Solvers for Systems of Nonlinear Algebraic Equations- Their Sensitivity to Starting Functions
by
Anna Kucaba-Pietal
Technical University of Rzeszow
Coauthors: Deborah Dent (Uniiversity of Southern Mississippi), Marcin Paprzycki (University of Southern Mississippi)

Recently we observe a growing interest in finding solutions to engineering problems resulting in large systems of nonlinear algebraic equations. While the mathematical theory and computational practice are well established when a system of linear algebraic equations or a single nonlinear equation is to be solved this is clearly not the case for systems of nonlinear algebraic equations. Our current research has shown both lack of library of solvers and a standardized set of test problems (different researchers use different test problems with only a minimal overlap). In this context we have to remember that, until recently, in the engineering practice, only systems with relatively few equations have been solved. This explains one of the problems of existing "popular" test cases. Most of them have a very small number of equations (2-10) and only very few are defined so that they can reach 100 equations.

In our earlier work we have reported on our efforts to collect most of the existing solvers and apply them to a large collection of popular test problems. We were able to locate solvers based on: Newton's method and its modifications, bisection, continuation, hybrid methods, homotopy, tensor method and applied them to 22 test problems collected from the literature and the Internet. The results of these tests allowed us to observe that proper choice of the starting vector is crucial to the success of the solution process (bad selection often result in lack of solution). Because the sensitivity of the starting vector depends on the solution method and the problem to be solved, we decided to experimentally compare the behavior of all of the solvers for each of the test problems using a perturbed solution as the starting vector. We have tested the case when all solution components have been moved away from the solution and when each solution component has been changed separately, while the remaining components remain unchanged (equal to the computed solution). Based on these experiments we will discuss the nature of the difficulty of each test problem and the reason for given solvers' success or failure to find the solution.

Date received: February 1, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeb-87.