pi}(x)} represents the NAG Library function X01AAF which returns \\indented{1}{an approximation to the value of \\spad{pi}}")) (|tanh| (($ $) "\\spad{tanh(x)} represents the Fortran intrinsic function TANH")) (|cosh| (($ $) "\\spad{cosh(x)} represents the Fortran intrinsic function COSH")) (|sinh| (($ $) "\\spad{sinh(x)} represents the Fortran intrinsic function SINH")) (|atan| (($ $) "\\spad{atan(x)} represents the Fortran intrinsic function ATAN")) (|acos| (($ $) "\\spad{acos(x)} represents the Fortran intrinsic function ACOS")) (|asin| (($ $) "\\spad{asin(x)} represents the Fortran intrinsic function ASIN")) (|tan| (($ $) "\\spad{tan(x)} represents the Fortran intrinsic function TAN")) (|cos| (($ $) "\\spad{cos(x)} represents the Fortran intrinsic function COS")) (|sin| (($ $) "\\spad{sin(x)} represents the Fortran intrinsic function SIN")) (|log10| (($ $) "\\spad{log10(x)} represents the Fortran intrinsic function \\spad{LOG10}")) (|log| (($ $) "\\spad{log(x)} represents the Fortran intrinsic function LOG")) (|exp| (($ $) "\\spad{exp(x)} represents the Fortran intrinsic function EXP")) (|sqrt| (($ $) "\\spad{sqrt(x)} represents the Fortran intrinsic function SQRT")) (|abs| (($ $) "\\spad{abs(x)} represents the Fortran intrinsic function ABS")) (|coerce| (((|Expression| |#3|) $) "\\spad{coerce(x)} is not documented")) (|retractIfCan| (((|Union| $ "failed") (|Polynomial| (|Float|))) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a FortranExpression checking that it contains no non-Fortran functions,{} and that it only contains the given basic symbols and subscripted symbols which correspond to scalar and array parameters respectively.") (((|Union| $ "failed") (|Fraction| (|Polynomial| (|Float|)))) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a FortranExpression checking that it contains no non-Fortran functions,{} and that it only contains the given basic symbols and subscripted symbols which correspond to scalar and array parameters respectively.") (((|Union| $ "failed") (|Expression| (|Float|))) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a FortranExpression checking that it contains no non-Fortran functions,{} and that it only contains the given basic symbols and subscripted symbols which correspond to scalar and array parameters respectively.") (((|Union| $ "failed") (|Polynomial| (|Integer|))) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a FortranExpression checking that it contains no non-Fortran functions,{} and that it only contains the given basic symbols and subscripted symbols which correspond to scalar and array parameters respectively.") (((|Union| $ "failed") (|Fraction| (|Polynomial| (|Integer|)))) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a FortranExpression checking that it contains no non-Fortran functions,{} and that it only contains the given basic symbols and subscripted symbols which correspond to scalar and array parameters respectively.") (((|Union| $ "failed") (|Expression| (|Integer|))) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a FortranExpression checking that it contains no non-Fortran functions,{} and that it only contains the given basic symbols and subscripted symbols which correspond to scalar and array parameters respectively.") (((|Union| $ "failed") (|Symbol|)) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a FortranExpression checking that it is one of the given basic symbols or subscripted symbols which correspond to scalar and array parameters respectively.") (((|Union| $ "failed") (|Expression| |#3|)) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a FortranExpression checking that it contains no non-Fortran functions,{} and that it only contains the given basic symbols and subscripted symbols which correspond to scalar and array parameters respectively.")) (|retract| (($ (|Polynomial| (|Float|))) "\\spad{retract(e)} takes \\spad{e} and transforms it into a FortranExpression checking that it contains no non-Fortran functions,{} and that it only contains the given basic symbols and subscripted symbols which correspond to scalar and array parameters respectively.") (($ (|Fraction| (|Polynomial| (|Float|)))) "\\spad{retract(e)} takes \\spad{e} and transforms it into a FortranExpression checking that it contains no non-Fortran functions,{} and that it only contains the given basic symbols and subscripted symbols which correspond to scalar and array parameters respectively.") (($ (|Expression| (|Float|))) "\\spad{retract(e)} takes \\spad{e} and transforms it into a FortranExpression checking that it contains no non-Fortran functions,{} and that it only contains the given basic symbols and subscripted symbols which correspond to scalar and array parameters respectively.") (($ (|Polynomial| (|Integer|))) "\\spad{retract(e)} takes \\spad{e} and transforms it into a FortranExpression checking that it contains no non-Fortran functions,{} and that it only contains the given basic symbols and subscripted symbols which correspond to scalar and array parameters respectively.") (($ (|Fraction| (|Polynomial| (|Integer|)))) "\\spad{retract(e)} takes \\spad{e} and transforms it into a FortranExpression checking that it contains no non-Fortran functions,{} and that it only contains the given basic symbols and subscripted symbols which correspond to scalar and array parameters respectively.") (($ (|Expression| (|Integer|))) "\\spad{retract(e)} takes \\spad{e} and transforms it into a FortranExpression checking that it contains no non-Fortran functions,{} and that it only contains the given basic symbols and subscripted symbols which correspond to scalar and array parameters respectively.") (($ (|Symbol|)) "\\spad{retract(e)} takes \\spad{e} and transforms it into a FortranExpression checking that it is one of the given basic symbols or subscripted symbols which correspond to scalar and array parameters respectively.") (($ (|Expression| |#3|)) "\\spad{retract(e)} takes \\spad{e} and transforms it into a FortranExpression checking that it contains no non-Fortran functions,{} and that it only contains the given basic symbols and subscripted symbols which correspond to scalar and array parameters respectively."))) 
((-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-560)))) (|HasCategory| |#3| (LIST (QUOTE -1029) (QUOTE (-375)))) (|HasCategory| $ (QUOTE (-1039))) (|HasCategory| $ (LIST (QUOTE -1029) (QUOTE (-560))))) 
(-332 R1 UP1 UPUP1 F1 R2 UP2 UPUP2 F2) 
((|constructor| (NIL "Lifts a map from rings to function fields over them.")) (|map| ((|#8| (|Mapping| |#5| |#1|) |#4|) "\\spad{map(f,{} p)} lifts \\spad{f} to \\spad{F1} and applies it to \\spad{p}."))) 
NIL 
NIL 
(-333 S -2262 UP UPUP) 
((|constructor| (NIL "This category is a model for the function field of a plane algebraic curve.")) (|rationalPoints| (((|List| (|List| |#2|))) "\\indented{1}{rationalPoints() returns the list of all the affine} rational points.")) (|nonSingularModel| (((|List| (|Polynomial| |#2|)) (|Symbol|)) "\\spad{nonSingularModel(u)} returns the equations in \\spad{u1},{}...,{}un of an affine non-singular model for the curve.")) (|algSplitSimple| (((|Record| (|:| |num| $) (|:| |den| |#3|) (|:| |derivden| |#3|) (|:| |gd| |#3|)) $ (|Mapping| |#3| |#3|)) "\\spad{algSplitSimple(f,{} D)} returns \\spad{[h,{}d,{}d',{}g]} such that \\spad{f=h/d},{} \\spad{h} is integral at all the normal places \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} \\spad{d' = Dd},{} \\spad{g = gcd(d,{} discriminant())} and \\spad{D} is the derivation to use. \\spad{f} must have at most simple finite poles.")) (|hyperelliptic| (((|Union| |#3| "failed")) "\\spad{hyperelliptic()} returns \\spad{p(x)} if the curve is the hyperelliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elliptic| (((|Union| |#3| "failed")) "\\spad{elliptic()} returns \\spad{p(x)} if the curve is the elliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elt| ((|#2| $ |#2| |#2|) "\\spad{elt(f,{}a,{}b)} or \\spad{f}(a,{} \\spad{b}) returns the value of \\spad{f} at the point \\spad{(x = a,{} y = b)} if it is not singular.")) (|primitivePart| (($ $) "\\spad{primitivePart(f)} removes the content of the denominator and the common content of the numerator of \\spad{f}.")) (|differentiate| (($ $ (|Mapping| |#3| |#3|)) "\\spad{differentiate(x,{} d)} extends the derivation \\spad{d} from UP to \\$ and applies it to \\spad{x}.")) (|integralDerivationMatrix| (((|Record| (|:| |num| (|Matrix| |#3|)) (|:| |den| |#3|)) (|Mapping| |#3| |#3|)) "\\spad{integralDerivationMatrix(d)} extends the derivation \\spad{d} from UP to \\$ and returns (\\spad{M},{} \\spad{Q}) such that the i^th row of \\spad{M} divided by \\spad{Q} form the coordinates of \\spad{d(\\spad{wi})} with respect to \\spad{(w1,{}...,{}wn)} where \\spad{(w1,{}...,{}wn)} is the integral basis returned by integralBasis().")) (|integralRepresents| (($ (|Vector| |#3|) |#3|) "\\spad{integralRepresents([A1,{}...,{}An],{} D)} returns \\spad{(A1 w1+...+An wn)/D} where \\spad{(w1,{}...,{}wn)} is the integral basis of \\spad{integralBasis()}.")) (|integralCoordinates| (((|Record| (|:| |num| (|Vector| |#3|)) (|:| |den| |#3|)) $) "\\spad{integralCoordinates(f)} returns \\spad{[[A1,{}...,{}An],{} D]} such that \\spad{f = (A1 w1 +...+ An wn) / D} where \\spad{(w1,{}...,{}wn)} is the integral basis returned by \\spad{integralBasis()}.")) (|represents| (($ (|Vector| |#3|) |#3|) "\\spad{represents([A0,{}...,{}A(n-1)],{}D)} returns \\spad{(A0 + A1 y +...+ A(n-1)*y**(n-1))/D}.") (($ (|Vector| |#3|) |#3|) "\\spad{represents([A0,{}...,{}A(n-1)],{}D)} returns \\spad{(A0 + A1 y +...+ A(n-1)*y**(n-1))/D}.")) (|yCoordinates| (((|Record| (|:| |num| (|Vector| |#3|)) (|:| |den| |#3|)) $) "\\spad{yCoordinates(f)} returns \\spad{[[A1,{}...,{}An],{} D]} such that \\spad{f = (A1 + A2 y +...+ An y**(n-1)) / D}.")) (|inverseIntegralMatrixAtInfinity| (((|Matrix| (|Fraction| |#3|))) "\\indented{1}{inverseIntegralMatrixAtInfinity() returns \\spad{M} such} \\indented{1}{that \\spad{M (v1,{}...,{}vn) = (1,{} y,{} ...,{} y**(n-1))}} \\indented{1}{where \\spad{(v1,{}...,{}vn)} is the local integral basis at infinity} \\indented{1}{returned by \\spad{infIntBasis()}.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} inverseIntegralMatrixAtInfinity()\\$\\spad{R}")) (|integralMatrixAtInfinity| (((|Matrix| (|Fraction| |#3|))) "\\indented{1}{integralMatrixAtInfinity() returns \\spad{M} such that} \\indented{1}{\\spad{(v1,{}...,{}vn) = M (1,{} y,{} ...,{} y**(n-1))}} \\indented{1}{where \\spad{(v1,{}...,{}vn)} is the local integral basis at infinity} \\indented{1}{returned by \\spad{infIntBasis()}.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} integralMatrixAtInfinity()\\$\\spad{R}")) (|inverseIntegralMatrix| (((|Matrix| (|Fraction| |#3|))) "\\indented{1}{inverseIntegralMatrix() returns \\spad{M} such that} \\indented{1}{\\spad{M (w1,{}...,{}wn) = (1,{} y,{} ...,{} y**(n-1))}} \\indented{1}{where \\spad{(w1,{}...,{}wn)} is the integral basis of} \\indented{1}{\\spadfunFrom{integralBasis}{FunctionFieldCategory}.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} inverseIntegralMatrix()\\$\\spad{R}")) (|integralMatrix| (((|Matrix| (|Fraction| |#3|))) "\\indented{1}{integralMatrix() returns \\spad{M} such that} \\indented{1}{\\spad{(w1,{}...,{}wn) = M (1,{} y,{} ...,{} y**(n-1))},{}} \\indented{1}{where \\spad{(w1,{}...,{}wn)} is the integral basis of} \\indented{1}{\\spadfunFrom{integralBasis}{FunctionFieldCategory}.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} integralMatrix()\\$\\spad{R}")) (|reduceBasisAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{reduceBasisAtInfinity(b1,{}...,{}bn)} returns \\spad{(x**i * bj)} for all \\spad{i},{}\\spad{j} such that \\spad{x**i*bj} is locally integral at infinity.")) (|normalizeAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{normalizeAtInfinity(v)} makes \\spad{v} normal at infinity.")) (|complementaryBasis| (((|Vector| $) (|Vector| $)) "\\spad{complementaryBasis(b1,{}...,{}bn)} returns the complementary basis \\spad{(b1',{}...,{}bn')} of \\spad{(b1,{}...,{}bn)}.")) (|integral?| (((|Boolean|) $ |#3|) "\\spad{integral?(f,{} p)} tests whether \\spad{f} is locally integral at \\spad{p(x) = 0}") (((|Boolean|) $ |#2|) "\\spad{integral?(f,{} a)} tests whether \\spad{f} is locally integral at \\spad{x = a}.") (((|Boolean|) $) "\\spad{integral?()} tests if \\spad{f} is integral over \\spad{k[x]}.")) (|integralAtInfinity?| (((|Boolean|) $) "\\spad{integralAtInfinity?()} tests if \\spad{f} is locally integral at infinity.")) (|integralBasisAtInfinity| (((|Vector| $)) "\\indented{1}{integralBasisAtInfinity() returns the local integral basis} \\indented{1}{at infinity} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} integralBasisAtInfinity()\\$\\spad{R}")) (|integralBasis| (((|Vector| $)) "\\indented{1}{integralBasis() returns the integral basis for the curve.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} integralBasis()\\$\\spad{R}")) (|ramified?| (((|Boolean|) |#3|) "\\spad{ramified?(p)} tests whether \\spad{p(x) = 0} is ramified.") (((|Boolean|) |#2|) "\\spad{ramified?(a)} tests whether \\spad{x = a} is ramified.")) (|ramifiedAtInfinity?| (((|Boolean|)) "\\spad{ramifiedAtInfinity?()} tests if infinity is ramified.")) (|singular?| (((|Boolean|) |#3|) "\\spad{singular?(p)} tests whether \\spad{p(x) = 0} is singular.") (((|Boolean|) |#2|) "\\spad{singular?(a)} tests whether \\spad{x = a} is singular.")) (|singularAtInfinity?| (((|Boolean|)) "\\spad{singularAtInfinity?()} tests if there is a singularity at infinity.")) (|branchPoint?| (((|Boolean|) |#3|) "\\spad{branchPoint?(p)} tests whether \\spad{p(x) = 0} is a branch point.") (((|Boolean|) |#2|) "\\spad{branchPoint?(a)} tests whether \\spad{x = a} is a branch point.")) (|branchPointAtInfinity?| (((|Boolean|)) "\\indented{1}{branchPointAtInfinity?() tests if there is a branch point} \\indented{1}{at infinity.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} branchPointAtInfinity?()\\$\\spad{R} \\spad{X} \\spad{R2} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 2 * \\spad{x**2},{} 4) \\spad{X} branchPointAtInfinity?()\\$\\spad{R}")) (|rationalPoint?| (((|Boolean|) |#2| |#2|) "\\indented{1}{rationalPoint?(a,{} \\spad{b}) tests if \\spad{(x=a,{}y=b)} is on the curve.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} rationalPoint?(0,{}0)\\$\\spad{R} \\spad{X} \\spad{R2} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 2 * \\spad{x**2},{} 4) \\spad{X} rationalPoint?(0,{}0)\\$\\spad{R2}")) (|absolutelyIrreducible?| (((|Boolean|)) "\\indented{1}{absolutelyIrreducible?() tests if the curve absolutely irreducible?} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R2} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 2 * \\spad{x**2},{} 4) \\spad{X} absolutelyIrreducible?()\\$\\spad{R2}")) (|genus| (((|NonNegativeInteger|)) "\\indented{1}{genus() returns the genus of one absolutely irreducible component} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} genus()\\$\\spad{R}")) (|numberOfComponents| (((|NonNegativeInteger|)) "\\indented{1}{numberOfComponents() returns the number of absolutely irreducible} \\indented{1}{components.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} numberOfComponents()\\$\\spad{R}"))) 
NIL 
((|HasCategory| |#2| (QUOTE (-364))) (|HasCategory| |#2| (QUOTE (-359)))) 
(-334 -2262 UP UPUP) 
