Numerical Homotopies to Compute Generic Points on Positive Dimensional Algebraic Sets
Source: Journal of Complexity, Volume 16, Number 3, September 2000 , pp. 572-602(31)
Publisher: Academic Press
Abstract:Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to various polynomial systems, such as the cyclic n-roots problem. Copyright 2000 Academic Press.
Document Type: Research Article
Affiliations: 1: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana, 46556 2: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan (M/C 249), Chicago, Illinois, 60607
Publication date: September 2000