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Symbolic solvers
*INTRO By solving one tries to find a mathematical object that meets certain criteria. This chapter documents the functions that are available to help find solutions to specific types of problems.
*CMD Solve  solve an equation
*STD
*CALL
Solve(eq, var)
*PARMS
{eq}  equation to solve
{var}  variable to solve for
*DESC
This command tries to solve an equation. If {eq} does not contain the
{==} operator, it is assumed that the user wants to solve $eq ==
0$. The result is a list of equations of the form {var == value}, each
representing a solution of the given equation. The {Where} operator
can be used to substitute this solution in another expression. If the
given equation {eq} does not have any solutions, or if {Solve} is
unable to find any, then an empty list is returned.
The current implementation is far from perfect. In particular, the
user should keep the following points in mind:
* {Solve} cannot solve all equations. If it is given a equation
it can not solve, it raises an error via {Check}. Unfortunately, this
is not displayed by the inline prettyprinter; call {PrettyPrinter'Set} to
change this. If an equation cannot be solved analytically, you may
want to call {Newton} to get a numerical solution.
* Systems of equations are not handled yet. For linear systems,
{MatrixSolve} can be used. The old version of {Solve}, with the name
{OldSolve} might be able to solve nonlinear systems of equations.
* The periodicity of the trigonometric functions {Sin}, {Cos},
and {Tan} is not taken into account. The same goes for the (imaginary)
periodicity of {Exp}. This causes {Solve} to miss solutions.
* It is assumed that all denominators are nonzero. Hence, a
solution reported by {Solve} may in fact fail to be a solution because
a denominator vanishes.
* In general, it is wise not to have blind trust in the results
returned by {Solve}. A good strategy is to substitute the solutions
back in the equation.
*E.G. notest
First a simple example, where everything works as it should. The
quadratic equation $x^2 + x == 0$ is solved. Then the result is
checked by substituting it back in the quadratic.
In> quadratic := x^2+x;
Out> x^2+x;
In> Solve(quadratic, x);
Out> {x==0,x==(1)};
In> quadratic Where %;
Out> {0,0};
If one tries to solve the equation $Exp(x) == Sin(x)$, one finds that
{Solve} can not do this.
In> PrettyPrinter'Set("DefaultPrint");
Out> True;
In> Solve(Exp(x) == Sin(x), x);
Error: Solve'Fails: cannot solve equation Exp(x)Sin(x) for x
Out> {};
The equation $Cos(x) == 1/2$ has an infinite number of solutions,
namely $x == (2*k + 1/3) * Pi$ and $x == (2*k  1/3) * Pi$ for any
integer $k$. However, {Solve} only reports the solutions with $k == 0$.
In> Solve(Cos(x) == 1/2, x);
Out> {x==Pi/3,x== Pi/3};
For the equation $x/Sin(x) == 0$, a spurious solution at $x == 0$ is
returned. However, the fraction is undefined at that point.
In> Solve(x / Sin(x) == 0, x);
Out> {x==0};
At first sight, the equation $Sqrt(x) == a$ seems to have the solution
$x == a^2$. However, this is not true for eg. $a == 1$.
In> PrettyPrinter'Set("DefaultPrint");
Out> True;
In> Solve(Sqrt(x) == a, x);
Error: Solve'Fails: cannot solve equation Sqrt(x)a for x
Out> {};
In> Solve(Sqrt(x) == 2, x);
Out> {x==4};
In> Solve(Sqrt(x) == 1, x);
Out> {};
*SEE Check, MatrixSolve, Newton, OldSolve, PrettyPrinter'Set, PSolve, Where, ==
*CMD OldSolve  old version of {Solve}
*STD
*CALL
OldSolve(eq, var)
OldSolve(eqlist, varlist)
*PARMS
{eq}  single identity equation
{var}  single variable
{eqlist}  list of identity equations
{varlist}  list of variables
*DESC
This is an older version of {Solve}. It is retained for two
reasons. The first one is philosophical: it is good to have multiple
algorithms available. The second reason is more practical: the newer
version cannot handle systems of equations, but {OldSolve} can.