((|constructor| (NIL "This category is a model for the function field of a plane algebraic curve.")) (|rationalPoints| (((|List| (|List| |#1|))) "\\indented{1}{rationalPoints() returns the list of all the affine} rational points.")) (|nonSingularModel| (((|List| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{nonSingularModel(u)} returns the equations in \\spad{u1},{}...,{}un of an affine non-singular model for the curve.")) (|algSplitSimple| (((|Record| (|:| |num| $) (|:| |den| |#2|) (|:| |derivden| |#2|) (|:| |gd| |#2|)) $ (|Mapping| |#2| |#2|)) "\\spad{algSplitSimple(f,{} D)} returns \\spad{[h,{}d,{}d',{}g]} such that \\spad{f=h/d},{} \\spad{h} is integral at all the normal places \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} \\spad{d' = Dd},{} \\spad{g = gcd(d,{} discriminant())} and \\spad{D} is the derivation to use. \\spad{f} must have at most simple finite poles.")) (|hyperelliptic| (((|Union| |#2| "failed")) "\\spad{hyperelliptic()} returns \\spad{p(x)} if the curve is the hyperelliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elliptic| (((|Union| |#2| "failed")) "\\spad{elliptic()} returns \\spad{p(x)} if the curve is the elliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elt| ((|#1| $ |#1| |#1|) "\\spad{elt(f,{}a,{}b)} or \\spad{f}(a,{} \\spad{b}) returns the value of \\spad{f} at the point \\spad{(x = a,{} y = b)} if it is not singular.")) (|primitivePart| (($ $) "\\spad{primitivePart(f)} removes the content of the denominator and the common content of the numerator of \\spad{f}.")) (|differentiate| (($ $ (|Mapping| |#2| |#2|)) "\\spad{differentiate(x,{} d)} extends the derivation \\spad{d} from UP to \\$ and applies it to \\spad{x}.")) (|integralDerivationMatrix| (((|Record| (|:| |num| (|Matrix| |#2|)) (|:| |den| |#2|)) (|Mapping| |#2| |#2|)) "\\spad{integralDerivationMatrix(d)} extends the derivation \\spad{d} from UP to \\$ and returns (\\spad{M},{} \\spad{Q}) such that the i^th row of \\spad{M} divided by \\spad{Q} form the coordinates of \\spad{d(\\spad{wi})} with respect to \\spad{(w1,{}...,{}wn)} where \\spad{(w1,{}...,{}wn)} is the integral basis returned by integralBasis().")) (|integralRepresents| (($ (|Vector| |#2|) |#2|) "\\spad{integralRepresents([A1,{}...,{}An],{} D)} returns \\spad{(A1 w1+...+An wn)/D} where \\spad{(w1,{}...,{}wn)} is the integral basis of \\spad{integralBasis()}.")) (|integralCoordinates| (((|Record| (|:| |num| (|Vector| |#2|)) (|:| |den| |#2|)) $) "\\spad{integralCoordinates(f)} returns \\spad{[[A1,{}...,{}An],{} D]} such that \\spad{f = (A1 w1 +...+ An wn) / D} where \\spad{(w1,{}...,{}wn)} is the integral basis returned by \\spad{integralBasis()}.")) (|represents| (($ (|Vector| |#2|) |#2|) "\\spad{represents([A0,{}...,{}A(n-1)],{}D)} returns \\spad{(A0 + A1 y +...+ A(n-1)*y**(n-1))/D}.") (($ (|Vector| |#2|) |#2|) "\\spad{represents([A0,{}...,{}A(n-1)],{}D)} returns \\spad{(A0 + A1 y +...+ A(n-1)*y**(n-1))/D}.")) (|yCoordinates| (((|Record| (|:| |num| (|Vector| |#2|)) (|:| |den| |#2|)) $) "\\spad{yCoordinates(f)} returns \\spad{[[A1,{}...,{}An],{} D]} such that \\spad{f = (A1 + A2 y +...+ An y**(n-1)) / D}.")) (|inverseIntegralMatrixAtInfinity| (((|Matrix| (|Fraction| |#2|))) "\\indented{1}{inverseIntegralMatrixAtInfinity() returns \\spad{M} such} \\indented{1}{that \\spad{M (v1,{}...,{}vn) = (1,{} y,{} ...,{} y**(n-1))}} \\indented{1}{where \\spad{(v1,{}...,{}vn)} is the local integral basis at infinity} \\indented{1}{returned by \\spad{infIntBasis()}.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} inverseIntegralMatrixAtInfinity()\\$\\spad{R}")) (|integralMatrixAtInfinity| (((|Matrix| (|Fraction| |#2|))) "\\indented{1}{integralMatrixAtInfinity() returns \\spad{M} such that} \\indented{1}{\\spad{(v1,{}...,{}vn) = M (1,{} y,{} ...,{} y**(n-1))}} \\indented{1}{where \\spad{(v1,{}...,{}vn)} is the local integral basis at infinity} \\indented{1}{returned by \\spad{infIntBasis()}.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} integralMatrixAtInfinity()\\$\\spad{R}")) (|inverseIntegralMatrix| (((|Matrix| (|Fraction| |#2|))) "\\indented{1}{inverseIntegralMatrix() returns \\spad{M} such that} \\indented{1}{\\spad{M (w1,{}...,{}wn) = (1,{} y,{} ...,{} y**(n-1))}} \\indented{1}{where \\spad{(w1,{}...,{}wn)} is the integral basis of} \\indented{1}{\\spadfunFrom{integralBasis}{FunctionFieldCategory}.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} inverseIntegralMatrix()\\$\\spad{R}")) (|integralMatrix| (((|Matrix| (|Fraction| |#2|))) "\\indented{1}{integralMatrix() returns \\spad{M} such that} \\indented{1}{\\spad{(w1,{}...,{}wn) = M (1,{} y,{} ...,{} y**(n-1))},{}} \\indented{1}{where \\spad{(w1,{}...,{}wn)} is the integral basis of} \\indented{1}{\\spadfunFrom{integralBasis}{FunctionFieldCategory}.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} integralMatrix()\\$\\spad{R}")) (|reduceBasisAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{reduceBasisAtInfinity(b1,{}...,{}bn)} returns \\spad{(x**i * bj)} for all \\spad{i},{}\\spad{j} such that \\spad{x**i*bj} is locally integral at infinity.")) (|normalizeAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{normalizeAtInfinity(v)} makes \\spad{v} normal at infinity.")) (|complementaryBasis| (((|Vector| $) (|Vector| $)) "\\spad{complementaryBasis(b1,{}...,{}bn)} returns the complementary basis \\spad{(b1',{}...,{}bn')} of \\spad{(b1,{}...,{}bn)}.")) (|integral?| (((|Boolean|) $ |#2|) "\\spad{integral?(f,{} p)} tests whether \\spad{f} is locally integral at \\spad{p(x) = 0}") (((|Boolean|) $ |#1|) "\\spad{integral?(f,{} a)} tests whether \\spad{f} is locally integral at \\spad{x = a}.") (((|Boolean|) $) "\\spad{integral?()} tests if \\spad{f} is integral over \\spad{k[x]}.")) (|integralAtInfinity?| (((|Boolean|) $) "\\spad{integralAtInfinity?()} tests if \\spad{f} is locally integral at infinity.")) (|integralBasisAtInfinity| (((|Vector| $)) "\\indented{1}{integralBasisAtInfinity() returns the local integral basis} \\indented{1}{at infinity} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} integralBasisAtInfinity()\\$\\spad{R}")) (|integralBasis| (((|Vector| $)) "\\indented{1}{integralBasis() returns the integral basis for the curve.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} integralBasis()\\$\\spad{R}")) (|ramified?| (((|Boolean|) |#2|) "\\spad{ramified?(p)} tests whether \\spad{p(x) = 0} is ramified.") (((|Boolean|) |#1|) "\\spad{ramified?(a)} tests whether \\spad{x = a} is ramified.")) (|ramifiedAtInfinity?| (((|Boolean|)) "\\spad{ramifiedAtInfinity?()} tests if infinity is ramified.")) (|singular?| (((|Boolean|) |#2|) "\\spad{singular?(p)} tests whether \\spad{p(x) = 0} is singular.") (((|Boolean|) |#1|) "\\spad{singular?(a)} tests whether \\spad{x = a} is singular.")) (|singularAtInfinity?| (((|Boolean|)) "\\spad{singularAtInfinity?()} tests if there is a singularity at infinity.")) (|branchPoint?| (((|Boolean|) |#2|) "\\spad{branchPoint?(p)} tests whether \\spad{p(x) = 0} is a branch point.") (((|Boolean|) |#1|) "\\spad{branchPoint?(a)} tests whether \\spad{x = a} is a branch point.")) (|branchPointAtInfinity?| (((|Boolean|)) "\\indented{1}{branchPointAtInfinity?() tests if there is a branch point} \\indented{1}{at infinity.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} branchPointAtInfinity?()\\$\\spad{R} \\spad{X} \\spad{R2} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 2 * \\spad{x**2},{} 4) \\spad{X} branchPointAtInfinity?()\\$\\spad{R}")) (|rationalPoint?| (((|Boolean|) |#1| |#1|) "\\indented{1}{rationalPoint?(a,{} \\spad{b}) tests if \\spad{(x=a,{}y=b)} is on the curve.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} rationalPoint?(0,{}0)\\$\\spad{R} \\spad{X} \\spad{R2} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 2 * \\spad{x**2},{} 4) \\spad{X} rationalPoint?(0,{}0)\\$\\spad{R2}")) (|absolutelyIrreducible?| (((|Boolean|)) "\\indented{1}{absolutelyIrreducible?() tests if the curve absolutely irreducible?} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R2} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 2 * \\spad{x**2},{} 4) \\spad{X} absolutelyIrreducible?()\\$\\spad{R2}")) (|genus| (((|NonNegativeInteger|)) "\\indented{1}{genus() returns the genus of one absolutely irreducible component} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} genus()\\$\\spad{R}")) (|numberOfComponents| (((|NonNegativeInteger|)) "\\indented{1}{numberOfComponents() returns the number of absolutely irreducible} \\indented{1}{components.} \\blankline \\spad{X} \\spad{P0} \\spad{:=} UnivariatePolynomial(\\spad{x},{} Integer) \\spad{X} \\spad{P1} \\spad{:=} UnivariatePolynomial(\\spad{y},{} Fraction \\spad{P0}) \\spad{X} \\spad{R} \\spad{:=} RadicalFunctionField(INT,{} \\spad{P0},{} \\spad{P1},{} 1 - \\spad{x**20},{} 20) \\spad{X} numberOfComponents()\\$\\spad{R}"))) 
((-4498 |has| (-403 |#2|) (-359)) (-4503 |has| (-403 |#2|) (-359)) (-4497 |has| (-403 |#2|) (-359)) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-335 |p| |extdeg|) 
((|constructor| (NIL "FiniteFieldCyclicGroup(\\spad{p},{}\\spad{n}) implements a finite field extension of degee \\spad{n} over the prime field with \\spad{p} elements. Its elements are represented by powers of a primitive element,{} \\spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial,{} which is created by createPrimitivePoly from \\spadtype{FiniteFieldPolynomialPackage}. The Zech logarithms are stored in a table of size half of the field size,{} and use \\spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.")) (|getZechTable| (((|PrimitiveArray| (|SingleInteger|))) "\\spad{getZechTable()} returns the zech logarithm table of the field. This table is used to perform additions in the field quickly."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| (-897 |#1|) (QUOTE (-148))) (|HasCategory| (-897 |#1|) (QUOTE (-364))) (|HasCategory| (-897 |#1|) (QUOTE (-146))) (-3322 (|HasCategory| (-897 |#1|) (QUOTE (-146))) (|HasCategory| (-897 |#1|) (QUOTE (-364))))) 
(-336 GF |defpol|) 
((|constructor| (NIL "FiniteFieldCyclicGroupExtensionByPolynomial(\\spad{GF},{}defpol) implements a finite extension field of the ground field \\spad{GF}. Its elements are represented by powers of a primitive element,{} \\spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial defpol,{} which MUST be primitive (user responsibility). Zech logarithms are stored in a table of size half of the field size,{} and use \\spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.")) (|getZechTable| (((|PrimitiveArray| (|SingleInteger|))) "\\spad{getZechTable()} returns the zech logarithm table of the field it is used to perform additions in the field quickly."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-364))) (|HasCategory| |#1| (QUOTE (-146))) (-3322 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-364))))) 
(-337 GF |extdeg|) 
((|constructor| (NIL "FiniteFieldCyclicGroupExtension(\\spad{GF},{}\\spad{n}) implements a extension of degree \\spad{n} over the ground field \\spad{GF}. Its elements are represented by powers of a primitive element,{} \\spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial,{} which is created by createPrimitivePoly from \\spadtype{FiniteFieldPolynomialPackage}. Zech logarithms are stored in a table of size half of the field size,{} and use \\spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.")) (|getZechTable| (((|PrimitiveArray| (|SingleInteger|))) "\\spad{getZechTable()} returns the zech logarithm table of the field. This table is used to perform additions in the field quickly."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-364))) (|HasCategory| |#1| (QUOTE (-146))) (-3322 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-364))))) 
(-338 K |PolK|) 
((|constructor| (NIL "Part of the package for Algebraic Function Fields in one variable (\\spad{PAFF}) It has been modified (very slitely) so that each time the \"factor\" function is used,{} the variable related to the size of the field over which the polynomial is factorized is reset. This is done in order to be used with a \"dynamic extension field\" which size is not fixed but set before calling the \"factor\" function and which is parse by side effect to this package via the function \"size\". See the local function \"initialize\" of this package."))) 
NIL 
NIL 
(-339 -3468 V VF) 
((|constructor| (NIL "This package lifts the interpolation functions from \\spadtype{FractionFreeFastGaussian} to fractions. The packages defined in this file provide fast fraction free rational interpolation algorithms. (see \\spad{FAMR2},{} FFFG,{} FFFGF,{} NEWTON)")) (|generalInterpolation| (((|Stream| (|Matrix| (|SparseUnivariatePolynomial| |#1|))) (|List| |#1|) (|Mapping| |#1| (|NonNegativeInteger|) (|NonNegativeInteger|) |#2|) (|Vector| |#3|) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{generalInterpolation(l,{} CA,{} f,{} sumEta,{} maxEta)} applies generalInterpolation(\\spad{l},{} \\spad{CA},{} \\spad{f},{} eta) for all possible eta with maximal entry maxEta and sum of entries \\spad{sumEta}") (((|Matrix| (|SparseUnivariatePolynomial| |#1|)) (|List| |#1|) (|Mapping| |#1| (|NonNegativeInteger|) (|NonNegativeInteger|) |#2|) (|Vector| |#3|) (|List| (|NonNegativeInteger|))) "\\spad{generalInterpolation(l,{} CA,{} f,{} eta)} performs Hermite-Pade approximation using the given action \\spad{CA} of polynomials on the elements of \\spad{f}. The result is guaranteed to be correct up to order |eta|-1. Given that eta is a \"normal\" point,{} the degrees on the diagonal are given by eta. The degrees of column \\spad{i} are in this case eta + \\spad{e}.\\spad{i} - [1,{}1,{}...,{}1],{} where the degree of zero is \\spad{-1}."))) 
NIL 
NIL 
(-340 -3468 V) 
((|constructor| (NIL "This package implements the interpolation algorithm proposed in Beckermann,{} Bernhard and Labahn,{} George,{} Fraction-free computation of matrix rational interpolants and matrix GCDs,{} SIAM Journal on Matrix Analysis and Applications 22. The packages defined in this file provide fast fraction free rational interpolation algorithms. (see \\spad{FAMR2},{} FFFG,{} FFFGF,{} NEWTON)")) (|qShiftC| (((|List| |#1|) |#1| (|NonNegativeInteger|)) "\\spad{qShiftC} gives the coefficients \\spad{c_}{\\spad{k},{}\\spad{k}} in the expansion <x^k> \\spad{z} \\spad{g}(\\spad{x}) = sum_{\\spad{i=0}}\\spad{^k} \\spad{c_}{\\spad{k},{}\\spad{i}} <x^i> \\spad{g}(\\spad{x}),{} where \\spad{z} acts on \\spad{g}(\\spad{x}) by shifting. In fact,{} the result is [1,{}\\spad{q},{}\\spad{q^2},{}...]")) (|qShiftAction| ((|#1| |#1| (|NonNegativeInteger|) (|NonNegativeInteger|) |#2|) "\\spad{qShiftAction(q,{} k,{} l,{} g)} gives the coefficient of \\spad{x^k} in \\spad{z^l} \\spad{g}(\\spad{x}),{} where \\spad{z*}(a+b*x+c*x^2+d*x^3+...) = (a+q*b*x+q^2*c*x^2+q^3*d*x^3+...). In terms of sequences,{} z*u(\\spad{n})=q^n*u(\\spad{n}).")) (|DiffC| (((|List| |#1|) (|NonNegativeInteger|)) "\\spad{DiffC} gives the coefficients \\spad{c_}{\\spad{k},{}\\spad{k}} in the expansion <x^k> \\spad{z} \\spad{g}(\\spad{x}) = sum_{\\spad{i=0}}\\spad{^k} \\spad{c_}{\\spad{k},{}\\spad{i}} <x^i> \\spad{g}(\\spad{x}),{} where \\spad{z} acts on \\spad{g}(\\spad{x}) by shifting. In fact,{} the result is [0,{}0,{}0,{}...]")) (|DiffAction| ((|#1| (|NonNegativeInteger|) (|NonNegativeInteger|) |#2|) "\\spad{DiffAction(k,{} l,{} g)} gives the coefficient of \\spad{x^k} in \\spad{z^l} \\spad{g}(\\spad{x}),{} where \\spad{z*}(a+b*x+c*x^2+d*x^3+...) = (a*x+b*x^2+c*x^3+...),{} \\spadignore{i.e.} multiplication with \\spad{x}.")) (|ShiftC| (((|List| |#1|) (|NonNegativeInteger|)) "\\spad{ShiftC} gives the coefficients \\spad{c_}{\\spad{k},{}\\spad{k}} in the expansion <x^k> \\spad{z} \\spad{g}(\\spad{x}) = sum_{\\spad{i=0}}\\spad{^k} \\spad{c_}{\\spad{k},{}\\spad{i}} <x^i> \\spad{g}(\\spad{x}),{} where \\spad{z} acts on \\spad{g}(\\spad{x}) by shifting. In fact,{} the result is [0,{}1,{}2,{}...]")) (|ShiftAction| ((|#1| (|NonNegativeInteger|) (|NonNegativeInteger|) |#2|) "\\spad{ShiftAction(k,{} l,{} g)} gives the coefficient of \\spad{x^k} in \\spad{z^l} \\spad{g}(\\spad{x}),{} where \\spad{z*(a+b*x+c*x^2+d*x^3+...) = (b*x+2*c*x^2+3*d*x^3+...)}. In terms of sequences,{} z*u(\\spad{n})=n*u(\\spad{n}).")) (|generalCoefficient| ((|#1| (|Mapping| |#1| (|NonNegativeInteger|) (|NonNegativeInteger|) |#2|) (|Vector| |#2|) (|NonNegativeInteger|) (|Vector| (|SparseUnivariatePolynomial| |#1|))) "\\spad{generalCoefficient(action,{} f,{} k,{} p)} gives the coefficient of \\spad{x^k} in \\spad{p}(\\spad{z})\\dot \\spad{f}(\\spad{x}),{} where the \\spad{action} of \\spad{z^l} on a polynomial in \\spad{x} is given by \\spad{action},{} \\spadignore{i.e.} \\spad{action}(\\spad{k},{} \\spad{l},{} \\spad{f}) should return the coefficient of \\spad{x^k} in \\spad{z^l} \\spad{f}(\\spad{x}).")) (|generalInterpolation| (((|Stream| (|Matrix| (|SparseUnivariatePolynomial| |#1|))) (|List| |#1|) (|Mapping| |#1| (|NonNegativeInteger|) (|NonNegativeInteger|) |#2|) (|Vector| |#2|) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{generalInterpolation(C,{} CA,{} f,{} sumEta,{} maxEta)} applies \\spad{generalInterpolation(C,{} CA,{} f,{} eta)} for all possible \\spad{eta} with maximal entry \\spad{maxEta} and sum of entries at most \\spad{sumEta}. \\blankline The first argument \\spad{C} is the list of coefficients \\spad{c_}{\\spad{k},{}\\spad{k}} in the expansion <x^k> \\spad{z} \\spad{g}(\\spad{x}) = sum_{\\spad{i=0}}\\spad{^k} \\spad{c_}{\\spad{k},{}\\spad{i}} <x^i> \\spad{g}(\\spad{x}). \\blankline The second argument,{} \\spad{CA}(\\spad{k},{} \\spad{l},{} \\spad{f}),{} should return the coefficient of \\spad{x^k} in \\spad{z^l} \\spad{f}(\\spad{x}).") (((|Matrix| (|SparseUnivariatePolynomial| |#1|)) (|List| |#1|) (|Mapping| |#1| (|NonNegativeInteger|) (|NonNegativeInteger|) |#2|) (|Vector| |#2|) (|List| (|NonNegativeInteger|))) "\\spad{generalInterpolation(C,{} CA,{} f,{} eta)} performs Hermite-Pade approximation using the given action \\spad{CA} of polynomials on the elements of \\spad{f}. The result is guaranteed to be correct up to order |eta|-1. Given that eta is a \"normal\" point,{} the degrees on the diagonal are given by eta. The degrees of column \\spad{i} are in this case eta + \\spad{e}.\\spad{i} - [1,{}1,{}...,{}1],{} where the degree of zero is \\spad{-1}. \\blankline The first argument \\spad{C} is the list of coefficients \\spad{c_}{\\spad{k},{}\\spad{k}} in the expansion <x^k> \\spad{z} \\spad{g}(\\spad{x}) = sum_{\\spad{i=0}}\\spad{^k} \\spad{c_}{\\spad{k},{}\\spad{i}} <x^i> \\spad{g}(\\spad{x}). \\blankline The second argument,{} \\spad{CA}(\\spad{k},{} \\spad{l},{} \\spad{f}),{} should return the coefficient of \\spad{x^k} in \\spad{z^l} \\spad{f}(\\spad{x}).")) (|interpolate| (((|Fraction| (|SparseUnivariatePolynomial| |#1|)) (|List| (|Fraction| |#1|)) (|List| (|Fraction| |#1|)) (|NonNegativeInteger|)) "\\spad{interpolate(xlist,{} ylist,{} deg} returns the rational function with numerator degree \\spad{deg} that interpolates the given points using fraction free arithmetic.") (((|Fraction| (|SparseUnivariatePolynomial| |#1|)) (|List| |#1|) (|List| |#1|) (|NonNegativeInteger|)) "\\spad{interpolate(xlist,{} ylist,{} deg} returns the rational function with numerator degree at most \\spad{deg} and denominator degree at most \\spad{\\#xlist-deg-1} that interpolates the given points using fraction free arithmetic. Note that rational interpolation does not guarantee that all given points are interpolated correctly: unattainable points may make this impossible.")) (|fffg| (((|Matrix| (|SparseUnivariatePolynomial| |#1|)) (|List| |#1|) (|Mapping| |#1| (|NonNegativeInteger|) (|Vector| (|SparseUnivariatePolynomial| |#1|))) (|List| (|NonNegativeInteger|))) "\\spad{fffg} is the general algorithm as proposed by Beckermann and Labahn. \\blankline The first argument is the list of \\spad{c_}{\\spad{i},{}\\spad{i}}. These are the only values of \\spad{C} explicitely needed in \\spad{fffg}. \\blankline The second argument \\spad{c},{} computes \\spad{c_k}(\\spad{M}),{} \\spadignore{i.e.} \\spad{c_k}(.) is the dual basis of the vector space \\spad{V},{} but also knows about the special multiplication rule as descibed in Equation (2). Note that the information about \\spad{f} is therefore encoded in \\spad{c}. \\blankline The third argument is the vector of degree bounds \\spad{n},{} as introduced in Definition 2.1. In particular,{} the sum of the entries is the order of the Mahler system computed."))) 
