On the convergence of Broyden's method and some accelerated schemes for singular problems

TL;DR

This paper analyzes Broyden's method and accelerated schemes for singular problems, demonstrating enlarged convergence domains, q-linear convergence of matrix updates, and the violation of uniform linear independence, supported by numerical experiments.

Contribution

It extends the understanding of Broyden's method convergence for singular problems, showing q-linear convergence of matrices and the impact of initial steps, with numerical validation.

Findings

Enlarged domain of convergence with Newton-like steps

Matrix updates of Broyden's method converge q-linearly

Broyden directions violate uniform linear independence

Abstract

We consider Broyden's method and some accelerated schemes for nonlinear equations having a strongly regular singularity of first order with a one-dimensional nullspace. Our two main results are as follows. First, we show that the use of a preceding Newton-like step ensures convergence for starting points in a starlike domain with density 1. This extends the domain of convergence of these methods significantly. Second, we establish that the matrix updates of Broyden's method converge q-linearly with the same asymptotic factor as the iterates. This contributes to the long-standing question whether the Broyden matrices converge by showing that this is indeed the case for the setting at hand. Furthermore, we prove that the Broyden directions violate uniform linear independence, which implies that existing results for convergence of the Broyden matrices cannot be applied. Numerical experiments of high precision confirm the enlarged domain of convergence, the q-linear convergence of the matrix updates, and the lack of uniform linear independence. In addition, they suggest that these results can be extended to singularities of higher order and that Broyden's method can converge r-linearly without converging q-linearly. The underlying code is freely available.