This command tries to solve one or more equations. Use the first form
to solve a single equation and the second one for systems of
equations.
The first calling sequence solves the equation "eq" for the variable
"var". Use the {==} operator to form the equation.
The value of "var" which satisfies the equation, is returned. Note
that only one solution is found and returned.
To solve a system of equations, the second form should be used. It
solves the system of equations contained in the list "eqlist" for
the variables appearing in the list "varlist". A list of results is
returned, and each result is a list containing the values of the
variables in "varlist". Again, at most a single solution is
returned.
The task of solving a single equation is simply delegated to {SuchThat}. Multiple equations are solved recursively:
firstly, an equation is sought in which one of the variables occurs
exactly once; then this equation is solved with {SuchThat}; and finally the solution is substituted in the
other equations by {Eliminate} decreasing the number
of equations by one. This suffices for all linear equations and a
large group of simple nonlinear equations.
*E.G.
In> OldSolve(a+x*y==z,x)
Out> (za)/y;
In> OldSolve({a*x+y==0,x+z==0},{x,y})
Out> {{z,z*a}};
This means that "x = (za)/y" is a solution of the first equation
and that "x = z", "y = z*a" is a solution of the systems of
equations in the second command.
An example which {OldSolve} cannot solve:
In> OldSolve({x^2x == y^2y,x^2x == y^3+y},{x,y});
Out> {};
*SEE Solve, SuchThat, Eliminate, PSolve, ==
*CMD SuchThat  special purpose solver
*STD
*CALL
SuchThat(expr, var)
*PARMS
{expr}  expression to make zero
{var}  variable (or subexpression) to solve for
*DESC
This functions tries to find a value of the variable "var" which
makes the expression "expr" zero. It is also possible to pass a
subexpression as "var", in which case {SuchThat}
will try to solve for that subexpression.
Basically, only expressions in which "var" occurs only once are
handled; in fact, {SuchThat} may even give wrong
results if the variables occurs more than once. This is a consequence
of the implementation, which repeatedly applies the inverse of the top
function until the variable "var" is reached.
*E.G.
In> SuchThat(a+b*x, x)
Out> (a)/b;
In> SuchThat(Cos(a)+Cos(b)^2, Cos(b))
Out> Cos(a)^(1/2);
In> A:=Expand(a*x+b*x+c, x)
Out> (a+b)*x+c;
In> SuchThat(A, x)
Out> (c)/(a+b);
*SEE Solve, OldSolve, Subst, Simplify
*CMD Eliminate  substitute and simplify
*STD
*CALL
Eliminate(var, value, expr)
*PARMS
{var}  variable (or subexpression) to substitute
{value}  new value of "var"
{expr}  expression in which the substitution should take place
*DESC
This function uses {Subst} to replace all instances
of the variable (or subexpression) "var" in the expression "expr"
with "value", calls {Simplify} to simplify the
resulting expression, and returns the result.
*E.G.
In> Subst(Cos(b), c) (Sin(a)+Cos(b)^2/c)
Out> Sin(a)+c^2/c;
In> Eliminate(Cos(b), c, Sin(a)+Cos(b)^2/c)
Out> Sin(a)+c;
*SEE SuchThat, Subst, Simplify
*CMD PSolve  solve a polynomial equation
*STD
*CALL
PSolve(poly, var)
*PARMS
{poly}  a polynomial in "var"
{var}  a variable
*DESC
This commands returns a list containing the roots of "poly",
considered as a polynomial in the variable "var". If there is only
one root, it is not returned as a oneentry list but just by
itself. A double root occurs twice in the result, and similarly for
roots of higher multiplicity. All polynomials of degree up to 4 are
handled.
*E.G.
In> PSolve(b*x+a,x)
Out> a/b;
In> PSolve(c*x^2+b*x+a,x)
Out> {(Sqrt(b^24*c*a)b)/(2*c),((b+
Sqrt(b^24*c*a)))/(2*c)};
*SEE Solve, Factor
*CMD MatrixSolve  solve a system of equations
*STD
*CALL
MatrixSolve(A,b)
*PARMS
{A}  coefficient matrix
{b}  row vector
*DESC
{MatrixSolve} solves the matrix equations {A*x = b} using Gaussian Elimination
with Backward substitution. If your matrix is triangular or diagonal, it will
be recognized as such and a faster algorithm will be used.