NIL 
NIL 
(-341 GF) 
((|constructor| (NIL "FiniteFieldFunctions(\\spad{GF}) is a package with functions concerning finite extension fields of the finite ground field \\spad{GF},{} \\spadignore{e.g.} Zech logarithms.")) (|createLowComplexityNormalBasis| (((|Union| (|SparseUnivariatePolynomial| |#1|) (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) (|PositiveInteger|)) "\\spad{createLowComplexityNormalBasis(n)} tries to find a a low complexity normal basis of degree \\spad{n} over \\spad{GF} and returns its multiplication matrix If no low complexity basis is found it calls \\axiomFunFrom{createNormalPoly}{FiniteFieldPolynomialPackage}(\\spad{n}) to produce a normal polynomial of degree \\spad{n} over \\spad{GF}")) (|createLowComplexityTable| (((|Union| (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|))))) "failed") (|PositiveInteger|)) "\\spad{createLowComplexityTable(n)} tries to find a low complexity normal basis of degree \\spad{n} over \\spad{GF} and returns its multiplication matrix Fails,{} if it does not find a low complexity basis")) (|sizeMultiplication| (((|NonNegativeInteger|) (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{sizeMultiplication(m)} returns the number of entries of the multiplication table \\spad{m}.")) (|createMultiplicationMatrix| (((|Matrix| |#1|) (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{createMultiplicationMatrix(m)} forms the multiplication table \\spad{m} into a matrix over the ground field.")) (|createMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|))))) (|SparseUnivariatePolynomial| |#1|)) "\\spad{createMultiplicationTable(f)} generates a multiplication table for the normal basis of the field extension determined by \\spad{f}. This is needed to perform multiplications between elements represented as coordinate vectors to this basis. See \\spadtype{FFNBP},{} \\spadtype{FFNBX}.")) (|createZechTable| (((|PrimitiveArray| (|SingleInteger|)) (|SparseUnivariatePolynomial| |#1|)) "\\spad{createZechTable(f)} generates a Zech logarithm table for the cyclic group representation of a extension of the ground field by the primitive polynomial \\spad{f}(\\spad{x}),{} \\spadignore{i.e.} \\spad{Z(i)},{} defined by x**Z(\\spad{i}) = 1+x**i is stored at index \\spad{i}. This is needed in particular to perform addition of field elements in finite fields represented in this way. See \\spadtype{FFCGP},{} \\spadtype{FFCGX}."))) 
NIL 
NIL 
(-342 F1 GF F2) 
((|constructor| (NIL "FiniteFieldHomomorphisms(\\spad{F1},{}\\spad{GF},{}\\spad{F2}) exports coercion functions of elements between the fields \\spad{F1} and \\spad{F2},{} which both must be finite simple algebraic extensions of the finite ground field \\spad{GF}.")) (|coerce| ((|#1| |#3|) "\\spad{coerce(x)} is the homomorphic image of \\spad{x} from \\spad{F2} in \\spad{F1},{} where coerce is a field homomorphism between the fields extensions \\spad{F2} and \\spad{F1} both over ground field \\spad{GF} (the second argument to the package). Error: if the extension degree of \\spad{F2} doesn\\spad{'t} divide the extension degree of \\spad{F1}. Note that the other coercion function in the \\spadtype{FiniteFieldHomomorphisms} is a left inverse.") ((|#3| |#1|) "\\spad{coerce(x)} is the homomorphic image of \\spad{x} from \\spad{F1} in \\spad{F2}. Thus coerce is a field homomorphism between the fields extensions \\spad{F1} and \\spad{F2} both over ground field \\spad{GF} (the second argument to the package). Error: if the extension degree of \\spad{F1} doesn\\spad{'t} divide the extension degree of \\spad{F2}. Note that the other coercion function in the \\spadtype{FiniteFieldHomomorphisms} is a left inverse."))) 
NIL 
NIL 
(-343 S) 
((|constructor| (NIL "FiniteFieldCategory is the category of finite fields")) (|representationType| (((|Union| "prime" "polynomial" "normal" "cyclic")) "\\spad{representationType()} returns the type of the representation,{} one of: \\spad{prime},{} \\spad{polynomial},{} \\spad{normal},{} or \\spad{cyclic}.")) (|order| (((|PositiveInteger|) $) "\\spad{order(b)} computes the order of an element \\spad{b} in the multiplicative group of the field. Error: if \\spad{b} equals 0.")) (|discreteLog| (((|NonNegativeInteger|) $) "\\spad{discreteLog(a)} computes the discrete logarithm of \\spad{a} with respect to \\spad{primitiveElement()} of the field.")) (|primitive?| (((|Boolean|) $) "\\spad{primitive?(b)} tests whether the element \\spad{b} is a generator of the (cyclic) multiplicative group of the field,{} \\spadignore{i.e.} is a primitive element. Implementation Note that see \\spad{ch}.IX.1.3,{} th.2 in \\spad{D}. Lipson.")) (|primitiveElement| (($) "\\spad{primitiveElement()} returns a primitive element stored in a global variable in the domain. At first call,{} the primitive element is computed by calling \\spadfun{createPrimitiveElement}.")) (|createPrimitiveElement| (($) "\\spad{createPrimitiveElement()} computes a generator of the (cyclic) multiplicative group of the field.")) (|tableForDiscreteLogarithm| (((|Table| (|PositiveInteger|) (|NonNegativeInteger|)) (|Integer|)) "\\spad{tableForDiscreteLogarithm(a,{}n)} returns a table of the discrete logarithms of \\spad{a**0} up to \\spad{a**(n-1)} which,{} called with key \\spad{lookup(a**i)} returns \\spad{i} for \\spad{i} in \\spad{0..n-1}. Error: if not called for prime divisors of order of \\indented{7}{multiplicative group.}")) (|factorsOfCyclicGroupSize| (((|List| (|Record| (|:| |factor| (|Integer|)) (|:| |exponent| (|Integer|))))) "\\spad{factorsOfCyclicGroupSize()} returns the factorization of size()\\spad{-1}")) (|conditionP| (((|Union| (|Vector| $) "failed") (|Matrix| $)) "\\spad{conditionP(mat)},{} given a matrix representing a homogeneous system of equations,{} returns a vector whose characteristic'th powers is a non-trivial solution,{} or \"failed\" if no such vector exists.")) (|charthRoot| (($ $) "\\spad{charthRoot(a)} takes the characteristic'th root of a. Note that such a root is alway defined in finite fields."))) 
NIL 
NIL 
(-344) 
((|constructor| (NIL "FiniteFieldCategory is the category of finite fields")) (|representationType| (((|Union| "prime" "polynomial" "normal" "cyclic")) "\\spad{representationType()} returns the type of the representation,{} one of: \\spad{prime},{} \\spad{polynomial},{} \\spad{normal},{} or \\spad{cyclic}.")) (|order| (((|PositiveInteger|) $) "\\spad{order(b)} computes the order of an element \\spad{b} in the multiplicative group of the field. Error: if \\spad{b} equals 0.")) (|discreteLog| (((|NonNegativeInteger|) $) "\\spad{discreteLog(a)} computes the discrete logarithm of \\spad{a} with respect to \\spad{primitiveElement()} of the field.")) (|primitive?| (((|Boolean|) $) "\\spad{primitive?(b)} tests whether the element \\spad{b} is a generator of the (cyclic) multiplicative group of the field,{} \\spadignore{i.e.} is a primitive element. Implementation Note that see \\spad{ch}.IX.1.3,{} th.2 in \\spad{D}. Lipson.")) (|primitiveElement| (($) "\\spad{primitiveElement()} returns a primitive element stored in a global variable in the domain. At first call,{} the primitive element is computed by calling \\spadfun{createPrimitiveElement}.")) (|createPrimitiveElement| (($) "\\spad{createPrimitiveElement()} computes a generator of the (cyclic) multiplicative group of the field.")) (|tableForDiscreteLogarithm| (((|Table| (|PositiveInteger|) (|NonNegativeInteger|)) (|Integer|)) "\\spad{tableForDiscreteLogarithm(a,{}n)} returns a table of the discrete logarithms of \\spad{a**0} up to \\spad{a**(n-1)} which,{} called with key \\spad{lookup(a**i)} returns \\spad{i} for \\spad{i} in \\spad{0..n-1}. Error: if not called for prime divisors of order of \\indented{7}{multiplicative group.}")) (|factorsOfCyclicGroupSize| (((|List| (|Record| (|:| |factor| (|Integer|)) (|:| |exponent| (|Integer|))))) "\\spad{factorsOfCyclicGroupSize()} returns the factorization of size()\\spad{-1}")) (|conditionP| (((|Union| (|Vector| $) "failed") (|Matrix| $)) "\\spad{conditionP(mat)},{} given a matrix representing a homogeneous system of equations,{} returns a vector whose characteristic'th powers is a non-trivial solution,{} or \"failed\" if no such vector exists.")) (|charthRoot| (($ $) "\\spad{charthRoot(a)} takes the characteristic'th root of a. Note that such a root is alway defined in finite fields."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-345 R UP -2262) 
((|constructor| (NIL "Integral bases for function fields of dimension one In this package \\spad{R} is a Euclidean domain and \\spad{F} is a framed algebra over \\spad{R}. The package provides functions to compute the integral closure of \\spad{R} in the quotient field of \\spad{F}. It is assumed that \\spad{char(R/P) = char(R)} for any prime \\spad{P} of \\spad{R}. A typical instance of this is when \\spad{R = K[x]} and \\spad{F} is a function field over \\spad{R}.")) (|localIntegralBasis| (((|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|))) |#1|) "\\spad{integralBasis(p)} returns a record \\spad{[basis,{}basisDen,{}basisInv]} containing information regarding the local integral closure of \\spad{R} at the prime \\spad{p} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,{}w2,{}...,{}wn}. If \\spad{basis} is the matrix \\spad{(aij,{} i = 1..n,{} j = 1..n)},{} then the \\spad{i}th element of the local integral basis is \\spad{\\spad{vi} = (1/basisDen) * sum(aij * wj,{} j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{\\spad{wi}} with respect to the basis \\spad{v1,{}...,{}vn}: if \\spad{basisInv} is the matrix \\spad{(bij,{} i = 1..n,{} j = 1..n)},{} then \\spad{\\spad{wi} = sum(bij * vj,{} j = 1..n)}.")) (|integralBasis| (((|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|)))) "\\spad{integralBasis()} returns a record \\spad{[basis,{}basisDen,{}basisInv]} containing information regarding the integral closure of \\spad{R} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,{}w2,{}...,{}wn}. If \\spad{basis} is the matrix \\spad{(aij,{} i = 1..n,{} j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{\\spad{vi} = (1/basisDen) * sum(aij * wj,{} j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{\\spad{wi}} with respect to the basis \\spad{v1,{}...,{}vn}: if \\spad{basisInv} is the matrix \\spad{(bij,{} i = 1..n,{} j = 1..n)},{} then \\spad{\\spad{wi} = sum(bij * vj,{} j = 1..n)}.")) (|squareFree| (((|Factored| $) $) "\\spad{squareFree(x)} returns a square-free factorisation of \\spad{x}"))) 
NIL 
NIL 
(-346 |p| |extdeg|) 
((|constructor| (NIL "FiniteFieldNormalBasis(\\spad{p},{}\\spad{n}) implements a finite extension field of degree \\spad{n} over the prime field with \\spad{p} elements. The elements are represented by coordinate vectors with respect to a normal basis,{} \\spadignore{i.e.} a basis consisting of the conjugates (\\spad{q}-powers) of an element,{} in this case called normal element. This is chosen as a root of the extension polynomial created by createNormalPoly")) (|sizeMultiplication| (((|NonNegativeInteger|)) "\\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: The time of multiplication of field elements depends on this size.")) (|getMultiplicationMatrix| (((|Matrix| (|PrimeField| |#1|))) "\\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.")) (|getMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| (|PrimeField| |#1|)) (|:| |index| (|SingleInteger|)))))) "\\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| (-897 |#1|) (QUOTE (-148))) (|HasCategory| (-897 |#1|) (QUOTE (-364))) (|HasCategory| (-897 |#1|) (QUOTE (-146))) (-3322 (|HasCategory| (-897 |#1|) (QUOTE (-146))) (|HasCategory| (-897 |#1|) (QUOTE (-364))))) 
(-347 GF |uni|) 
((|constructor| (NIL "FiniteFieldNormalBasisExtensionByPolynomial(\\spad{GF},{}uni) implements a finite extension of the ground field \\spad{GF}. The elements are represented by coordinate vectors with respect to. a normal basis,{} \\spadignore{i.e.} a basis consisting of the conjugates (\\spad{q}-powers) of an element,{} in this case called normal element,{} where \\spad{q} is the size of \\spad{GF}. The normal element is chosen as a root of the extension polynomial,{} which MUST be normal over \\spad{GF} (user responsibility)")) (|sizeMultiplication| (((|NonNegativeInteger|)) "\\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: the time of multiplication of field elements depends on this size.")) (|getMultiplicationMatrix| (((|Matrix| |#1|)) "\\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.")) (|getMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-364))) (|HasCategory| |#1| (QUOTE (-146))) (-3322 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-364))))) 
(-348 GF |extdeg|) 
((|constructor| (NIL "FiniteFieldNormalBasisExtensionByPolynomial(\\spad{GF},{}\\spad{n}) implements a finite extension field of degree \\spad{n} over the ground field \\spad{GF}. The elements are represented by coordinate vectors with respect to a normal basis,{} \\spadignore{i.e.} a basis consisting of the conjugates (\\spad{q}-powers) of an element,{} in this case called normal element. This is chosen as a root of the extension polynomial,{} created by createNormalPoly from \\spadtype{FiniteFieldPolynomialPackage}")) (|sizeMultiplication| (((|NonNegativeInteger|)) "\\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: the time of multiplication of field elements depends on this size.")) (|getMultiplicationMatrix| (((|Matrix| |#1|)) "\\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.")) (|getMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-364))) (|HasCategory| |#1| (QUOTE (-146))) (-3322 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-364))))) 
(-349 |p| |n|) 
((|constructor| (NIL "FiniteField(\\spad{p},{}\\spad{n}) implements finite fields with p**n elements. This packages checks that \\spad{p} is prime. For a non-checking version,{} see \\spadtype{InnerFiniteField}."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| (-897 |#1|) (QUOTE (-148))) (|HasCategory| (-897 |#1|) (QUOTE (-364))) (|HasCategory| (-897 |#1|) (QUOTE (-146))) (-3322 (|HasCategory| (-897 |#1|) (QUOTE (-146))) (|HasCategory| (-897 |#1|) (QUOTE (-364))))) 
(-350 GF |defpol|) 
((|constructor| (NIL "FiniteFieldExtensionByPolynomial(\\spad{GF},{} defpol) implements the extension of the finite field \\spad{GF} generated by the extension polynomial defpol which MUST be irreducible."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-364))) (|HasCategory| |#1| (QUOTE (-146))) (-3322 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-364))))) 
(-351 -2262 GF) 
((|constructor| (NIL "\\spad{FiniteFieldPolynomialPackage2}(\\spad{F},{}\\spad{GF}) exports some functions concerning finite fields,{} which depend on a finite field \\spad{GF} and an algebraic extension \\spad{F} of \\spad{GF},{} \\spadignore{e.g.} a zero of a polynomial over \\spad{GF} in \\spad{F}.")) (|rootOfIrreduciblePoly| ((|#1| (|SparseUnivariatePolynomial| |#2|)) "\\spad{rootOfIrreduciblePoly(f)} computes one root of the monic,{} irreducible polynomial \\spad{f},{} which degree must divide the extension degree of \\spad{F} over \\spad{GF},{} \\spadignore{i.e.} \\spad{f} splits into linear factors over \\spad{F}.")) (|Frobenius| ((|#1| |#1|) "\\spad{Frobenius(x)} \\undocumented{}")) (|basis| (((|Vector| |#1|) (|PositiveInteger|)) "\\spad{basis(n)} \\undocumented{}")) (|lookup| (((|PositiveInteger|) |#1|) "\\spad{lookup(x)} \\undocumented{}")) (|coerce| ((|#1| |#2|) "\\spad{coerce(x)} \\undocumented{}"))) 
NIL 
NIL 
(-352 GF) 
((|constructor| (NIL "This package provides a number of functions for generating,{} counting and testing irreducible,{} normal,{} primitive,{} random polynomials over finite fields.")) (|reducedQPowers| (((|PrimitiveArray| (|SparseUnivariatePolynomial| |#1|)) (|SparseUnivariatePolynomial| |#1|)) "\\spad{reducedQPowers(f)} generates \\spad{[x,{}x**q,{}x**(q**2),{}...,{}x**(q**(n-1))]} reduced modulo \\spad{f} where \\spad{q = size()\\$GF} and \\spad{n = degree f}.")) (|leastAffineMultiple| (((|SparseUnivariatePolynomial| |#1|) (|SparseUnivariatePolynomial| |#1|)) "\\spad{leastAffineMultiple(f)} computes the least affine polynomial which is divisible by the polynomial \\spad{f} over the finite field \\spad{GF},{} \\spadignore{i.e.} a polynomial whose exponents are 0 or a power of \\spad{q},{} the size of \\spad{GF}.")) (|random| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|) (|PositiveInteger|)) "\\spad{random(m,{}n)}\\$FFPOLY(\\spad{GF}) generates a random monic polynomial of degree \\spad{d} over the finite field \\spad{GF},{} \\spad{d} between \\spad{m} and \\spad{n}.") (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{random(n)}\\$FFPOLY(\\spad{GF}) generates a random monic polynomial of degree \\spad{n} over the finite field \\spad{GF}.")) (|nextPrimitiveNormalPoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextPrimitiveNormalPoly(f)} yields the next primitive normal polynomial over a finite field \\spad{GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note that the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the lookup of the constant term of \\spad{f} is less than this number for \\spad{g} or,{} in case these numbers are equal,{} if the lookup of the coefficient of the term of degree \\spad{n}-1 of \\spad{f} is less than this number for \\spad{g}. If these numbers are equals,{} \\spad{f < g} if the number of monomials of \\spad{f} is less than that for \\spad{g},{} or if the lists of exponents for \\spad{f} are lexicographically less than those for \\spad{g}. If these lists are also equal,{} the lists of coefficients are coefficients according to the lexicographic ordering induced by the ordering of the elements of \\spad{GF} given by lookup. This operation is equivalent to nextNormalPrimitivePoly(\\spad{f}).")) (|nextNormalPrimitivePoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextNormalPrimitivePoly(f)} yields the next normal primitive polynomial over a finite field \\spad{GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note that the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the lookup of the constant term of \\spad{f} is less than this number for \\spad{g} or if lookup of the coefficient of the term of degree \\spad{n}-1 of \\spad{f} is less than this number for \\spad{g}. Otherwise,{} \\spad{f < g} if the number of monomials of \\spad{f} is less than that for \\spad{g} or if the lists of exponents for \\spad{f} are lexicographically less than those for \\spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of \\spad{GF} given by lookup. This operation is equivalent to nextPrimitiveNormalPoly(\\spad{f}).")) (|nextNormalPoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextNormalPoly(f)} yields the next normal polynomial over a finite field \\spad{GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note that the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the lookup of the coefficient of the term of degree \\spad{n}-1 of \\spad{f} is less than that for \\spad{g}. In case these numbers are equal,{} \\spad{f < g} if if the number of monomials of \\spad{f} is less that for \\spad{g} or if the list of exponents of \\spad{f} are lexicographically less than the corresponding list for \\spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of \\spad{GF} given by lookup.")) (|nextPrimitivePoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextPrimitivePoly(f)} yields the next primitive polynomial over a finite field \\spad{GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note that the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the lookup of the constant term of \\spad{f} is less than this number for \\spad{g}. If these values are equal,{} then \\spad{f < g} if if the number of monomials of \\spad{f} is less than that for \\spad{g} or if the lists of exponents of \\spad{f} are lexicographically less than the corresponding list for \\spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of \\spad{GF} given by lookup.")) (|nextIrreduciblePoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextIrreduciblePoly(f)} yields the next monic irreducible polynomial over a finite field \\spad{GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note that the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the number of monomials of \\spad{f} is less than this number for \\spad{g}. If \\spad{f} and \\spad{g} have the same number of monomials,{} the lists of exponents are compared lexicographically. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of \\spad{GF} given by lookup.")) (|createPrimitiveNormalPoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createPrimitiveNormalPoly(n)}\\$FFPOLY(\\spad{GF}) generates a normal and primitive polynomial of degree \\spad{n} over the field \\spad{GF}. polynomial of degree \\spad{n} over the field \\spad{GF}.")) (|createNormalPrimitivePoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createNormalPrimitivePoly(n)}\\$FFPOLY(\\spad{GF}) generates a normal and primitive polynomial of degree \\spad{n} over the field \\spad{GF}. Note that this function is equivalent to createPrimitiveNormalPoly(\\spad{n})")) (|createNormalPoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createNormalPoly(n)}\\$FFPOLY(\\spad{GF}) generates a normal polynomial of degree \\spad{n} over the finite field \\spad{GF}.")) (|createPrimitivePoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createPrimitivePoly(n)}\\$FFPOLY(\\spad{GF}) generates a primitive polynomial of degree \\spad{n} over the finite field \\spad{GF}.")) (|createIrreduciblePoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createIrreduciblePoly(n)}\\$FFPOLY(\\spad{GF}) generates a monic irreducible univariate polynomial of degree \\spad{n} over the finite field \\spad{GF}.")) (|numberOfNormalPoly| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{numberOfNormalPoly(n)}\\$FFPOLY(\\spad{GF}) yields the number of normal polynomials of degree \\spad{n} over the finite field \\spad{GF}.")) (|numberOfPrimitivePoly| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{numberOfPrimitivePoly(n)}\\$FFPOLY(\\spad{GF}) yields the number of primitive polynomials of degree \\spad{n} over the finite field \\spad{GF}.")) (|numberOfIrreduciblePoly| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{numberOfIrreduciblePoly(n)}\\$FFPOLY(\\spad{GF}) yields the number of monic irreducible univariate polynomials of degree \\spad{n} over the finite field \\spad{GF}.")) (|normal?| (((|Boolean|) (|SparseUnivariatePolynomial| |#1|)) "\\spad{normal?(f)} tests whether the polynomial \\spad{f} over a finite field is normal,{} \\spadignore{i.e.} its roots are linearly independent over the field.")) (|primitive?| (((|Boolean|) (|SparseUnivariatePolynomial| |#1|)) "\\spad{primitive?(f)} tests whether the polynomial \\spad{f} over a finite field is primitive,{} \\spadignore{i.e.} all its roots are primitive."))) 
