A Cayley-free Two-Step Algorithm for Inverse Singular Value Problems
Jiechang Fan, Weiping Shen, Yusong Luo, Enping Lou
TL;DR
The paper tackles inverse singular value problems by introducing a Cayley-free two-step algorithm that obviates Cayley transforms and the associated solution of $2(m+n)$ linear systems per outer iteration. Under the assumption that the Jacobian at the solution is nonsingular, it proves cubic root-convergence for the iterates toward the target singular values. The method constructs first-order perturbations and uses a two-step Ulm-Chebyshev-like update for the coefficient vector, achieving substantial computational savings while preserving convergence. Numerical experiments on large-scale ISVP instances demonstrate superior efficiency relative to existing Cayley-based and Newton-type approaches, highlighting the method’s practical impact for applications requiring fast, reliable recovery of matrices from prescribed singular spectra.
Abstract
In this paper, we investigate numerical solutions for inverse singular value problems (for short, ISVPs) arising in various applications. Inspired by the methodologies employed for inverse eigenvalue problems, we propose a Cayley-free two-step algorithm for solving the ISVP. Compared to the existing two-step algorithms for the ISVP, our algorithm eliminates the need for Cayley transformations and consequently avoids solving $2(m+n)$ linear systems during the computation of approximate singular vectors at each outer iteration. Under the assumption that the Jacobian matrix at a solution is nonsingular, we present a convergence analysis for the proposed algorithm and prove a cubic root-convergence rate. Numerical experiments are conducted to validate the effectiveness of our algorithm.