Verified error bounds for multiple roots of systems of nonlinear equations

It is well known that it is an ill-posed problem to decide whether a function has a multiple root. Even for a univariate polynomial an arbitrary small perturbation of a polynomial coefficient may change the answer from yes to no. Let a system of nonlinear equations be given. In this paper we describe an algorithm for computing verified and narrow error bounds with the property that a slightly perturbed system is proved to have a double root within the computed bounds. For a univariate nonlinear function f we give a similar method also for a multiple root. A narrow error bound for the perturbation is computed as well. Computational results for systems with up to 1000 unknowns demonstrate the performance of the methods. Key words. nonlinear equations, double roots, multiple roots, verification, error bounds, INTLAB AMS subject classifications. 65H10, 65G20, 65H05, 65-04

• Type: Journal
• Year: 2010
• Where: NA
• Authors: Siegfried M. Rump, Stef Graillat
• Keywords: Multiple Roots, NA 2010, Narrow Error, Nonlinear Equations