NIL 
NIL 
(-353 -2262 FP FPP) 
((|constructor| (NIL "This package solves linear diophantine equations for Bivariate polynomials over finite fields")) (|solveLinearPolynomialEquation| (((|Union| (|List| |#3|) "failed") (|List| |#3|) |#3|) "\\spad{solveLinearPolynomialEquation([f1,{} ...,{} fn],{} g)} (where the \\spad{fi} are relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g/prod \\spad{fi} = sum ai/fi} or returns \"failed\" if no such list of \\spad{ai}\\spad{'s} exists."))) 
NIL 
NIL 
(-354 K |PolK|) 
((|constructor| (NIL "Part of the package for Algebraic Function Fields in one variable (\\spad{PAFF})"))) 
NIL 
NIL 
(-355 GF |n|) 
((|constructor| (NIL "FiniteFieldExtensionByPolynomial(\\spad{GF},{} \\spad{n}) implements an extension of the finite field \\spad{GF} of degree \\spad{n} generated by the extension polynomial constructed by createIrreduciblePoly from \\spadtype{FiniteFieldPolynomialPackage}."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-364))) (|HasCategory| |#1| (QUOTE (-146))) (-3322 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-364))))) 
(-356 R |ls|) 
((|constructor| (NIL "This is just an interface between several packages and domains. The goal is to compute lexicographical Groebner bases of sets of polynomial with type \\spadtype{Polynomial R} by the FGLM algorithm if this is possible (\\spadignore{i.e.} if the input system generates a zero-dimensional ideal).")) (|groebner| (((|List| (|Polynomial| |#1|)) (|List| (|Polynomial| |#1|))) "\\axiom{groebner(\\spad{lq1})} returns the lexicographical Groebner basis of \\axiom{\\spad{lq1}}. If \\axiom{\\spad{lq1}} generates a zero-dimensional ideal then the FGLM strategy is used,{} otherwise the Sugar strategy is used.")) (|fglmIfCan| (((|Union| (|List| (|Polynomial| |#1|)) "failed") (|List| (|Polynomial| |#1|))) "\\axiom{fglmIfCan(\\spad{lq1})} returns the lexicographical Groebner basis of \\axiom{\\spad{lq1}} by using the FGLM strategy,{} if \\axiom{zeroDimensional?(\\spad{lq1})} holds.")) (|zeroDimensional?| (((|Boolean|) (|List| (|Polynomial| |#1|))) "\\axiom{zeroDimensional?(\\spad{lq1})} returns \\spad{true} iff \\axiom{\\spad{lq1}} generates a zero-dimensional ideal \\spad{w}.\\spad{r}.\\spad{t}. the variables of \\axiom{\\spad{ls}}."))) 
NIL 
NIL 
(-357 S) 
((|constructor| (NIL "The free group on a set \\spad{S} is the group of finite products of the form \\spad{reduce(*,{}[\\spad{si} ** \\spad{ni}])} where the \\spad{si}\\spad{'s} are in \\spad{S},{} and the \\spad{ni}\\spad{'s} are integers. The multiplication is not commutative.")) (|factors| (((|List| (|Record| (|:| |gen| |#1|) (|:| |exp| (|Integer|)))) $) "\\spad{factors(a1\\^e1,{}...,{}an\\^en)} returns \\spad{[[a1,{} e1],{}...,{}[an,{} en]]}.")) (|mapGen| (($ (|Mapping| |#1| |#1|) $) "\\spad{mapGen(f,{} a1\\^e1 ... an\\^en)} returns \\spad{f(a1)\\^e1 ... f(an)\\^en}.")) (|mapExpon| (($ (|Mapping| (|Integer|) (|Integer|)) $) "\\spad{mapExpon(f,{} a1\\^e1 ... an\\^en)} returns \\spad{a1\\^f(e1) ... an\\^f(en)}.")) (|nthFactor| ((|#1| $ (|Integer|)) "\\spad{nthFactor(x,{} n)} returns the factor of the n^th monomial of \\spad{x}.")) (|nthExpon| (((|Integer|) $ (|Integer|)) "\\spad{nthExpon(x,{} n)} returns the exponent of the n^th monomial of \\spad{x}.")) (|size| (((|NonNegativeInteger|) $) "\\spad{size(x)} returns the number of monomials in \\spad{x}.")) (** (($ |#1| (|Integer|)) "\\spad{s ** n} returns the product of \\spad{s} by itself \\spad{n} times.")) (* (($ $ |#1|) "\\spad{x * s} returns the product of \\spad{x} by \\spad{s} on the right.") (($ |#1| $) "\\spad{s * x} returns the product of \\spad{x} by \\spad{s} on the left."))) 
((-4502 . T)) 
NIL 
(-358 S) 
((|constructor| (NIL "The category of commutative fields,{} \\spadignore{i.e.} commutative rings where all non-zero elements have multiplicative inverses. The \\spadfun{factor} operation while trivial is useful to have defined. \\blankline Axioms\\spad{\\br} \\tab{5}\\spad{a*(b/a) = b}\\spad{\\br} \\tab{5}\\spad{inv(a) = 1/a}")) (|canonicalsClosed| ((|attribute|) "since \\spad{0*0=0},{} \\spad{1*1=1}")) (|canonicalUnitNormal| ((|attribute|) "either 0 or 1.")) (/ (($ $ $) "\\spad{x/y} divides the element \\spad{x} by the element \\spad{y}. Error: if \\spad{y} is 0."))) 
NIL 
NIL 
(-359) 
((|constructor| (NIL "The category of commutative fields,{} \\spadignore{i.e.} commutative rings where all non-zero elements have multiplicative inverses. The \\spadfun{factor} operation while trivial is useful to have defined. \\blankline Axioms\\spad{\\br} \\tab{5}\\spad{a*(b/a) = b}\\spad{\\br} \\tab{5}\\spad{inv(a) = 1/a}")) (|canonicalsClosed| ((|attribute|) "since \\spad{0*0=0},{} \\spad{1*1=1}")) (|canonicalUnitNormal| ((|attribute|) "either 0 or 1.")) (/ (($ $ $) "\\spad{x/y} divides the element \\spad{x} by the element \\spad{y}. Error: if \\spad{y} is 0."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-360 |Name| S) 
((|constructor| (NIL "This category provides an interface to operate on files in the computer\\spad{'s} file system. The precise method of naming files is determined by the Name parameter. The type of the contents of the file is determined by \\spad{S}.")) (|flush| (((|Void|) $) "\\spad{flush(f)} makes sure that buffered data is written out")) (|write!| ((|#2| $ |#2|) "\\spad{write!(f,{}s)} puts the value \\spad{s} into the file \\spad{f}. The state of \\spad{f} is modified so subsequents call to \\spad{write!} will append one after another.")) (|read!| ((|#2| $) "\\spad{read!(f)} extracts a value from file \\spad{f}. The state of \\spad{f} is modified so a subsequent call to \\spadfun{read!} will return the next element.")) (|iomode| (((|String|) $) "\\spad{iomode(f)} returns the status of the file \\spad{f}. The input/output status of \\spad{f} may be \"input\",{} \"output\" or \"closed\" mode.")) (|name| ((|#1| $) "\\spad{name(f)} returns the external name of the file \\spad{f}.")) (|close!| (($ $) "\\spad{close!(f)} returns the file \\spad{f} closed to input and output.")) (|reopen!| (($ $ (|String|)) "\\spad{reopen!(f,{}mode)} returns a file \\spad{f} reopened for operation in the indicated mode: \"input\" or \"output\". \\spad{reopen!(f,{}\"input\")} will reopen the file \\spad{f} for input.")) (|open| (($ |#1| (|String|)) "\\spad{open(s,{}mode)} returns a file \\spad{s} open for operation in the indicated mode: \"input\" or \"output\".") (($ |#1|) "\\spad{open(s)} returns the file \\spad{s} open for input."))) 
NIL 
NIL 
(-361 S) 
((|constructor| (NIL "This domain provides a basic model of files to save arbitrary values. The operations provide sequential access to the contents.")) (|readIfCan!| (((|Union| |#1| "failed") $) "\\spad{readIfCan!(f)} returns a value from the file \\spad{f},{} if possible. If \\spad{f} is not open for reading,{} or if \\spad{f} is at the end of file then \\spad{\"failed\"} is the result."))) 
NIL 
NIL 
(-362 S R) 
((|constructor| (NIL "A FiniteRankNonAssociativeAlgebra is a non associative algebra over a commutative ring \\spad{R} which is a free \\spad{R}-module of finite rank.")) (|unitsKnown| ((|attribute|) "unitsKnown means that \\spadfun{recip} truly yields reciprocal or \\spad{\"failed\"} if not a unit,{} similarly for \\spadfun{leftRecip} and \\spadfun{rightRecip}. The reason is that we use left,{} respectively right,{} minimal polynomials to decide this question.")) (|unit| (((|Union| $ "failed")) "\\spad{unit()} returns a unit of the algebra (necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|rightUnit| (((|Union| $ "failed")) "\\spad{rightUnit()} returns a right unit of the algebra (not necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|leftUnit| (((|Union| $ "failed")) "\\spad{leftUnit()} returns a left unit of the algebra (not necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|rightUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{rightUnits()} returns the affine space of all right units of the algebra,{} or \\spad{\"failed\"} if there is none.")) (|leftUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{leftUnits()} returns the affine space of all left units of the algebra,{} or \\spad{\"failed\"} if there is none.")) (|rightMinimalPolynomial| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{rightMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of right powers of \\spad{a}. Note that the polynomial never has a constant term as in general the algebra has no unit.")) (|leftMinimalPolynomial| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{leftMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of left powers of \\spad{a}. Note that the polynomial never has a constant term as in general the algebra has no unit.")) (|associatorDependence| (((|List| (|Vector| |#2|))) "\\spad{associatorDependence()} looks for the associator identities,{} \\spadignore{i.e.} finds a basis of the solutions of the linear combinations of the six permutations of \\spad{associator(a,{}b,{}c)} which yield 0,{} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra. The order of the permutations is \\spad{123 231 312 132 321 213}.")) (|rightRecip| (((|Union| $ "failed") $) "\\spad{rightRecip(a)} returns an element,{} which is a right inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|leftRecip| (((|Union| $ "failed") $) "\\spad{leftRecip(a)} returns an element,{} which is a left inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(a)} returns an element,{} which is both a left and a right inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|lieAlgebra?| (((|Boolean|)) "\\spad{lieAlgebra?()} tests if the algebra is anticommutative and \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra (Jacobi identity). Example: for every associative algebra \\spad{(A,{}+,{}@)} we can construct a Lie algebra \\spad{(A,{}+,{}*)},{} where \\spad{a*b := a@b-b@a}.")) (|jordanAlgebra?| (((|Boolean|)) "\\spad{jordanAlgebra?()} tests if the algebra is commutative,{} characteristic is not 2,{} and \\spad{(a*b)*a**2 - a*(b*a**2) = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra (Jordan identity). Example: for every associative algebra \\spad{(A,{}+,{}@)} we can construct a Jordan algebra \\spad{(A,{}+,{}*)},{} where \\spad{a*b := (a@b+b@a)/2}.")) (|noncommutativeJordanAlgebra?| (((|Boolean|)) "\\spad{noncommutativeJordanAlgebra?()} tests if the algebra is flexible and Jordan admissible.")) (|jordanAdmissible?| (((|Boolean|)) "\\spad{jordanAdmissible?()} tests if 2 is invertible in the coefficient domain and the multiplication defined by \\spad{(1/2)(a*b+b*a)} determines a Jordan algebra,{} \\spadignore{i.e.} satisfies the Jordan identity. The property of \\spadatt{commutative(\\spad{\"*\"})} follows from by definition.")) (|lieAdmissible?| (((|Boolean|)) "\\spad{lieAdmissible?()} tests if the algebra defined by the commutators is a Lie algebra,{} \\spadignore{i.e.} satisfies the Jacobi identity. The property of anticommutativity follows from definition.")) (|jacobiIdentity?| (((|Boolean|)) "\\spad{jacobiIdentity?()} tests if \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra. For example,{} this holds for crossed products of 3-dimensional vectors.")) (|powerAssociative?| (((|Boolean|)) "\\spad{powerAssociative?()} tests if all subalgebras generated by a single element are associative.")) (|alternative?| (((|Boolean|)) "\\spad{alternative?()} tests if \\spad{2*associator(a,{}a,{}b) = 0 = 2*associator(a,{}b,{}b)} for all \\spad{a},{} \\spad{b} in the algebra. Note that we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|flexible?| (((|Boolean|)) "\\spad{flexible?()} tests if \\spad{2*associator(a,{}b,{}a) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note that we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|rightAlternative?| (((|Boolean|)) "\\spad{rightAlternative?()} tests if \\spad{2*associator(a,{}b,{}b) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note that we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|leftAlternative?| (((|Boolean|)) "\\spad{leftAlternative?()} tests if \\spad{2*associator(a,{}a,{}b) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note that we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|antiAssociative?| (((|Boolean|)) "\\spad{antiAssociative?()} tests if multiplication in algebra is anti-associative,{} \\spadignore{i.e.} \\spad{(a*b)*c + a*(b*c) = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra.")) (|associative?| (((|Boolean|)) "\\spad{associative?()} tests if multiplication in algebra is associative.")) (|antiCommutative?| (((|Boolean|)) "\\spad{antiCommutative?()} tests if \\spad{a*a = 0} for all \\spad{a} in the algebra. Note that this implies \\spad{a*b + b*a = 0} for all \\spad{a} and \\spad{b}.")) (|commutative?| (((|Boolean|)) "\\spad{commutative?()} tests if multiplication in the algebra is commutative.")) (|rightCharacteristicPolynomial| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{rightCharacteristicPolynomial(a)} returns the characteristic polynomial of the right regular representation of \\spad{a} with respect to any basis.")) (|leftCharacteristicPolynomial| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{leftCharacteristicPolynomial(a)} returns the characteristic polynomial of the left regular representation of \\spad{a} with respect to any basis.")) (|rightTraceMatrix| (((|Matrix| |#2|) (|Vector| $)) "\\spad{rightTraceMatrix([v1,{}...,{}vn])} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj}.")) (|leftTraceMatrix| (((|Matrix| |#2|) (|Vector| $)) "\\spad{leftTraceMatrix([v1,{}...,{}vn])} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj}.")) (|rightDiscriminant| ((|#2| (|Vector| $)) "\\spad{rightDiscriminant([v1,{}...,{}vn])} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj}. Note that this is the same as \\spad{determinant(rightTraceMatrix([v1,{}...,{}vn]))}.")) (|leftDiscriminant| ((|#2| (|Vector| $)) "\\spad{leftDiscriminant([v1,{}...,{}vn])} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj}. Note that this is the same as \\spad{determinant(leftTraceMatrix([v1,{}...,{}vn]))}.")) (|represents| (($ (|Vector| |#2|) (|Vector| $)) "\\spad{represents([a1,{}...,{}am],{}[v1,{}...,{}vm])} returns the linear combination \\spad{a1*vm + ... + an*vm}.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $) (|Vector| $)) "\\spad{coordinates([a1,{}...,{}am],{}[v1,{}...,{}vn])} returns a matrix whose \\spad{i}-th row is formed by the coordinates of \\spad{\\spad{ai}} with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.") (((|Vector| |#2|) $ (|Vector| $)) "\\spad{coordinates(a,{}[v1,{}...,{}vn])} returns the coordinates of \\spad{a} with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.")) (|rightNorm| ((|#2| $) "\\spad{rightNorm(a)} returns the determinant of the right regular representation of \\spad{a}.")) (|leftNorm| ((|#2| $) "\\spad{leftNorm(a)} returns the determinant of the left regular representation of \\spad{a}.")) (|rightTrace| ((|#2| $) "\\spad{rightTrace(a)} returns the trace of the right regular representation of \\spad{a}.")) (|leftTrace| ((|#2| $) "\\spad{leftTrace(a)} returns the trace of the left regular representation of \\spad{a}.")) (|rightRegularRepresentation| (((|Matrix| |#2|) $ (|Vector| $)) "\\spad{rightRegularRepresentation(a,{}[v1,{}...,{}vn])} returns the matrix of the linear map defined by right multiplication by \\spad{a} with respect to the \\spad{R}-module basis \\spad{[v1,{}...,{}vn]}.")) (|leftRegularRepresentation| (((|Matrix| |#2|) $ (|Vector| $)) "\\spad{leftRegularRepresentation(a,{}[v1,{}...,{}vn])} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the \\spad{R}-module basis \\spad{[v1,{}...,{}vn]}.")) (|structuralConstants| (((|Vector| (|Matrix| |#2|)) (|Vector| $)) "\\spad{structuralConstants([v1,{}v2,{}...,{}vm])} calculates the structural constants \\spad{[(gammaijk) for k in 1..m]} defined by \\spad{\\spad{vi} * vj = gammaij1 * v1 + ... + gammaijm * vm},{} where \\spad{[v1,{}...,{}vm]} is an \\spad{R}-module basis of a subalgebra.")) (|conditionsForIdempotents| (((|List| (|Polynomial| |#2|)) (|Vector| $)) "\\spad{conditionsForIdempotents([v1,{}...,{}vn])} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.")) (|rank| (((|PositiveInteger|)) "\\spad{rank()} returns the rank of the algebra as \\spad{R}-module.")) (|someBasis| (((|Vector| $)) "\\spad{someBasis()} returns some \\spad{R}-module basis."))) 
