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-- also see comodule-doc.m2
-- also see packages/Saturation/quotient-doc.m2
document { Key => quotient, Headline => "quotient or division" }
document { Key => {(quotient, Matrix, GroebnerBasis), (quotient, Matrix, Matrix)},
Headline => "matrix quotient",
Usage => "q = quotient(f,g)",
Inputs => { "f" => Matrix, "g" => {ofClass{GroebnerBasis,Matrix}, ", with the same target as ", TT "f"}},
Outputs => {
"q" => {"the quotient of ", TT "f", " upon division by ", TT "g"}
},
"The equation ", TT "g*q+r == f", " will hold, where ", TT "r", " is the map provided by ", TO "remainder", ". The source of ", TT "f", " should be a free module.",
EXAMPLE lines ///
R = ZZ[x,y]
f = random(R^2,R^{2:-1})
g = vars R ++ vars R
quotient(f,g)
f = f + map(target f, source f, id_(R^2))
quotient(f,g)
///,
SeeAlso => {quotientRemainder,quotient'}
}
document { Key => {quotient',(quotient', Matrix, Matrix)},
Headline => "matrix quotient (opposite)",
Usage => "q = quotient'(f,g)",
Inputs => { "f" => Matrix, "g" => {ofClass{GroebnerBasis,Matrix}, ", with the same source as ", TT "f"}},
Outputs => {
"q" => {"the quotient of ", TT "f", " upon (opposite) division by ", TT "g"},
},
"The equation ", TT "q*g+r == f", " will hold, where ", TT "r", " is the map provided by ", TO "remainder'", " . The sources and targets of the maps should be free modules.
This function is obtained from ", TT "quotient", " by transposing the inputs and outputs.",
EXAMPLE lines ///
R = ZZ[x,y]
f = random(R^{2:1},R^2)
g = transpose (vars R ++ vars R)
quotient'(f,g)
f = f + map(target f, source f, id_(R^2))
quotient'(f,g)
///,
SeeAlso => {quotientRemainder,quotient},
SourceCode => {quotient'}
}
document { Key => {quotientRemainder,(quotientRemainder, Matrix, GroebnerBasis), (quotientRemainder, Matrix, Matrix)},
Headline => "matrix quotient and remainder",
Usage => "(q,r) = quotientRemainder(f,g)",
Inputs => { "f" => Matrix, "g" => {ofClass{GroebnerBasis,Matrix}, ", with the same target as ", TT "f"}},
Outputs => {
"q" => {"the quotient of ", TT "f", " upon division by ", TT "g"},
"r" => {"the remainder of ", TT "f", " upon division by ", TT "g"}
},
"The equation ", TT "g*q+r == f", " will hold. The source of ", TT "f", " should be a free module.",
EXAMPLE lines ///
R = ZZ[x,y]
f = random(R^2,R^{2:-1})
g = vars R ++ vars R
(q,r) = quotientRemainder(f,g)
g*q+r == f
f = f + map(target f, source f, id_(R^2))
(q,r) = quotientRemainder(f,g)
g*q+r == f
///,
SeeAlso => {quotientRemainder'}
}
document { Key => {quotientRemainder',(quotientRemainder', Matrix, Matrix)},
Headline => "matrix quotient and remainder (opposite)",
Usage => "(q,r) = quotientRemainder'(f,g)",
Inputs => { "f" => Matrix, "g" => {ofClass{GroebnerBasis,Matrix}, ", with the same source as ", TT "f"}},
Outputs => {
"q" => {"the quotient of ", TT "f", " upon (opposite) division by ", TT "g"},
"r" => {"the remainder of ", TT "f", " upon (opposite) division by ", TT "g"}
},
"The equation ", TT "q*g+r == f", " will hold. The sources and targets of the maps should be free modules.
This function is obtained from ", TT "quotientRemainder", " by transposing the inputs and outputs.",
EXAMPLE lines ///
R = ZZ[x,y]
f = random(R^{2:1},R^2)
g = transpose (vars R ++ vars R)
(q,r) = quotientRemainder'(f,g)
q*g+r == f
f = f + map(target f, source f, id_(R^2))
(q,r) = quotientRemainder'(f,g)
q*g+r == f
///,
SeeAlso => {quotientRemainder},
SourceCode => {quotientRemainder'}
}
document { Key => {remainder,(remainder, Matrix, GroebnerBasis), (remainder, Matrix, Matrix)},
Headline => "matrix remainder",
Usage => "r = remainder(f,g)",
Inputs => { "f" => Matrix, "g" => {ofClass{GroebnerBasis,Matrix}, ", with the same target as ", TT "f"}},
Outputs => {
"r" => {"the remainder of ", TT "f", " upon division by ", TT "g"}
},
PARA{"This operation is the same as ", TO (symbol %, Matrix, GroebnerBasis), "."},
PARA{"The equation ", TT "g*q+r == f", " will hold, where ", TT "q", " is the map provided by ", TO "quotient", ". The source of ", TT "f", " should be a free module."},
EXAMPLE lines ///
R = ZZ[x,y]
f = random(R^2,R^{2:-1})
g = vars R ++ vars R
remainder(f,g)
f = f + map(target f, source f, id_(R^2))
remainder(f,g)
///,
SeeAlso => {quotientRemainder,remainder'}
}
document { Key => {remainder',(remainder', Matrix, Matrix)},
Headline => "matrix quotient and remainder (opposite)",
Usage => "r = remainder'(f,g)",
Inputs => { "f" => Matrix, "g" => {ofClass{GroebnerBasis,Matrix}, ", with the same source as ", TT "f"}},
Outputs => {
"r" => {"the remainder of ", TT "f", " upon (opposite) division by ", TT "g"}
},
"The equation ", TT "q*g+r == f", " will hold, where ", TT "q", " is the map provided by ", TO "quotient'", ". The sources and targets of the maps should be free modules.
This function is obtained from ", TT "remainder", " by transposing the inputs and outputs.",
EXAMPLE lines ///
R = ZZ[x,y]
f = random(R^{2:1},R^2)
g = transpose (vars R ++ vars R)
remainder'(f,g)
f = f + map(target f, source f, id_(R^2))
remainder'(f,g)
///,
SeeAlso => {quotientRemainder,remainder},
SourceCode => {remainder'}
}
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