Solution of Systems of Polynomial Equations
Given a system of equations of rational functions with exact coefficients
p1(x1,...,xn)
.
.
pm(x1,...,xn)
Axiom can find numeric or symbolic solutions. The system is first split
into irreducible components, then for each component, a triangular system
of equations is found that reduces the problem to sequential solutions of
univariate polynomials resulting from substitution of partial solutions
from the previous stage.
q1(x1,...,xn)
.
.
qm(xn)
Symbolic solutions can be presented using "implicit" algebraic numbers
defined as roots of irreducible polynomials or in terms of radicals. Axiom
can also find approximations to the real or complex roots of a system of
polynomial equations to any user specified accuracy.
The operation solve for systems is used in
a way similar to solve for single equations.
Instead of a polynomial equation, one has to give a list of equations and
instead of a single variable to solve for, a list of variables. For
solutions of single equations see
Solution of a Single Polynomial Equation
Use the operation solve if you want
implicitly presented solutions.
Use radicalSolve if you want your
solutions expressed in terms of radicals.
To get numeric solutions you only need to give the list of equations and
the precision desired. The list of variables would be redundant information
since there can be no parameters for the numerical solver.
If the precision is expressed as a floating point number you get results
expressed as floats.
To get complex numeric solutions, use the operation
complexSolve, which takes the same
arguments as in the real case.
It is also possible to solve systems of equations in rational functions
over the rational numbers. Note that [x=0.0,a=0.0] is not returned as
a solution since the denominator vanishes there.
When solving equations with denominators, all solutions where the
denominator vanishes are discarded.