NIL 
((|HasCategory| |#2| (QUOTE (-550)))) 
(-363 R) 
((|constructor| (NIL "A FiniteRankNonAssociativeAlgebra is a non associative algebra over a commutative ring \\spad{R} which is a free \\spad{R}-module of finite rank.")) (|unitsKnown| ((|attribute|) "unitsKnown means that \\spadfun{recip} truly yields reciprocal or \\spad{\"failed\"} if not a unit,{} similarly for \\spadfun{leftRecip} and \\spadfun{rightRecip}. The reason is that we use left,{} respectively right,{} minimal polynomials to decide this question.")) (|unit| (((|Union| $ "failed")) "\\spad{unit()} returns a unit of the algebra (necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|rightUnit| (((|Union| $ "failed")) "\\spad{rightUnit()} returns a right unit of the algebra (not necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|leftUnit| (((|Union| $ "failed")) "\\spad{leftUnit()} returns a left unit of the algebra (not necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|rightUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{rightUnits()} returns the affine space of all right units of the algebra,{} or \\spad{\"failed\"} if there is none.")) (|leftUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{leftUnits()} returns the affine space of all left units of the algebra,{} or \\spad{\"failed\"} if there is none.")) (|rightMinimalPolynomial| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{rightMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of right powers of \\spad{a}. Note that the polynomial never has a constant term as in general the algebra has no unit.")) (|leftMinimalPolynomial| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{leftMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of left powers of \\spad{a}. Note that the polynomial never has a constant term as in general the algebra has no unit.")) (|associatorDependence| (((|List| (|Vector| |#1|))) "\\spad{associatorDependence()} looks for the associator identities,{} \\spadignore{i.e.} finds a basis of the solutions of the linear combinations of the six permutations of \\spad{associator(a,{}b,{}c)} which yield 0,{} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra. The order of the permutations is \\spad{123 231 312 132 321 213}.")) (|rightRecip| (((|Union| $ "failed") $) "\\spad{rightRecip(a)} returns an element,{} which is a right inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|leftRecip| (((|Union| $ "failed") $) "\\spad{leftRecip(a)} returns an element,{} which is a left inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(a)} returns an element,{} which is both a left and a right inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|lieAlgebra?| (((|Boolean|)) "\\spad{lieAlgebra?()} tests if the algebra is anticommutative and \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra (Jacobi identity). Example: for every associative algebra \\spad{(A,{}+,{}@)} we can construct a Lie algebra \\spad{(A,{}+,{}*)},{} where \\spad{a*b := a@b-b@a}.")) (|jordanAlgebra?| (((|Boolean|)) "\\spad{jordanAlgebra?()} tests if the algebra is commutative,{} characteristic is not 2,{} and \\spad{(a*b)*a**2 - a*(b*a**2) = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra (Jordan identity). Example: for every associative algebra \\spad{(A,{}+,{}@)} we can construct a Jordan algebra \\spad{(A,{}+,{}*)},{} where \\spad{a*b := (a@b+b@a)/2}.")) (|noncommutativeJordanAlgebra?| (((|Boolean|)) "\\spad{noncommutativeJordanAlgebra?()} tests if the algebra is flexible and Jordan admissible.")) (|jordanAdmissible?| (((|Boolean|)) "\\spad{jordanAdmissible?()} tests if 2 is invertible in the coefficient domain and the multiplication defined by \\spad{(1/2)(a*b+b*a)} determines a Jordan algebra,{} \\spadignore{i.e.} satisfies the Jordan identity. The property of \\spadatt{commutative(\\spad{\"*\"})} follows from by definition.")) (|lieAdmissible?| (((|Boolean|)) "\\spad{lieAdmissible?()} tests if the algebra defined by the commutators is a Lie algebra,{} \\spadignore{i.e.} satisfies the Jacobi identity. The property of anticommutativity follows from definition.")) (|jacobiIdentity?| (((|Boolean|)) "\\spad{jacobiIdentity?()} tests if \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra. For example,{} this holds for crossed products of 3-dimensional vectors.")) (|powerAssociative?| (((|Boolean|)) "\\spad{powerAssociative?()} tests if all subalgebras generated by a single element are associative.")) (|alternative?| (((|Boolean|)) "\\spad{alternative?()} tests if \\spad{2*associator(a,{}a,{}b) = 0 = 2*associator(a,{}b,{}b)} for all \\spad{a},{} \\spad{b} in the algebra. Note that we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|flexible?| (((|Boolean|)) "\\spad{flexible?()} tests if \\spad{2*associator(a,{}b,{}a) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note that we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|rightAlternative?| (((|Boolean|)) "\\spad{rightAlternative?()} tests if \\spad{2*associator(a,{}b,{}b) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note that we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|leftAlternative?| (((|Boolean|)) "\\spad{leftAlternative?()} tests if \\spad{2*associator(a,{}a,{}b) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note that we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|antiAssociative?| (((|Boolean|)) "\\spad{antiAssociative?()} tests if multiplication in algebra is anti-associative,{} \\spadignore{i.e.} \\spad{(a*b)*c + a*(b*c) = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra.")) (|associative?| (((|Boolean|)) "\\spad{associative?()} tests if multiplication in algebra is associative.")) (|antiCommutative?| (((|Boolean|)) "\\spad{antiCommutative?()} tests if \\spad{a*a = 0} for all \\spad{a} in the algebra. Note that this implies \\spad{a*b + b*a = 0} for all \\spad{a} and \\spad{b}.")) (|commutative?| (((|Boolean|)) "\\spad{commutative?()} tests if multiplication in the algebra is commutative.")) (|rightCharacteristicPolynomial| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{rightCharacteristicPolynomial(a)} returns the characteristic polynomial of the right regular representation of \\spad{a} with respect to any basis.")) (|leftCharacteristicPolynomial| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{leftCharacteristicPolynomial(a)} returns the characteristic polynomial of the left regular representation of \\spad{a} with respect to any basis.")) (|rightTraceMatrix| (((|Matrix| |#1|) (|Vector| $)) "\\spad{rightTraceMatrix([v1,{}...,{}vn])} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj}.")) (|leftTraceMatrix| (((|Matrix| |#1|) (|Vector| $)) "\\spad{leftTraceMatrix([v1,{}...,{}vn])} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj}.")) (|rightDiscriminant| ((|#1| (|Vector| $)) "\\spad{rightDiscriminant([v1,{}...,{}vn])} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj}. Note that this is the same as \\spad{determinant(rightTraceMatrix([v1,{}...,{}vn]))}.")) (|leftDiscriminant| ((|#1| (|Vector| $)) "\\spad{leftDiscriminant([v1,{}...,{}vn])} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj}. Note that this is the same as \\spad{determinant(leftTraceMatrix([v1,{}...,{}vn]))}.")) (|represents| (($ (|Vector| |#1|) (|Vector| $)) "\\spad{represents([a1,{}...,{}am],{}[v1,{}...,{}vm])} returns the linear combination \\spad{a1*vm + ... + an*vm}.")) (|coordinates| (((|Matrix| |#1|) (|Vector| $) (|Vector| $)) "\\spad{coordinates([a1,{}...,{}am],{}[v1,{}...,{}vn])} returns a matrix whose \\spad{i}-th row is formed by the coordinates of \\spad{\\spad{ai}} with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.") (((|Vector| |#1|) $ (|Vector| $)) "\\spad{coordinates(a,{}[v1,{}...,{}vn])} returns the coordinates of \\spad{a} with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.")) (|rightNorm| ((|#1| $) "\\spad{rightNorm(a)} returns the determinant of the right regular representation of \\spad{a}.")) (|leftNorm| ((|#1| $) "\\spad{leftNorm(a)} returns the determinant of the left regular representation of \\spad{a}.")) (|rightTrace| ((|#1| $) "\\spad{rightTrace(a)} returns the trace of the right regular representation of \\spad{a}.")) (|leftTrace| ((|#1| $) "\\spad{leftTrace(a)} returns the trace of the left regular representation of \\spad{a}.")) (|rightRegularRepresentation| (((|Matrix| |#1|) $ (|Vector| $)) "\\spad{rightRegularRepresentation(a,{}[v1,{}...,{}vn])} returns the matrix of the linear map defined by right multiplication by \\spad{a} with respect to the \\spad{R}-module basis \\spad{[v1,{}...,{}vn]}.")) (|leftRegularRepresentation| (((|Matrix| |#1|) $ (|Vector| $)) "\\spad{leftRegularRepresentation(a,{}[v1,{}...,{}vn])} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the \\spad{R}-module basis \\spad{[v1,{}...,{}vn]}.")) (|structuralConstants| (((|Vector| (|Matrix| |#1|)) (|Vector| $)) "\\spad{structuralConstants([v1,{}v2,{}...,{}vm])} calculates the structural constants \\spad{[(gammaijk) for k in 1..m]} defined by \\spad{\\spad{vi} * vj = gammaij1 * v1 + ... + gammaijm * vm},{} where \\spad{[v1,{}...,{}vm]} is an \\spad{R}-module basis of a subalgebra.")) (|conditionsForIdempotents| (((|List| (|Polynomial| |#1|)) (|Vector| $)) "\\spad{conditionsForIdempotents([v1,{}...,{}vn])} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.")) (|rank| (((|PositiveInteger|)) "\\spad{rank()} returns the rank of the algebra as \\spad{R}-module.")) (|someBasis| (((|Vector| $)) "\\spad{someBasis()} returns some \\spad{R}-module basis."))) 
((-4502 |has| |#1| (-550)) (-4500 . T) (-4499 . T)) 
NIL 
(-364) 
((|constructor| (NIL "The category of domains composed of a finite set of elements. We include the functions \\spadfun{lookup} and \\spadfun{index} to give a bijection between the finite set and an initial segment of positive integers. \\blankline")) (|random| (($) "\\spad{random()} returns a random element from the set.")) (|lookup| (((|PositiveInteger|) $) "\\spad{lookup(x)} returns a positive integer such that \\spad{x = index lookup x}.")) (|index| (($ (|PositiveInteger|)) "\\spad{index(i)} takes a positive integer \\spad{i} less than or equal to \\spad{size()} and returns the \\spad{i}\\spad{-}th element of the set. This operation establishs a bijection between the elements of the finite set and \\spad{1..size()}.")) (|size| (((|NonNegativeInteger|)) "\\spad{size()} returns the number of elements in the set."))) 
NIL 
NIL 
(-365 S R UP) 
((|constructor| (NIL "A FiniteRankAlgebra is an algebra over a commutative ring \\spad{R} which is a free \\spad{R}-module of finite rank.")) (|minimalPolynomial| ((|#3| $) "\\spad{minimalPolynomial(a)} returns the minimal polynomial of \\spad{a}.")) (|characteristicPolynomial| ((|#3| $) "\\spad{characteristicPolynomial(a)} returns the characteristic polynomial of the regular representation of \\spad{a} with respect to any basis.")) (|traceMatrix| (((|Matrix| |#2|) (|Vector| $)) "\\spad{traceMatrix([v1,{}..,{}vn])} is the \\spad{n}-by-\\spad{n} matrix ( \\spad{Tr}(\\spad{vi} * \\spad{vj}) )")) (|discriminant| ((|#2| (|Vector| $)) "\\spad{discriminant([v1,{}..,{}vn])} returns \\spad{determinant(traceMatrix([v1,{}..,{}vn]))}.")) (|represents| (($ (|Vector| |#2|) (|Vector| $)) "\\spad{represents([a1,{}..,{}an],{}[v1,{}..,{}vn])} returns \\spad{a1*v1+...+an*vn}.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $) (|Vector| $)) "\\spad{coordinates([v1,{}...,{}vm],{} basis)} returns the coordinates of the \\spad{vi}\\spad{'s} with to the basis \\spad{basis}. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#2|) $ (|Vector| $)) "\\spad{coordinates(a,{}basis)} returns the coordinates of \\spad{a} with respect to the \\spad{basis} \\spad{basis}.")) (|norm| ((|#2| $) "\\spad{norm(a)} returns the determinant of the regular representation of \\spad{a} with respect to any basis.")) (|trace| ((|#2| $) "\\spad{trace(a)} returns the trace of the regular representation of \\spad{a} with respect to any basis.")) (|regularRepresentation| (((|Matrix| |#2|) $ (|Vector| $)) "\\spad{regularRepresentation(a,{}basis)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the \\spad{basis} \\spad{basis}.")) (|rank| (((|PositiveInteger|)) "\\spad{rank()} returns the rank of the algebra."))) 
NIL 
((|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-359)))) 
(-366 R UP) 
((|constructor| (NIL "A FiniteRankAlgebra is an algebra over a commutative ring \\spad{R} which is a free \\spad{R}-module of finite rank.")) (|minimalPolynomial| ((|#2| $) "\\spad{minimalPolynomial(a)} returns the minimal polynomial of \\spad{a}.")) (|characteristicPolynomial| ((|#2| $) "\\spad{characteristicPolynomial(a)} returns the characteristic polynomial of the regular representation of \\spad{a} with respect to any basis.")) (|traceMatrix| (((|Matrix| |#1|) (|Vector| $)) "\\spad{traceMatrix([v1,{}..,{}vn])} is the \\spad{n}-by-\\spad{n} matrix ( \\spad{Tr}(\\spad{vi} * \\spad{vj}) )")) (|discriminant| ((|#1| (|Vector| $)) "\\spad{discriminant([v1,{}..,{}vn])} returns \\spad{determinant(traceMatrix([v1,{}..,{}vn]))}.")) (|represents| (($ (|Vector| |#1|) (|Vector| $)) "\\spad{represents([a1,{}..,{}an],{}[v1,{}..,{}vn])} returns \\spad{a1*v1+...+an*vn}.")) (|coordinates| (((|Matrix| |#1|) (|Vector| $) (|Vector| $)) "\\spad{coordinates([v1,{}...,{}vm],{} basis)} returns the coordinates of the \\spad{vi}\\spad{'s} with to the basis \\spad{basis}. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#1|) $ (|Vector| $)) "\\spad{coordinates(a,{}basis)} returns the coordinates of \\spad{a} with respect to the \\spad{basis} \\spad{basis}.")) (|norm| ((|#1| $) "\\spad{norm(a)} returns the determinant of the regular representation of \\spad{a} with respect to any basis.")) (|trace| ((|#1| $) "\\spad{trace(a)} returns the trace of the regular representation of \\spad{a} with respect to any basis.")) (|regularRepresentation| (((|Matrix| |#1|) $ (|Vector| $)) "\\spad{regularRepresentation(a,{}basis)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the \\spad{basis} \\spad{basis}.")) (|rank| (((|PositiveInteger|)) "\\spad{rank()} returns the rank of the algebra."))) 
((-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-367 S A R B) 
((|constructor| (NIL "\\spad{FiniteLinearAggregateFunctions2} provides functions involving two FiniteLinearAggregates where the underlying domains might be different. An example of this might be creating a list of rational numbers by mapping a function across a list of integers where the function divides each integer by 1000.")) (|scan| ((|#4| (|Mapping| |#3| |#1| |#3|) |#2| |#3|) "\\spad{scan(f,{}a,{}r)} successively applies \\spad{reduce(f,{}x,{}r)} to more and more leading sub-aggregates \\spad{x} of aggregrate \\spad{a}. More precisely,{} if \\spad{a} is \\spad{[a1,{}a2,{}...]},{} then \\spad{scan(f,{}a,{}r)} returns \\spad{[reduce(f,{}[a1],{}r),{}reduce(f,{}[a1,{}a2],{}r),{}...]}.")) (|reduce| ((|#3| (|Mapping| |#3| |#1| |#3|) |#2| |#3|) "\\spad{reduce(f,{}a,{}r)} applies function \\spad{f} to each successive element of the aggregate \\spad{a} and an accumulant initialized to \\spad{r}. For example,{} \\spad{reduce(_+\\$Integer,{}[1,{}2,{}3],{}0)} does \\spad{3+(2+(1+0))}. Note that third argument \\spad{r} may be regarded as the identity element for the function \\spad{f}.")) (|map| ((|#4| (|Mapping| |#3| |#1|) |#2|) "\\spad{map(f,{}a)} applies function \\spad{f} to each member of aggregate \\spad{a} resulting in a new aggregate over a possibly different underlying domain."))) 
NIL 
NIL 
(-368 A S) 
((|constructor| (NIL "A finite linear aggregate is a linear aggregate of finite length. The finite property of the aggregate adds several exports to the list of exports from \\spadtype{LinearAggregate} such as \\spadfun{reverse},{} \\spadfun{sort},{} and so on.")) (|sort!| (($ $) "\\spad{sort!(u)} returns \\spad{u} with its elements in ascending order.") (($ (|Mapping| (|Boolean|) |#2| |#2|) $) "\\spad{sort!(p,{}u)} returns \\spad{u} with its elements ordered by \\spad{p}.")) (|reverse!| (($ $) "\\spad{reverse!(u)} returns \\spad{u} with its elements in reverse order.")) (|copyInto!| (($ $ $ (|Integer|)) "\\spad{copyInto!(u,{}v,{}i)} returns aggregate \\spad{u} containing a copy of \\spad{v} inserted at element \\spad{i}.")) (|position| (((|Integer|) |#2| $ (|Integer|)) "\\spad{position(x,{}a,{}n)} returns the index \\spad{i} of the first occurrence of \\spad{x} in \\axiom{a} where \\axiom{\\spad{i} \\spad{>=} \\spad{n}},{} and \\axiom{minIndex(a) - 1} if no such \\spad{x} is found.") (((|Integer|) |#2| $) "\\spad{position(x,{}a)} returns the index \\spad{i} of the first occurrence of \\spad{x} in a,{} and \\axiom{minIndex(a) - 1} if there is no such \\spad{x}.") (((|Integer|) (|Mapping| (|Boolean|) |#2|) $) "\\spad{position(p,{}a)} returns the index \\spad{i} of the first \\spad{x} in \\axiom{a} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true},{} and \\axiom{minIndex(a) - 1} if there is no such \\spad{x}.")) (|sorted?| (((|Boolean|) $) "\\spad{sorted?(u)} tests if the elements of \\spad{u} are in ascending order.") (((|Boolean|) (|Mapping| (|Boolean|) |#2| |#2|) $) "\\spad{sorted?(p,{}a)} tests if \\axiom{a} is sorted according to predicate \\spad{p}.")) (|sort| (($ $) "\\spad{sort(u)} returns an \\spad{u} with elements in ascending order. Note that \\axiom{sort(\\spad{u}) = sort(\\spad{<=},{}\\spad{u})}.") (($ (|Mapping| (|Boolean|) |#2| |#2|) $) "\\spad{sort(p,{}a)} returns a copy of \\axiom{a} sorted using total ordering predicate \\spad{p}.")) (|reverse| (($ $) "\\spad{reverse(a)} returns a copy of \\axiom{a} with elements in reverse order.")) (|merge| (($ $ $) "\\spad{merge(u,{}v)} merges \\spad{u} and \\spad{v} in ascending order. Note that \\axiom{merge(\\spad{u},{}\\spad{v}) = merge(\\spad{<=},{}\\spad{u},{}\\spad{v})}.") (($ (|Mapping| (|Boolean|) |#2| |#2|) $ $) "\\spad{merge(p,{}a,{}b)} returns an aggregate \\spad{c} which merges \\axiom{a} and \\spad{b}. The result is produced by examining each element \\spad{x} of \\axiom{a} and \\spad{y} of \\spad{b} successively. If \\axiom{\\spad{p}(\\spad{x},{}\\spad{y})} is \\spad{true},{} then \\spad{x} is inserted into the result; otherwise \\spad{y} is inserted. If \\spad{x} is chosen,{} the next element of \\axiom{a} is examined,{} and so on. When all the elements of one aggregate are examined,{} the remaining elements of the other are appended. For example,{} \\axiom{merge(<,{}[1,{}3],{}[2,{}7,{}5])} returns \\axiom{[1,{}2,{}3,{}7,{}5]}."))) 
