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An exploration of homotopy solving in Maple. (English) Zbl 1047.65033

Li, Ziming (ed.) et al., Computer mathematics. Proceedings of the sixth Asian symposium (ASCM 2003), Beijing, China, April 17–19, 2003. River Edge, NJ: World Scientific (ISBN 981-238-220-8/hbk). Lect. Notes Ser. Comput. 10, 145-162 (2003).
Summary: Homotopy continuation methods find approximate solutions of a given system by a continuous deformation of the solutions of a related exactly solvable system. There has been much recent progress in the theory and implementation of such path following methods for polynomial systems. In particular, exactly solvable related systems can be given which enable the computation of all isolated roots of a given polynomial system.
Extension of such methods to determine manifolds of solutions has also been recently achieved. This progress, and our own research on extending continuation methods to identifying missing constraints for systems of differential equations, motivated us to implement higher-order continuation methods in the computer algebra language Maple. By higher-order, we refer to the iterative scheme used to solve for the roots of the homotopy equation at each step. We provide examples for which the higher-order iterative scheme achieves a speed up when compared with the standard second-order scheme.
We also demonstrate how existing Maple numerical solvers of ordinary differential equations can be used to give a predictor only continuation method for solving polynomial systems. We apply homotopy continuation to determine the missing constraints in a system of nonlinear partial differential equations which is to our knowledge, the first published instance of such a calculation.
For the entire collection see [Zbl 1062.68006].

MSC:

65H10 Numerical computation of solutions to systems of equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
68W30 Symbolic computation and algebraic computation
34A34 Nonlinear ordinary differential equations and systems
12Y05 Computational aspects of field theory and polynomials (MSC2010)
26C10 Real polynomials: location of zeros
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)

Software:

PHCpack; Maple
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