*E.G.
In> A:={{2,4,2,2},{1,2,4,3},{3,3,8,2},{1,1,6,3}};
Out> {{2,4,2,2},{1,2,4,3},{3,3,8,2},{1,1,6,3}};
In> b:={4,5,7,7};
Out> {4,5,7,7};
In> MatrixSolve(A,b);
Out> {1,2,3,4};
Numeric solvers
*CMD Newton  solve an equation numerically with Newton's method
*STD
*CALL
Newton(expr, var, initial, accuracy)
Newton(expr, var, initial, accuracy,min,max)
*PARMS
{expr}  an expression to find a zero for
{var}  free variable to adjust to find a zero
{initial}  initial value for "var" to use in the search
{accuracy}  minimum required accuracy of the result
{min}  minimum value for "var" to use in the search
{max}  maximum value for "var" to use in the search
*DESC
This function tries to numerically find a zero of the expression
{expr}, which should depend only on the variable {var}. It uses
the value {initial} as an initial guess.
The function will iterate using Newton's method until it estimates
that it has come within a distance {accuracy} of the correct
solution, and then it will return its best guess. In particular, it
may loop forever if the algorithm does not converge.
When {min} and {max} are supplied, the Newton iteration takes them
into account by returning {Fail} if it failed to find a root in
the given range. Note this doesn't mean there isn't a root, just
that this algorithm failed to find it due to the trial values
going outside of the bounds.
*E.G.
In> Newton(Sin(x),x,3,0.0001)
Out> 3.1415926535;
In> Newton(x^21,x,2,0.0001,5,5)
Out> 1;
In> Newton(x^2+1,x,2,0.0001,5,5)
Out> Fail;
*SEE Solve, NewtonNum
*CMD FindRealRoots  find the real roots of a polynomial
*STD
*CALL
FindRealRoots(p)
*PARMS
{p}  a polynomial in {x}
*DESC
Return a list with the real roots of $ p $. It tries to find the realvalued
roots, and thus requires numeric floating point calculations. The precision
of the result can be improved by increasing the calculation precision.
*E.G. notest
In> p:=Expand((x+3.1)^5*(x6.23))
Out> x^6+9.27*x^50.465*x^4300.793*x^3
1394.2188*x^22590.476405*x1783.5961073;
In> FindRealRoots(p)
Out> {3.1,6.23};
*SEE SquareFree, NumRealRoots, MinimumBound, MaximumBound, Factor
*CMD NumRealRoots  return the number of real roots of a polynomial
*STD
*CALL
NumRealRoots(p)
*PARMS
{p}  a polynomial in {x}
*DESC
Returns the number of real roots of a polynomial $ p $.
The polynomial must use the variable {x} and no other variables.
*E.G.
In> NumRealRoots(x^21)
Out> 2;
In> NumRealRoots(x^2+1)
Out> 0;
*SEE FindRealRoots, SquareFree, MinimumBound, MaximumBound, Factor
*CMD MinimumBound  return lower bounds on the absolute values of real roots of a polynomial
*CMD MaximumBound  return upper bounds on the absolute values of real roots of a polynomial
*STD
*CALL
MinimumBound(p)
MaximumBound(p)
*PARMS
{p}  a polynomial in $x$
*DESC
Return minimum and maximum bounds for the absolute values of the real
roots of a polynomial {p}. The polynomial has to be converted to one with
rational coefficients first, and be made squarefree.
The polynomial must use the variable {x}.
*E.G.
In> p:=SquareFree(Rationalize((x3.1)*(x+6.23)))
Out> (40000*x^2125200*x+772520)/870489;
In> MinimumBound(p)
Out> 5000000000/2275491039;
In> N(%)
Out> 2.1973279236;
In> MaximumBound(p)
Out> 10986639613/1250000000;
In> N(%)
Out> 8.7893116904;
*SEE SquareFree, NumRealRoots, FindRealRoots, Factor
*INCLUDE logic.chapt
*INCLUDE ode.chapt