NIL 
((|HasAttribute| |#1| (QUOTE -4506)) (|HasCategory| |#2| (QUOTE (-834))) (|HasCategory| |#2| (QUOTE (-1082)))) 
(-369 S) 
((|constructor| (NIL "A finite linear aggregate is a linear aggregate of finite length. The finite property of the aggregate adds several exports to the list of exports from \\spadtype{LinearAggregate} such as \\spadfun{reverse},{} \\spadfun{sort},{} and so on.")) (|sort!| (($ $) "\\spad{sort!(u)} returns \\spad{u} with its elements in ascending order.") (($ (|Mapping| (|Boolean|) |#1| |#1|) $) "\\spad{sort!(p,{}u)} returns \\spad{u} with its elements ordered by \\spad{p}.")) (|reverse!| (($ $) "\\spad{reverse!(u)} returns \\spad{u} with its elements in reverse order.")) (|copyInto!| (($ $ $ (|Integer|)) "\\spad{copyInto!(u,{}v,{}i)} returns aggregate \\spad{u} containing a copy of \\spad{v} inserted at element \\spad{i}.")) (|position| (((|Integer|) |#1| $ (|Integer|)) "\\spad{position(x,{}a,{}n)} returns the index \\spad{i} of the first occurrence of \\spad{x} in \\axiom{a} where \\axiom{\\spad{i} \\spad{>=} \\spad{n}},{} and \\axiom{minIndex(a) - 1} if no such \\spad{x} is found.") (((|Integer|) |#1| $) "\\spad{position(x,{}a)} returns the index \\spad{i} of the first occurrence of \\spad{x} in a,{} and \\axiom{minIndex(a) - 1} if there is no such \\spad{x}.") (((|Integer|) (|Mapping| (|Boolean|) |#1|) $) "\\spad{position(p,{}a)} returns the index \\spad{i} of the first \\spad{x} in \\axiom{a} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true},{} and \\axiom{minIndex(a) - 1} if there is no such \\spad{x}.")) (|sorted?| (((|Boolean|) $) "\\spad{sorted?(u)} tests if the elements of \\spad{u} are in ascending order.") (((|Boolean|) (|Mapping| (|Boolean|) |#1| |#1|) $) "\\spad{sorted?(p,{}a)} tests if \\axiom{a} is sorted according to predicate \\spad{p}.")) (|sort| (($ $) "\\spad{sort(u)} returns an \\spad{u} with elements in ascending order. Note that \\axiom{sort(\\spad{u}) = sort(\\spad{<=},{}\\spad{u})}.") (($ (|Mapping| (|Boolean|) |#1| |#1|) $) "\\spad{sort(p,{}a)} returns a copy of \\axiom{a} sorted using total ordering predicate \\spad{p}.")) (|reverse| (($ $) "\\spad{reverse(a)} returns a copy of \\axiom{a} with elements in reverse order.")) (|merge| (($ $ $) "\\spad{merge(u,{}v)} merges \\spad{u} and \\spad{v} in ascending order. Note that \\axiom{merge(\\spad{u},{}\\spad{v}) = merge(\\spad{<=},{}\\spad{u},{}\\spad{v})}.") (($ (|Mapping| (|Boolean|) |#1| |#1|) $ $) "\\spad{merge(p,{}a,{}b)} returns an aggregate \\spad{c} which merges \\axiom{a} and \\spad{b}. The result is produced by examining each element \\spad{x} of \\axiom{a} and \\spad{y} of \\spad{b} successively. If \\axiom{\\spad{p}(\\spad{x},{}\\spad{y})} is \\spad{true},{} then \\spad{x} is inserted into the result; otherwise \\spad{y} is inserted. If \\spad{x} is chosen,{} the next element of \\axiom{a} is examined,{} and so on. When all the elements of one aggregate are examined,{} the remaining elements of the other are appended. For example,{} \\axiom{merge(<,{}[1,{}3],{}[2,{}7,{}5])} returns \\axiom{[1,{}2,{}3,{}7,{}5]}."))) 
((-4505 . T) (-3576 . T)) 
NIL 
(-370 |VarSet| R) 
((|constructor| (NIL "The category of free Lie algebras. It is used by domains of non-commutative algebra: \\spadtype{LiePolynomial} and \\spadtype{XPBWPolynomial}.")) (|eval| (($ $ (|List| |#1|) (|List| $)) "\\axiom{eval(\\spad{p},{} [\\spad{x1},{}...,{}\\spad{xn}],{} [\\spad{v1},{}...,{}\\spad{vn}])} replaces \\axiom{\\spad{xi}} by \\axiom{\\spad{vi}} in \\axiom{\\spad{p}}.") (($ $ |#1| $) "\\axiom{eval(\\spad{p},{} \\spad{x},{} \\spad{v})} replaces \\axiom{\\spad{x}} by \\axiom{\\spad{v}} in \\axiom{\\spad{p}}.")) (|varList| (((|List| |#1|) $) "\\axiom{varList(\\spad{x})} returns the list of distinct entries of \\axiom{\\spad{x}}.")) (|trunc| (($ $ (|NonNegativeInteger|)) "\\axiom{trunc(\\spad{p},{}\\spad{n})} returns the polynomial \\axiom{\\spad{p}} truncated at order \\axiom{\\spad{n}}.")) (|mirror| (($ $) "\\axiom{mirror(\\spad{x})} returns \\axiom{Sum(r_i mirror(w_i))} if \\axiom{\\spad{x}} is \\axiom{Sum(r_i w_i)}.")) (|LiePoly| (($ (|LyndonWord| |#1|)) "\\axiom{LiePoly(\\spad{l})} returns the bracketed form of \\axiom{\\spad{l}} as a Lie polynomial.")) (|rquo| (((|XRecursivePolynomial| |#1| |#2|) (|XRecursivePolynomial| |#1| |#2|) $) "\\axiom{rquo(\\spad{x},{}\\spad{y})} returns the right simplification of \\axiom{\\spad{x}} by \\axiom{\\spad{y}}.")) (|lquo| (((|XRecursivePolynomial| |#1| |#2|) (|XRecursivePolynomial| |#1| |#2|) $) "\\axiom{lquo(\\spad{x},{}\\spad{y})} returns the left simplification of \\axiom{\\spad{x}} by \\axiom{\\spad{y}}.")) (|degree| (((|NonNegativeInteger|) $) "\\axiom{degree(\\spad{x})} returns the greatest length of a word in the support of \\axiom{\\spad{x}}.")) (|coerce| (((|XRecursivePolynomial| |#1| |#2|) $) "\\axiom{coerce(\\spad{x})} returns \\axiom{\\spad{x}} as a recursive polynomial.") (((|XDistributedPolynomial| |#1| |#2|) $) "\\axiom{coerce(\\spad{x})} returns \\axiom{\\spad{x}} as distributed polynomial.") (($ |#1|) "\\axiom{coerce(\\spad{x})} returns \\axiom{\\spad{x}} as a Lie polynomial.")) (|coef| ((|#2| (|XRecursivePolynomial| |#1| |#2|) $) "\\axiom{coef(\\spad{x},{}\\spad{y})} returns the scalar product of \\axiom{\\spad{x}} by \\axiom{\\spad{y}},{} the set of words being regarded as an orthogonal basis."))) 
((|JacobiIdentity| . T) (|NullSquare| . T) (-4500 . T) (-4499 . T)) 
NIL 
(-371 S V) 
((|constructor| (NIL "This package exports 3 sorting algorithms which work over FiniteLinearAggregates. Sort package (in-place) for shallowlyMutable Finite Linear Aggregates")) (|shellSort| ((|#2| (|Mapping| (|Boolean|) |#1| |#1|) |#2|) "\\spad{shellSort(f,{} agg)} sorts the aggregate agg with the ordering function \\spad{f} using the shellSort algorithm.")) (|heapSort| ((|#2| (|Mapping| (|Boolean|) |#1| |#1|) |#2|) "\\spad{heapSort(f,{} agg)} sorts the aggregate agg with the ordering function \\spad{f} using the heapsort algorithm.")) (|quickSort| ((|#2| (|Mapping| (|Boolean|) |#1| |#1|) |#2|) "\\spad{quickSort(f,{} agg)} sorts the aggregate agg with the ordering function \\spad{f} using the quicksort algorithm."))) 
NIL 
NIL 
(-372 S R) 
((|constructor| (NIL "\\spad{S} is \\spadtype{FullyLinearlyExplicitRingOver R} means that \\spad{S} is a \\spadtype{LinearlyExplicitRingOver R} and,{} in addition,{} if \\spad{R} is a \\spadtype{LinearlyExplicitRingOver Integer},{} then so is \\spad{S}"))) 
NIL 
((|HasCategory| |#2| (LIST (QUOTE -622) (QUOTE (-560))))) 
(-373 R) 
((|constructor| (NIL "\\spad{S} is \\spadtype{FullyLinearlyExplicitRingOver R} means that \\spad{S} is a \\spadtype{LinearlyExplicitRingOver R} and,{} in addition,{} if \\spad{R} is a \\spadtype{LinearlyExplicitRingOver Integer},{} then so is \\spad{S}"))) 
((-4502 . T)) 
NIL 
(-374 |Par|) 
((|constructor| (NIL "This is a package for the approximation of complex solutions for systems of equations of rational functions with complex rational coefficients. The results are expressed as either complex rational numbers or complex floats depending on the type of the precision parameter which can be either a rational number or a floating point number.")) (|complexRoots| (((|List| (|List| (|Complex| |#1|))) (|List| (|Fraction| (|Polynomial| (|Complex| (|Integer|))))) (|List| (|Symbol|)) |#1|) "\\spad{complexRoots(lrf,{} lv,{} eps)} finds all the complex solutions of a list of rational functions with rational number coefficients with respect the the variables appearing in \\spad{lv}. Each solution is computed to precision eps and returned as list corresponding to the order of variables in \\spad{lv}.") (((|List| (|Complex| |#1|)) (|Fraction| (|Polynomial| (|Complex| (|Integer|)))) |#1|) "\\spad{complexRoots(rf,{} eps)} finds all the complex solutions of a univariate rational function with rational number coefficients. The solutions are computed to precision eps.")) (|complexSolve| (((|List| (|Equation| (|Polynomial| (|Complex| |#1|)))) (|Equation| (|Fraction| (|Polynomial| (|Complex| (|Integer|))))) |#1|) "\\spad{complexSolve(eq,{}eps)} finds all the complex solutions of the equation \\spad{eq} of rational functions with rational rational coefficients with respect to all the variables appearing in \\spad{eq},{} with precision \\spad{eps}.") (((|List| (|Equation| (|Polynomial| (|Complex| |#1|)))) (|Fraction| (|Polynomial| (|Complex| (|Integer|)))) |#1|) "\\spad{complexSolve(p,{}eps)} find all the complex solutions of the rational function \\spad{p} with complex rational coefficients with respect to all the variables appearing in \\spad{p},{} with precision \\spad{eps}.") (((|List| (|List| (|Equation| (|Polynomial| (|Complex| |#1|))))) (|List| (|Equation| (|Fraction| (|Polynomial| (|Complex| (|Integer|)))))) |#1|) "\\spad{complexSolve(leq,{}eps)} finds all the complex solutions to precision \\spad{eps} of the system \\spad{leq} of equations of rational functions over complex rationals with respect to all the variables appearing in \\spad{lp}.") (((|List| (|List| (|Equation| (|Polynomial| (|Complex| |#1|))))) (|List| (|Fraction| (|Polynomial| (|Complex| (|Integer|))))) |#1|) "\\spad{complexSolve(lp,{}eps)} finds all the complex solutions to precision \\spad{eps} of the system \\spad{lp} of rational functions over the complex rationals with respect to all the variables appearing in \\spad{lp}."))) 
NIL 
NIL 
(-375) 
((|constructor| (NIL "\\spadtype{Float} implements arbitrary precision floating point arithmetic. The number of significant digits of each operation can be set to an arbitrary value (the default is 20 decimal digits). The operation \\spad{float(mantissa,{}exponent,{}base)} for integer \\spad{mantissa},{} \\spad{exponent} specifies the number \\spad{mantissa * base ** exponent} The underlying representation for floats is binary not decimal. The implications of this are described below. \\blankline The model adopted is that arithmetic operations are rounded to to nearest unit in the last place,{} that is,{} accurate to within \\spad{2**(-bits)}. Also,{} the elementary functions and constants are accurate to one unit in the last place. A float is represented as a record of two integers,{} the mantissa and the exponent. The base of the representation is binary,{} hence a \\spad{Record(m:mantissa,{}e:exponent)} represents the number \\spad{m * 2 ** e}. Though it is not assumed that the underlying integers are represented with a binary base,{} the code will be most efficient when this is the the case (this is \\spad{true} in most implementations of Lisp). The decision to choose the base to be binary has some unfortunate consequences. First,{} decimal numbers like 0.3 cannot be represented exactly. Second,{} there is a further loss of accuracy during conversion to decimal for output. To compensate for this,{} if \\spad{d} digits of precision are specified,{} \\spad{1 + ceiling(log2 d)} bits are used. Two numbers that are displayed identically may therefore be not equal. On the other hand,{} a significant efficiency loss would be incurred if we chose to use a decimal base when the underlying integer base is binary. \\blankline Algorithms used: For the elementary functions,{} the general approach is to apply identities so that the taylor series can be used,{} and,{} so that it will converge within \\spad{O( sqrt n )} steps. For example,{} using the identity \\spad{exp(x) = exp(x/2)**2},{} we can compute \\spad{exp(1/3)} to \\spad{n} digits of precision as follows. We have \\spad{exp(1/3) = exp(2 ** (-sqrt s) / 3) ** (2 ** sqrt s)}. The taylor series will converge in less than sqrt \\spad{n} steps and the exponentiation requires sqrt \\spad{n} multiplications for a total of \\spad{2 sqrt n} multiplications. Assuming integer multiplication costs \\spad{O( n**2 )} the overall running time is \\spad{O( sqrt(n) n**2 )}. This approach is the best known approach for precisions up to about 10,{}000 digits at which point the methods of Brent which are \\spad{O( log(n) n**2 )} become competitive. Note also that summing the terms of the taylor series for the elementary functions is done using integer operations. This avoids the overhead of floating point operations and results in efficient code at low precisions. This implementation makes no attempt to reuse storage,{} relying on the underlying system to do \\spadgloss{garbage collection}. \\spad{I} estimate that the efficiency of this package at low precisions could be improved by a factor of 2 if in-place operations were available. \\blankline Running times: in the following,{} \\spad{n} is the number of bits of precision\\spad{\\br} \\spad{*},{} \\spad{/},{} \\spad{sqrt},{} \\spad{\\spad{pi}},{} \\spad{exp1},{} \\spad{log2},{} \\spad{log10}: \\spad{ O( n**2 )} \\spad{\\br} \\spad{exp},{} \\spad{log},{} \\spad{sin},{} \\spad{atan}: \\spad{O(sqrt(n) n**2)}\\spad{\\br} The other elementary functions are coded in terms of the ones above.")) (|outputSpacing| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputSpacing(n)} inserts a space after \\spad{n} (default 10) digits on output; outputSpacing(0) means no spaces are inserted.")) (|outputGeneral| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputGeneral(n)} sets the output mode to general notation with \\spad{n} significant digits displayed.") (((|Void|)) "\\spad{outputGeneral()} sets the output mode (default mode) to general notation; numbers will be displayed in either fixed or floating (scientific) notation depending on the magnitude.")) (|outputFixed| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputFixed(n)} sets the output mode to fixed point notation,{} with \\spad{n} digits displayed after the decimal point.") (((|Void|)) "\\spad{outputFixed()} sets the output mode to fixed point notation; the output will contain a decimal point.")) (|outputFloating| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputFloating(n)} sets the output mode to floating (scientific) notation with \\spad{n} significant digits displayed after the decimal point.") (((|Void|)) "\\spad{outputFloating()} sets the output mode to floating (scientific) notation,{} \\spadignore{i.e.} \\spad{mantissa * 10 exponent} is displayed as \\spad{0.mantissa E exponent}.")) (|convert| (($ (|DoubleFloat|)) "\\spad{convert(x)} converts a \\spadtype{DoubleFloat} \\spad{x} to a \\spadtype{Float}.")) (|atan| (($ $ $) "\\spad{atan(x,{}y)} computes the arc tangent from \\spad{x} with phase \\spad{y}.")) (|exp1| (($) "\\spad{exp1()} returns exp 1: \\spad{2.7182818284...}.")) (|log10| (($ $) "\\spad{log10(x)} computes the logarithm for \\spad{x} to base 10.") (($) "\\spad{log10()} returns \\spad{ln 10}: \\spad{2.3025809299...}.")) (|log2| (($ $) "\\spad{log2(x)} computes the logarithm for \\spad{x} to base 2.") (($) "\\spad{log2()} returns \\spad{ln 2},{} \\spadignore{i.e.} \\spad{0.6931471805...}.")) (|rationalApproximation| (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{rationalApproximation(f,{} n,{} b)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< b**(-n)},{} that is \\spad{|(r-f)/f| < b**(-n)}.") (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|)) "\\spad{rationalApproximation(f,{} n)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< 10**(-n)}.")) (|shift| (($ $ (|Integer|)) "\\spad{shift(x,{}n)} adds \\spad{n} to the exponent of float \\spad{x}.")) (|relerror| (((|Integer|) $ $) "\\spad{relerror(x,{}y)} computes the absolute value of \\spad{x - y} divided by \\spad{y},{} when \\spad{y \\^= 0}.")) (|normalize| (($ $) "\\spad{normalize(x)} normalizes \\spad{x} at current precision.")) (** (($ $ $) "\\spad{x ** y} computes \\spad{exp(y log x)} where \\spad{x >= 0}.")) (/ (($ $ (|Integer|)) "\\spad{x / i} computes the division from \\spad{x} by an integer \\spad{i}."))) 
