Verified error bounds for multiple roots of systems of nonlinear equations

Abstract

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.

A new deflation method for verifying the isolated singular zeros of polynomial systems

“…In [23], by introducing a smoothing parameter to the original system and n − 1 new variables to the Jacobian matrix of the original system, which produces new equations, Rump and Graillat consider the case of the double zero of the original system. In [15], based on the parameterized multiplicity structure, Li and Zhi generalize the algorithm in [23] to deflate the breath-one isolated singular zero of the original system. Their final deflated regular system is of size µn × µn.…”

On isolation of simple multiple zeros and clusters of zeros of polynomial systems

“…The proof of the non-quantified quadratic convergence [24,Theorem 3.16] of Algorithm 1 in [24] has also been simplified. There are other approaches to compute isolated multiple zeros or zero clusters, e.g., corrected Newton methods [33,5,6,7,14,15,34,35,29], deflation techniques [32,48,31,20,4,21,22,45,3,36,24,27,11,26,18,16]. We refer to [10,16] for excellent introductions of previous works on approximating multiple zeros.…”

Verified Error Bounds for Isolated Singular Solutions of Polynomial Systems

“…In [33], by introducing a smoothing parameter, Rump and Graillat described a verification method for computing guaranteed (real or complex) error bounds such that a slightly perturbed system is proved to have a double root within the computed bounds. In [20], by adding a perturbed univariate polynomial in one selected variable with some smoothing parameters to one selected equation of the original system, we generalized the algorithm in [33] to compute guaranteed error bounds, such that a slightly perturbed system is proved to possess an isolated singular solution whose Jacobian matrix has corank one within the computed bounds.…”