Numerical solution of multivariate polynomial systems by homotopy continuation methods. (English) Zbl 0886.65054

Li, T. Y.

Iserles, A. (ed.), Acta Numerica Vol. 6, 1997. Cambridge: Cambridge University Press. 399-436 (1997).

Let \(P(x)= 0\) be a system of \(n\) polynomial equations with \(x\in\mathbb{R}^n\). Two different problems are considered:
Problem A: Solve the system of equations \(P(x) =0\).
Problem B: For each of several different choices of coefficients \(c\), solve the system of equations \(P(c, x)= 0\).
The author divides the discussion on dealing with and eliminating extraneous paths for Problem A and for Problem B.
Furthermore, an algorithm which, in some sense, uses the method for Problem B to treat Problem A, is presented. Some numerical considerations, the use of the projective coordinates and real homotopies are given.
For the entire collection see [Zbl 0868.00024].

Reviewer: J.Guddat (Berlin)

MSC:

65H10	Numerical computation of solutions to systems of equations
65H20	Global methods, including homotopy approaches to the numerical solution of nonlinear equations
12Y05	Computational aspects of field theory and polynomials (MSC2010)
26C10	Real polynomials: location of zeros

Keywords:

multivariate polynomial systems; homotopy continuation methods; numerical examples