arXiv subjects

Tom Werner

Publications and source records attributed to Tom Werner.

3 recordsLinked to original sources

arXiv2025

nlKrylov: A Unified Framework for Nonlinear GCR-type Krylov Subspace Methods

In this paper, we introduce a unified framework for nonlinear Krylov subspace methods (\textit{nlKrylov}) to solve systems of nonlinear equations. Building on classical GCR-like/type linear Krylov solvers such as GMRESR, we generalize these approaches to nonlinear problems via nested algorithmic structures. We present rigorous convergence results for problems, relying on relaxed assumptions that avoid the need for exact line searches. The framework is further extended to matrix-valued root finding problems using global nonlinear Krylov approaches. Extensive numerical experiments validate the theoretical insights and demonstrate the robustness and efficiency of our proposed algorithms.

math.NA↗

arXiv2023

An inexact Matrix-Newton method for solving NEPv

In this paper, an inexact Newton method for solving real-valued nonlinear eigenvalue problems with eigenvector dependency (NEPv) is introduced that is able to solve the problem on a matrix level. Our main contribution is to derive a variant of Newton's method that uses global Krylov methods such as global GMRES to solve the linear operator equation necessary to compute the Newton correction in a matrix-free way. The advantages that this second order method has over the well-established SCF algorithm are explained and visualized by a variety of numerical experiments.

math.NA↗

arXiv2021

On using the complex step method for the approximation of Fr\'echet derivatives of matrix functions in automorphism groups

We show, that the Complex Step approximation to the Fr\'echet derivative of real matrix functions is applicable to the matrix sign, square root and polar mapping using iterative schemes. While this property was already discovered for the matrix sign using Newton's method, we extend the research to the family of Pad\'e iterations, that allows us to introduce iterative schemes for finding function and derivative values while approximately preserving automorphism group structure.

math.NA↗