((-4488 . T) (-4496 . T) (-3580 . T) (-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-376 |Par|) 
((|constructor| (NIL "This is a package for the approximation of real solutions for systems of polynomial equations over the rational numbers. The results are expressed as either rational numbers or floats depending on the type of the precision parameter which can be either a rational number or a floating point number.")) (|realRoots| (((|List| |#1|) (|Fraction| (|Polynomial| (|Integer|))) |#1|) "\\spad{realRoots(rf,{} eps)} finds the real zeros of a univariate rational function with precision given by eps.") (((|List| (|List| |#1|)) (|List| (|Fraction| (|Polynomial| (|Integer|)))) (|List| (|Symbol|)) |#1|) "\\spad{realRoots(lp,{}lv,{}eps)} computes the list of the real solutions of the list \\spad{lp} of rational functions with rational coefficients with respect to the variables in \\spad{lv},{} with precision \\spad{eps}. Each solution is expressed as a list of numbers in order corresponding to the variables in \\spad{lv}.")) (|solve| (((|List| (|Equation| (|Polynomial| |#1|))) (|Equation| (|Fraction| (|Polynomial| (|Integer|)))) |#1|) "\\spad{solve(eq,{}eps)} finds all of the real solutions of the univariate equation \\spad{eq} of rational functions with respect to the unique variables appearing in \\spad{eq},{} with precision \\spad{eps}.") (((|List| (|Equation| (|Polynomial| |#1|))) (|Fraction| (|Polynomial| (|Integer|))) |#1|) "\\spad{solve(p,{}eps)} finds all of the real solutions of the univariate rational function \\spad{p} with rational coefficients with respect to the unique variable appearing in \\spad{p},{} with precision \\spad{eps}.") (((|List| (|List| (|Equation| (|Polynomial| |#1|)))) (|List| (|Equation| (|Fraction| (|Polynomial| (|Integer|))))) |#1|) "\\spad{solve(leq,{}eps)} finds all of the real solutions of the system \\spad{leq} of equationas of rational functions with respect to all the variables appearing in \\spad{lp},{} with precision \\spad{eps}.") (((|List| (|List| (|Equation| (|Polynomial| |#1|)))) (|List| (|Fraction| (|Polynomial| (|Integer|)))) |#1|) "\\spad{solve(lp,{}eps)} finds all of the real solutions of the system \\spad{lp} of rational functions over the rational numbers with respect to all the variables appearing in \\spad{lp},{} with precision \\spad{eps}."))) 
NIL 
NIL 
(-377 R S) 
((|constructor| (NIL "This domain implements linear combinations of elements from the domain \\spad{S} with coefficients in the domain \\spad{R} where \\spad{S} is an ordered set and \\spad{R} is a ring (which may be non-commutative). This domain is used by domains of non-commutative algebra such as: XDistributedPolynomial,{} XRecursivePolynomial.")) (* (($ |#2| |#1|) "\\spad{s*r} returns the product \\spad{r*s} used by \\spadtype{XRecursivePolynomial}"))) 
((-4500 . T) (-4499 . T)) 
((|HasCategory| |#1| (QUOTE (-170)))) 
(-378 R |Basis|) 
((|constructor| (NIL "A domain of this category implements formal linear combinations of elements from a domain \\spad{Basis} with coefficients in a domain \\spad{R}. The domain \\spad{Basis} needs only to belong to the category \\spadtype{SetCategory} and \\spad{R} to the category \\spadtype{Ring}. Thus the coefficient ring may be non-commutative. See the \\spadtype{XDistributedPolynomial} constructor for examples of domains built with the \\spadtype{FreeModuleCat} category constructor.")) (|reductum| (($ $) "\\spad{reductum(x)} returns \\spad{x} minus its leading term.")) (|leadingTerm| (((|Record| (|:| |k| |#2|) (|:| |c| |#1|)) $) "\\spad{leadingTerm(x)} returns the first term which appears in \\spad{listOfTerms(x)}.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(x)} returns the first coefficient which appears in \\spad{listOfTerms(x)}.")) (|leadingMonomial| ((|#2| $) "\\spad{leadingMonomial(x)} returns the first element from \\spad{Basis} which appears in \\spad{listOfTerms(x)}.")) (|numberOfMonomials| (((|NonNegativeInteger|) $) "\\spad{numberOfMonomials(x)} returns the number of monomials of \\spad{x}.")) (|monomials| (((|List| $) $) "\\spad{monomials(x)} returns the list of \\spad{r_i*b_i} whose sum is \\spad{x}.")) (|coefficients| (((|List| |#1|) $) "\\spad{coefficients(x)} returns the list of coefficients of \\spad{x}")) (|listOfTerms| (((|List| (|Record| (|:| |k| |#2|) (|:| |c| |#1|))) $) "\\spad{listOfTerms(x)} returns a list \\spad{lt} of terms with type \\spad{Record(k: Basis,{} c: R)} such that \\spad{x} equals \\spad{reduce(+,{} map(x +-> monom(x.k,{} x.c),{} lt))}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(x)} returns \\spad{true} if \\spad{x} contains a single monomial.")) (|monom| (($ |#2| |#1|) "\\spad{monom(b,{}r)} returns the element with the single monomial \\indented{1}{\\spad{b} and coefficient \\spad{r}.}")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(fn,{}u)} maps function \\spad{fn} onto the coefficients \\indented{1}{of the non-zero monomials of \\spad{u}.}")) (|coefficient| ((|#1| $ |#2|) "\\spad{coefficient(x,{}b)} returns the coefficient of \\spad{b} in \\spad{x}.")) (* (($ |#1| |#2|) "\\spad{r*b} returns the product of \\spad{r} by \\spad{b}."))) 
((-4500 . T) (-4499 . T)) 
NIL 
(-379) 
((|constructor| (NIL "\\axiomType{FortranMatrixCategory} provides support for producing Functions and Subroutines when the input to these is an AXIOM object of type \\axiomType{Matrix} or in domains involving \\axiomType{FortranCode}.")) (|coerce| (($ (|Record| (|:| |localSymbols| (|SymbolTable|)) (|:| |code| (|List| (|FortranCode|))))) "\\spad{coerce(e)} takes the component of \\spad{e} from \\spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \\spadtype{SymbolTable} component.") (($ (|FortranCode|)) "\\spad{coerce(e)} takes an object from \\spadtype{FortranCode} and \\indented{1}{uses it as the body of an ASP.}") (($ (|List| (|FortranCode|))) "\\spad{coerce(e)} takes an object from \\spadtype{List FortranCode} and \\indented{1}{uses it as the body of an ASP.}") (($ (|Matrix| (|MachineFloat|))) "\\spad{coerce(v)} produces an ASP which returns the value of \\spad{v}."))) 
((-3576 . T)) 
NIL 
(-380) 
((|constructor| (NIL "\\axiomType{FortranMatrixFunctionCategory} provides support for producing Functions and Subroutines representing matrices of expressions.")) (|retractIfCan| (((|Union| $ "failed") (|Matrix| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Matrix| (|Fraction| (|Polynomial| (|Float|))))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Matrix| (|Polynomial| (|Integer|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Matrix| (|Polynomial| (|Float|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Matrix| (|Expression| (|Integer|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Matrix| (|Expression| (|Float|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}")) (|retract| (($ (|Matrix| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Matrix| (|Fraction| (|Polynomial| (|Float|))))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Matrix| (|Polynomial| (|Integer|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Matrix| (|Polynomial| (|Float|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Matrix| (|Expression| (|Integer|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Matrix| (|Expression| (|Float|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}")) (|coerce| (($ (|Record| (|:| |localSymbols| (|SymbolTable|)) (|:| |code| (|List| (|FortranCode|))))) "\\spad{coerce(e)} takes the component of \\spad{e} from \\spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \\spadtype{SymbolTable} component.") (($ (|FortranCode|)) "\\spad{coerce(e)} takes an object from \\spadtype{FortranCode} and \\indented{1}{uses it as the body of an ASP.}") (($ (|List| (|FortranCode|))) "\\spad{coerce(e)} takes an object from \\spadtype{List FortranCode} and \\indented{1}{uses it as the body of an ASP.}"))) 
((-3576 . T)) 
NIL 
(-381 R S) 
((|constructor| (NIL "A \\spad{bi}-module is a free module over a ring with generators indexed by an ordered set. Each element can be expressed as a finite linear combination of generators. Only non-zero terms are stored."))) 
((-4500 . T) (-4499 . T)) 
((|HasCategory| |#1| (QUOTE (-170)))) 
(-382 S) 
((|constructor| (NIL "The free monoid on a set \\spad{S} is the monoid of finite products of the form \\spad{reduce(*,{}[\\spad{si} ** \\spad{ni}])} where the \\spad{si}\\spad{'s} are in \\spad{S},{} and the \\spad{ni}\\spad{'s} are nonnegative integers. The multiplication is not commutative.")) (|mapGen| (($ (|Mapping| |#1| |#1|) $) "\\spad{mapGen(f,{} a1\\^e1 ... an\\^en)} returns \\spad{f(a1)\\^e1 ... f(an)\\^en}.")) (|mapExpon| (($ (|Mapping| (|NonNegativeInteger|) (|NonNegativeInteger|)) $) "\\spad{mapExpon(f,{} a1\\^e1 ... an\\^en)} returns \\spad{a1\\^f(e1) ... an\\^f(en)}.")) (|nthFactor| ((|#1| $ (|Integer|)) "\\spad{nthFactor(x,{} n)} returns the factor of the n^th monomial of \\spad{x}.")) (|nthExpon| (((|NonNegativeInteger|) $ (|Integer|)) "\\spad{nthExpon(x,{} n)} returns the exponent of the n^th monomial of \\spad{x}.")) (|factors| (((|List| (|Record| (|:| |gen| |#1|) (|:| |exp| (|NonNegativeInteger|)))) $) "\\spad{factors(a1\\^e1,{}...,{}an\\^en)} returns \\spad{[[a1,{} e1],{}...,{}[an,{} en]]}.")) (|size| (((|NonNegativeInteger|) $) "\\spad{size(x)} returns the number of monomials in \\spad{x}.")) (|overlap| (((|Record| (|:| |lm| $) (|:| |mm| $) (|:| |rm| $)) $ $) "\\spad{overlap(x,{} y)} returns \\spad{[l,{} m,{} r]} such that \\spad{x = l * m},{} \\spad{y = m * r} and \\spad{l} and \\spad{r} have no overlap,{} \\spadignore{i.e.} \\spad{overlap(l,{} r) = [l,{} 1,{} r]}.")) (|divide| (((|Union| (|Record| (|:| |lm| $) (|:| |rm| $)) "failed") $ $) "\\spad{divide(x,{} y)} returns the left and right exact quotients of \\spad{x} by \\spad{y},{} \\spadignore{i.e.} \\spad{[l,{} r]} such that \\spad{x = l * y * r},{} \"failed\" if \\spad{x} is not of the form \\spad{l * y * r}.")) (|rquo| (((|Union| $ "failed") $ $) "\\spad{rquo(x,{} y)} returns the exact right quotient of \\spad{x} by \\spad{y} \\spadignore{i.e.} \\spad{q} such that \\spad{x = q * y},{} \"failed\" if \\spad{x} is not of the form \\spad{q * y}.")) (|lquo| (((|Union| $ "failed") $ $) "\\spad{lquo(x,{} y)} returns the exact left quotient of \\spad{x} by \\spad{y} \\spadignore{i.e.} \\spad{q} such that \\spad{x = y * q},{} \"failed\" if \\spad{x} is not of the form \\spad{y * q}.")) (|hcrf| (($ $ $) "\\spad{hcrf(x,{} y)} returns the highest common right factor of \\spad{x} and \\spad{y},{} \\spadignore{i.e.} the largest \\spad{d} such that \\spad{x = a d} and \\spad{y = b d}.")) (|hclf| (($ $ $) "\\spad{hclf(x,{} y)} returns the highest common left factor of \\spad{x} and \\spad{y},{} \\spadignore{i.e.} the largest \\spad{d} such that \\spad{x = d a} and \\spad{y = d b}.")) (** (($ |#1| (|NonNegativeInteger|)) "\\spad{s ** n} returns the product of \\spad{s} by itself \\spad{n} times.")) (* (($ $ |#1|) "\\spad{x * s} returns the product of \\spad{x} by \\spad{s} on the right.") (($ |#1| $) "\\spad{s * x} returns the product of \\spad{x} by \\spad{s} on the left."))) 
NIL 
((|HasCategory| |#1| (QUOTE (-834)))) 
(-383) 
((|constructor| (NIL "A category of domains which model machine arithmetic used by machines in the AXIOM-NAG link."))) 
((-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-384) 
((|constructor| (NIL "This domain provides an interface to names in the file system."))) 
NIL 
NIL 
(-385) 
((|constructor| (NIL "This category provides an interface to names in the file system.")) (|new| (($ (|String|) (|String|) (|String|)) "\\spad{new(d,{}pref,{}e)} constructs the name of a new writable file with \\spad{d} as its directory,{} \\spad{pref} as a prefix of its name and \\spad{e} as its extension. When \\spad{d} or \\spad{t} is the empty string,{} a default is used. An error occurs if a new file cannot be written in the given directory.")) (|writable?| (((|Boolean|) $) "\\spad{writable?(f)} tests if the named file be opened for writing. The named file need not already exist.")) (|readable?| (((|Boolean|) $) "\\spad{readable?(f)} tests if the named file exist and can it be opened for reading.")) (|exists?| (((|Boolean|) $) "\\spad{exists?(f)} tests if the file exists in the file system.")) (|extension| (((|String|) $) "\\spad{extension(f)} returns the type part of the file name.")) (|name| (((|String|) $) "\\spad{name(f)} returns the name part of the file name.")) (|directory| (((|String|) $) "\\spad{directory(f)} returns the directory part of the file name.")) (|filename| (($ (|String|) (|String|) (|String|)) "\\spad{filename(d,{}n,{}e)} creates a file name with \\spad{d} as its directory,{} \\spad{n} as its name and \\spad{e} as its extension. This is a portable way to create file names. When \\spad{d} or \\spad{t} is the empty string,{} a default is used.")) (|coerce| (((|String|) $) "\\spad{coerce(fn)} produces a string for a file name according to operating system-dependent conventions.") (($ (|String|)) "\\spad{coerce(s)} converts a string to a file name according to operating system-dependent conventions."))) 
NIL 
NIL 
(-386 |n| |class| R) 
((|constructor| (NIL "Generate the Free Lie Algebra over a ring \\spad{R} with identity; A \\spad{P}. Hall basis is generated by a package call to HallBasis.")) (|generator| (($ (|NonNegativeInteger|)) "\\spad{generator(i)} is the \\spad{i}th Hall Basis element")) (|shallowExpand| (((|OutputForm|) $) "\\spad{shallowExpand(x)} is not documented")) (|deepExpand| (((|OutputForm|) $) "\\spad{deepExpand(x)} is not documented")) (|dimension| (((|NonNegativeInteger|)) "\\spad{dimension()} is the rank of this Lie algebra"))) 
((-4500 . T) (-4499 . T)) 
NIL 
(-387) 
((|constructor| (NIL "Code to manipulate Fortran Output Stack")) (|topFortranOutputStack| (((|String|)) "\\spad{topFortranOutputStack()} returns the top element of the Fortran output stack")) (|pushFortranOutputStack| (((|Void|) (|String|)) "\\spad{pushFortranOutputStack(f)} pushes \\spad{f} onto the Fortran output stack") (((|Void|) (|FileName|)) "\\spad{pushFortranOutputStack(f)} pushes \\spad{f} onto the Fortran output stack")) (|popFortranOutputStack| (((|Void|)) "\\spad{popFortranOutputStack()} pops the Fortran output stack")) (|showFortranOutputStack| (((|Stack| (|String|))) "\\spad{showFortranOutputStack()} returns the Fortran output stack")) (|clearFortranOutputStack| (((|Stack| (|String|))) "\\spad{clearFortranOutputStack()} clears the Fortran output stack"))) 
NIL 
NIL 
(-388 -2262 UP UPUP R) 
((|constructor| (NIL "Finds the order of a divisor over a finite field")) (|order| (((|NonNegativeInteger|) (|FiniteDivisor| |#1| |#2| |#3| |#4|)) "\\spad{order(x)} \\undocumented"))) 
NIL 
NIL 
(-389 S) 
((|constructor| (NIL "\\spadtype{ScriptFormulaFormat1} provides a utility coercion for changing to SCRIPT formula format anything that has a coercion to the standard output format.")) (|coerce| (((|ScriptFormulaFormat|) |#1|) "\\spad{coerce(s)} provides a direct coercion from an expression \\spad{s} of domain \\spad{S} to SCRIPT formula format. This allows the user to skip the step of first manually coercing the object to standard output format before it is coerced to SCRIPT formula format."))) 
NIL 
NIL 
(-390) 
((|constructor| (NIL "\\spadtype{ScriptFormulaFormat} provides a coercion from \\spadtype{OutputForm} to IBM SCRIPT/VS Mathematical Formula Format. The basic SCRIPT formula format object consists of three parts: a prologue,{} a formula part and an epilogue. The functions \\spadfun{prologue},{} \\spadfun{formula} and \\spadfun{epilogue} extract these parts,{} respectively. The central parts of the expression go into the formula part. The other parts can be set (\\spadfun{setPrologue!},{} \\spadfun{setEpilogue!}) so that contain the appropriate tags for printing. For example,{} the prologue and epilogue might simply contain \":df.\" and \":edf.\" so that the formula section will be printed in display math mode.")) (|setPrologue!| (((|List| (|String|)) $ (|List| (|String|))) "\\spad{setPrologue!(t,{}strings)} sets the prologue section of a formatted object \\spad{t} to \\spad{strings}.")) (|setFormula!| (((|List| (|String|)) $ (|List| (|String|))) "\\spad{setFormula!(t,{}strings)} sets the formula section of a formatted object \\spad{t} to \\spad{strings}.")) (|setEpilogue!| (((|List| (|String|)) $ (|List| (|String|))) "\\spad{setEpilogue!(t,{}strings)} sets the epilogue section of a formatted object \\spad{t} to \\spad{strings}.")) (|prologue| (((|List| (|String|)) $) "\\spad{prologue(t)} extracts the prologue section of a formatted object \\spad{t}.")) (|new| (($) "\\spad{new()} create a new,{} empty object. Use \\spadfun{setPrologue!},{} \\spadfun{setFormula!} and \\spadfun{setEpilogue!} to set the various components of this object.")) (|formula| (((|List| (|String|)) $) "\\spad{formula(t)} extracts the formula section of a formatted object \\spad{t}.")) (|epilogue| (((|List| (|String|)) $) "\\spad{epilogue(t)} extracts the epilogue section of a formatted object \\spad{t}.")) (|display| (((|Void|) $) "\\spad{display(t)} outputs the formatted code \\spad{t} so that each line has length less than or equal to the value set by the system command \\spadsyscom{set output length}.") (((|Void|) $ (|Integer|)) "\\spad{display(t,{}width)} outputs the formatted code \\spad{t} so that each line has length less than or equal to \\spadvar{\\spad{width}}.")) (|convert| (($ (|OutputForm|) (|Integer|)) "\\spad{convert(o,{}step)} changes \\spad{o} in standard output format to SCRIPT formula format and also adds the given \\spad{step} number. This is useful if you want to create equations with given numbers or have the equation numbers correspond to the interpreter \\spad{step} numbers.")) (|coerce| (($ (|OutputForm|)) "\\spad{coerce(o)} changes \\spad{o} in the standard output format to SCRIPT formula format."))) 
NIL 
NIL 
(-391) 
((|constructor| (NIL "\\axiomType{FortranProgramCategory} provides various models of FORTRAN subprograms. These can be transformed into actual FORTRAN code.")) (|outputAsFortran| (((|Void|) $) "\\axiom{outputAsFortran(\\spad{u})} translates \\axiom{\\spad{u}} into a legal FORTRAN subprogram."))) 
((-3576 . T)) 
NIL 
(-392) 
((|constructor| (NIL "\\axiomType{FortranFunctionCategory} is the category of arguments to NAG Library routines which return (sets of) function values.")) (|retractIfCan| (((|Union| $ "failed") (|Fraction| (|Polynomial| (|Integer|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Fraction| (|Polynomial| (|Float|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Polynomial| (|Integer|))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Polynomial| (|Float|))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Expression| (|Integer|))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Expression| (|Float|))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}")) (|retract| (($ (|Fraction| (|Polynomial| (|Integer|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Fraction| (|Polynomial| (|Float|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Polynomial| (|Integer|))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Polynomial| (|Float|))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Expression| (|Integer|))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Expression| (|Float|))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}")) (|coerce| (($ (|Record| (|:| |localSymbols| (|SymbolTable|)) (|:| |code| (|List| (|FortranCode|))))) "\\spad{coerce(e)} takes the component of \\spad{e} from \\spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \\spadtype{SymbolTable} component.") (($ (|FortranCode|)) "\\spad{coerce(e)} takes an object from \\spadtype{FortranCode} and \\indented{1}{uses it as the body of an ASP.}") (($ (|List| (|FortranCode|))) "\\spad{coerce(e)} takes an object from \\spadtype{List FortranCode} and \\indented{1}{uses it as the body of an ASP.}"))) 
((-3576 . T)) 
NIL 
(-393) 
((|constructor| (NIL "provides an interface to the boot code for calling Fortran")) (|setLegalFortranSourceExtensions| (((|List| (|String|)) (|List| (|String|))) "\\spad{setLegalFortranSourceExtensions(l)} \\undocumented{}")) (|outputAsFortran| (((|Void|) (|FileName|)) "\\spad{outputAsFortran(fn)} \\undocumented{}")) (|linkToFortran| (((|SExpression|) (|Symbol|) (|List| (|Symbol|)) (|TheSymbolTable|) (|List| (|Symbol|))) "\\spad{linkToFortran(s,{}l,{}t,{}lv)} \\undocumented{}") (((|SExpression|) (|Symbol|) (|List| (|Union| (|:| |array| (|List| (|Symbol|))) (|:| |scalar| (|Symbol|)))) (|List| (|List| (|Union| (|:| |array| (|List| (|Symbol|))) (|:| |scalar| (|Symbol|))))) (|List| (|Symbol|)) (|Symbol|)) "\\spad{linkToFortran(s,{}l,{}ll,{}lv,{}t)} \\undocumented{}") (((|SExpression|) (|Symbol|) (|List| (|Union| (|:| |array| (|List| (|Symbol|))) (|:| |scalar| (|Symbol|)))) (|List| (|List| (|Union| (|:| |array| (|List| (|Symbol|))) (|:| |scalar| (|Symbol|))))) (|List| (|Symbol|))) "\\spad{linkToFortran(s,{}l,{}ll,{}lv)} \\undocumented{}"))) 
NIL 
NIL 
(-394 -3409 |returnType| |arguments| |symbols|) 
((|constructor| (NIL "\\axiomType{FortranProgram} allows the user to build and manipulate simple models of FORTRAN subprograms. These can then be transformed into actual FORTRAN notation.")) (|coerce| (($ (|Equation| (|Expression| (|Complex| (|Float|))))) "\\spad{coerce(eq)} is not documented") (($ (|Equation| (|Expression| (|Float|)))) "\\spad{coerce(eq)} is not documented") (($ (|Equation| (|Expression| (|Integer|)))) "\\spad{coerce(eq)} is not documented") (($ (|Expression| (|Complex| (|Float|)))) "\\spad{coerce(e)} is not documented") (($ (|Expression| (|Float|))) "\\spad{coerce(e)} is not documented") (($ (|Expression| (|Integer|))) "\\spad{coerce(e)} is not documented") (($ (|Equation| (|Expression| (|MachineComplex|)))) "\\spad{coerce(eq)} is not documented") (($ (|Equation| (|Expression| (|MachineFloat|)))) "\\spad{coerce(eq)} is not documented") (($ (|Equation| (|Expression| (|MachineInteger|)))) "\\spad{coerce(eq)} is not documented") (($ (|Expression| (|MachineComplex|))) "\\spad{coerce(e)} is not documented") (($ (|Expression| (|MachineFloat|))) "\\spad{coerce(e)} is not documented") (($ (|Expression| (|MachineInteger|))) "\\spad{coerce(e)} is not documented") (($ (|Record| (|:| |localSymbols| (|SymbolTable|)) (|:| |code| (|List| (|FortranCode|))))) "\\spad{coerce(r)} is not documented") (($ (|List| (|FortranCode|))) "\\spad{coerce(lfc)} is not documented") (($ (|FortranCode|)) "\\spad{coerce(fc)} is not documented"))) 
NIL 
NIL 
(-395 -2262 UP) 
((|constructor| (NIL "Full partial fraction expansion of rational functions")) (D (($ $ (|NonNegativeInteger|)) "\\spad{D(f,{} n)} returns the \\spad{n}-th derivative of \\spad{f}.") (($ $) "\\spad{D(f)} returns the derivative of \\spad{f}.")) (|differentiate| (($ $ (|NonNegativeInteger|)) "\\spad{differentiate(f,{} n)} returns the \\spad{n}-th derivative of \\spad{f}.") (($ $) "\\spad{differentiate(f)} returns the derivative of \\spad{f}.")) (|construct| (($ (|List| (|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |center| |#2|) (|:| |num| |#2|)))) "\\spad{construct(l)} is the inverse of fracPart.")) (|fracPart| (((|List| (|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |center| |#2|) (|:| |num| |#2|))) $) "\\spad{fracPart(f)} returns the list of summands of the fractional part of \\spad{f}.")) (|polyPart| ((|#2| $) "\\spad{polyPart(f)} returns the polynomial part of \\spad{f}.")) (|fullPartialFraction| (($ (|Fraction| |#2|)) "\\spad{fullPartialFraction(f)} returns \\spad{[p,{} [[j,{} Dj,{} Hj]...]]} such that \\spad{f = p(x) + sum_{[j,{}Dj,{}Hj] in l} sum_{Dj(a)=0} Hj(a)/(x - a)\\^j}.")) (+ (($ |#2| $) "\\spad{p + x} returns the sum of \\spad{p} and \\spad{x}"))) 
NIL 
NIL 
(-396 R) 
((|constructor| (NIL "A set \\spad{S} is PatternMatchable over \\spad{R} if \\spad{S} can lift the pattern-matching functions of \\spad{S} over the integers and float to itself (necessary for matching in towers)."))) 
((-3576 . T)) 
NIL 
(-397 S) 
((|constructor| (NIL "FieldOfPrimeCharacteristic is the category of fields of prime characteristic,{} \\spadignore{e.g.} finite fields,{} algebraic closures of fields of prime characteristic,{} transcendental extensions of of fields of prime characteristic.")) (|primeFrobenius| (($ $ (|NonNegativeInteger|)) "\\spad{primeFrobenius(a,{}s)} returns \\spad{a**(p**s)} where \\spad{p} is the characteristic.") (($ $) "\\spad{primeFrobenius(a)} returns \\spad{a**p} where \\spad{p} is the characteristic.")) (|discreteLog| (((|Union| (|NonNegativeInteger|) "failed") $ $) "\\spad{discreteLog(b,{}a)} computes \\spad{s} with \\spad{b**s = a} if such an \\spad{s} exists.")) (|order| (((|OnePointCompletion| (|PositiveInteger|)) $) "\\spad{order(a)} computes the order of an element in the multiplicative group of the field. Error: if \\spad{a} is 0."))) 
NIL 
NIL 
(-398) 
((|constructor| (NIL "FieldOfPrimeCharacteristic is the category of fields of prime characteristic,{} \\spadignore{e.g.} finite fields,{} algebraic closures of fields of prime characteristic,{} transcendental extensions of of fields of prime characteristic.")) (|primeFrobenius| (($ $ (|NonNegativeInteger|)) "\\spad{primeFrobenius(a,{}s)} returns \\spad{a**(p**s)} where \\spad{p} is the characteristic.") (($ $) "\\spad{primeFrobenius(a)} returns \\spad{a**p} where \\spad{p} is the characteristic.")) (|discreteLog| (((|Union| (|NonNegativeInteger|) "failed") $ $) "\\spad{discreteLog(b,{}a)} computes \\spad{s} with \\spad{b**s = a} if such an \\spad{s} exists.")) (|order| (((|OnePointCompletion| (|PositiveInteger|)) $) "\\spad{order(a)} computes the order of an element in the multiplicative group of the field. Error: if \\spad{a} is 0."))) 
((-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-399 S) 
((|constructor| (NIL "This category is intended as a model for floating point systems. A floating point system is a model for the real numbers. In fact,{} it is an approximation in the sense that not all real numbers are exactly representable by floating point numbers. A floating point system is characterized by the following: \\blankline 1: base of the exponent where the actual implemenations are usually binary or decimal)\\spad{\\br} 2: precision of the mantissa (arbitrary or fixed)\\spad{\\br} 3: rounding error for operations \\blankline Because a Float is an approximation to the real numbers,{} even though it is defined to be a join of a Field and OrderedRing,{} some of the attributes do not hold. In particular associative(\\spad{\"+\"}) does not hold. Algorithms defined over a field need special considerations when the field is a floating point system.")) (|max| (($) "\\spad{max()} returns the maximum floating point number.")) (|min| (($) "\\spad{min()} returns the minimum floating point number.")) (|decreasePrecision| (((|PositiveInteger|) (|Integer|)) "\\spad{decreasePrecision(n)} decreases the current \\spadfunFrom{precision}{FloatingPointSystem} precision by \\spad{n} decimal digits.")) (|increasePrecision| (((|PositiveInteger|) (|Integer|)) "\\spad{increasePrecision(n)} increases the current \\spadfunFrom{precision}{FloatingPointSystem} by \\spad{n} decimal digits.")) (|precision| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{precision(n)} set the precision in the base to \\spad{n} decimal digits.") (((|PositiveInteger|)) "\\spad{precision()} returns the precision in digits base.")) (|digits| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{digits(d)} set the \\spadfunFrom{precision}{FloatingPointSystem} to \\spad{d} digits.") (((|PositiveInteger|)) "\\spad{digits()} returns ceiling\\spad{'s} precision in decimal digits.")) (|bits| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{bits(n)} set the \\spadfunFrom{precision}{FloatingPointSystem} to \\spad{n} bits.") (((|PositiveInteger|)) "\\spad{bits()} returns ceiling\\spad{'s} precision in bits.")) (|mantissa| (((|Integer|) $) "\\spad{mantissa(x)} returns the mantissa part of \\spad{x}.")) (|exponent| (((|Integer|) $) "\\spad{exponent(x)} returns the \\spadfunFrom{exponent}{FloatingPointSystem} part of \\spad{x}.")) (|base| (((|PositiveInteger|)) "\\indented{1}{base() returns the base of the} \\spadfunFrom{exponent}{FloatingPointSystem}.")) (|order| (((|Integer|) $) "\\spad{order x} is the order of magnitude of \\spad{x}. Note that \\spad{base ** order x <= |x| < base ** (1 + order x)}.")) (|float| (($ (|Integer|) (|Integer|) (|PositiveInteger|)) "\\spad{float(a,{}e,{}b)} returns \\spad{a * b ** e}.") (($ (|Integer|) (|Integer|)) "\\spad{float(a,{}e)} returns \\spad{a * base() ** e}.")) (|approximate| ((|attribute|) "\\spad{approximate} means \"is an approximation to the real numbers\"."))) 
NIL 
((|HasAttribute| |#1| (QUOTE -4488)) (|HasAttribute| |#1| (QUOTE -4496))) 
(-400) 
((|constructor| (NIL "This category is intended as a model for floating point systems. A floating point system is a model for the real numbers. In fact,{} it is an approximation in the sense that not all real numbers are exactly representable by floating point numbers. A floating point system is characterized by the following: \\blankline 1: base of the exponent where the actual implemenations are usually binary or decimal)\\spad{\\br} 2: precision of the mantissa (arbitrary or fixed)\\spad{\\br} 3: rounding error for operations \\blankline Because a Float is an approximation to the real numbers,{} even though it is defined to be a join of a Field and OrderedRing,{} some of the attributes do not hold. In particular associative(\\spad{\"+\"}) does not hold. Algorithms defined over a field need special considerations when the field is a floating point system.")) (|max| (($) "\\spad{max()} returns the maximum floating point number.")) (|min| (($) "\\spad{min()} returns the minimum floating point number.")) (|decreasePrecision| (((|PositiveInteger|) (|Integer|)) "\\spad{decreasePrecision(n)} decreases the current \\spadfunFrom{precision}{FloatingPointSystem} precision by \\spad{n} decimal digits.")) (|increasePrecision| (((|PositiveInteger|) (|Integer|)) "\\spad{increasePrecision(n)} increases the current \\spadfunFrom{precision}{FloatingPointSystem} by \\spad{n} decimal digits.")) (|precision| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{precision(n)} set the precision in the base to \\spad{n} decimal digits.") (((|PositiveInteger|)) "\\spad{precision()} returns the precision in digits base.")) (|digits| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{digits(d)} set the \\spadfunFrom{precision}{FloatingPointSystem} to \\spad{d} digits.") (((|PositiveInteger|)) "\\spad{digits()} returns ceiling\\spad{'s} precision in decimal digits.")) (|bits| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{bits(n)} set the \\spadfunFrom{precision}{FloatingPointSystem} to \\spad{n} bits.") (((|PositiveInteger|)) "\\spad{bits()} returns ceiling\\spad{'s} precision in bits.")) (|mantissa| (((|Integer|) $) "\\spad{mantissa(x)} returns the mantissa part of \\spad{x}.")) (|exponent| (((|Integer|) $) "\\spad{exponent(x)} returns the \\spadfunFrom{exponent}{FloatingPointSystem} part of \\spad{x}.")) (|base| (((|PositiveInteger|)) "\\indented{1}{base() returns the base of the} \\spadfunFrom{exponent}{FloatingPointSystem}.")) (|order| (((|Integer|) $) "\\spad{order x} is the order of magnitude of \\spad{x}. Note that \\spad{base ** order x <= |x| < base ** (1 + order x)}.")) (|float| (($ (|Integer|) (|Integer|) (|PositiveInteger|)) "\\spad{float(a,{}e,{}b)} returns \\spad{a * b ** e}.") (($ (|Integer|) (|Integer|)) "\\spad{float(a,{}e)} returns \\spad{a * base() ** e}.")) (|approximate| ((|attribute|) "\\spad{approximate} means \"is an approximation to the real numbers\"."))) 
((-3580 . T) (-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
NIL 
(-401 R S) 
((|constructor| (NIL "\\spadtype{FactoredFunctions2} contains functions that involve factored objects whose underlying domains may not be the same. For example,{} \\spadfun{map} might be used to coerce an object of type \\spadtype{Factored(Integer)} to \\spadtype{Factored(Complex(Integer))}.")) (|map| (((|Factored| |#2|) (|Mapping| |#2| |#1|) (|Factored| |#1|)) "\\spad{map(fn,{}u)} is used to apply the function \\userfun{\\spad{fn}} to every factor of \\spadvar{\\spad{u}}. The new factored object will have all its information flags set to \"nil\". This function is used,{} for example,{} to coerce every factor base to another type."))) 
NIL 
NIL 
(-402 A B) 
((|constructor| (NIL "This package extends a map between integral domains to a map between Fractions over those domains by applying the map to the numerators and denominators.")) (|map| (((|Fraction| |#2|) (|Mapping| |#2| |#1|) (|Fraction| |#1|)) "\\spad{map(func,{}frac)} applies the function \\spad{func} to the numerator and denominator of the fraction \\spad{frac}."))) 
NIL 
NIL 
(-403 S) 
((|constructor| (NIL "Fraction takes an IntegralDomain \\spad{S} and produces the domain of Fractions with numerators and denominators from \\spad{S}. If \\spad{S} is also a GcdDomain,{} then \\spad{gcd}\\spad{'s} between numerator and denominator will be cancelled during all operations.")) (|canonical| ((|attribute|) "\\spad{canonical} means that equal elements are in fact identical."))) 
((-4492 -12 (|has| |#1| (-6 -4503)) (|has| |#1| (-447)) (|has| |#1| (-6 -4492))) (-4497 . T) (-4503 . T) (-4498 . T) ((-4507 "*") . T) (-4499 . T) (-4500 . T) (-4502 . T)) 
((|HasCategory| |#1| (QUOTE (-896))) (|HasCategory| |#1| (LIST (QUOTE -1029) (QUOTE (-1153)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-1013))) (|HasCategory| |#1| (QUOTE (-807))) (|HasCategory| |#1| (QUOTE (-1128))) (|HasCategory| |#1| (LIST (QUOTE -873) (QUOTE (-375)))) (|HasCategory| |#1| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-375))))) (|HasCategory| |#1| (QUOTE (-221))) (|HasCategory| |#1| (LIST (QUOTE -887) (QUOTE (-1153)))) (|HasCategory| |#1| (LIST (QUOTE -515) (QUOTE (-1153)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -298) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -276) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-296))) (|HasCategory| |#1| (QUOTE (-542))) (-12 (|HasCategory| |#1| (QUOTE (-542))) (|HasCategory| |#1| (QUOTE (-815)))) (-12 (|HasAttribute| |#1| (QUOTE -4503)) (|HasAttribute| |#1| (QUOTE -4492)) (|HasCategory| |#1| (QUOTE (-447)))) (|HasCategory| |#1| (LIST (QUOTE -601) (QUOTE (-533)))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -601) (QUOTE (-533)))) (-12 (|HasCategory| |#1| (QUOTE (-542))) (|HasCategory| |#1| (QUOTE (-815))))) (|HasCategory| |#1| (QUOTE (-834))) (-3322 (|HasCategory| |#1| (QUOTE (-807))) (|HasCategory| |#1| (QUOTE (-834)))) (|HasCategory| |#1| (LIST (QUOTE -1029) (QUOTE (-560)))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -1029) (QUOTE (-560)))) (-12 (|HasCategory| |#1| (QUOTE (-542))) (|HasCategory| |#1| (QUOTE (-815))))) (|HasCategory| |#1| (LIST (QUOTE -873) (QUOTE (-560)))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -873) (QUOTE (-560)))) (-12 (|HasCategory| |#1| (QUOTE (-542))) (|HasCategory| |#1| (QUOTE (-815))))) (|HasCategory| |#1| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-560))))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -601) (LIST (QUOTE -879) (QUOTE (-560))))) (-12 (|HasCategory| |#1| (QUOTE (-542))) (|HasCategory| |#1| (QUOTE (-815))))) (|HasCategory| |#1| (LIST (QUOTE -622) (QUOTE (-560)))) (-3322 (|HasCategory| |#1| (LIST (QUOTE -622) (QUOTE (-560)))) (-12 (|HasCategory| |#1| (QUOTE (-542))) (|HasCategory| |#1| (QUOTE (-815))))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-896)))) (-3322 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-896)))) (|HasCategory| |#1| (QUOTE (-146))))) 
(-404 S R UP) 
((|constructor| (NIL "A \\spadtype{FramedAlgebra} is a \\spadtype{FiniteRankAlgebra} together with a fixed \\spad{R}-module basis.")) (|regularRepresentation| (((|Matrix| |#2|) $) "\\spad{regularRepresentation(a)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the fixed basis.")) (|discriminant| ((|#2|) "\\spad{discriminant()} = determinant(traceMatrix()).")) (|traceMatrix| (((|Matrix| |#2|)) "\\spad{traceMatrix()} is the \\spad{n}-by-\\spad{n} matrix ( \\spad{Tr(\\spad{vi} * vj)} ),{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed basis.")) (|convert| (($ (|Vector| |#2|)) "\\spad{convert([a1,{}..,{}an])} returns \\spad{a1*v1 + ... + a