Two‐step Ulm–Chebyshev‐like method for inverse singular value problems

Abstract

In this article, a two‐step Ulm–Chebyshev‐like method is proposed for solving inverse singular value problems. Under some mild assumptions, we prove that the proposed method converges cubically. Numerical implementations demonstrate the effectiveness of the new method.

Two-step Ulm-Chebyshev-like method for inverse singular value problems with multiple singular values

“…To our knowledge, the two-step Ulm-Chebyshev-like methods for solving the ISVPs (1.4) with multiple and positive singular values have not been explored. In this paper, the convergence of two-step Ulm-Chebyshev-like method for solving the inverse singular value problems with multiple and positive singular values is obtained, which extends the result of Ma (2022) [1]. Under the some non-singularity assumption used in [25], We show that the new method is cubically convergent even when multiple eigenvalues are given.…”

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In this article, when the given singular values are positive and multiple, we study the convergence of two-step Ulm-Chebyshev-like method for solving the inverse singular value problems is obtained, which extends the result of Ma ( 2022) [1]. We show that the new method is cubically convergent.

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“…a two-step inexact Newton-like method [23] for solving ISVPs with super quadratical convergence rate has been proposed. Recently, a two-step Ulm-Chebyshev-like method [1] for solving ISVPs with cubical convergence rate has been proposed. Without the distinction of the given singular values.…”

“…By using the tool of the relative generalized Jacobian of the singular value function [24], Shen, Li, Jin and Yao [25] studied the Newton-type method and inexact Newton-type method, which was extended to the multiple case and established a quadratical convergence result. Natural question is whether the two-step Ulm-Chebyshev-like method [1] and its convergence results can be extended to the multiple case. To our knowledge, the two-step Ulm-Chebyshev-like methods for solving the ISVPs (1.4) with multiple and positive singular values have not been explored.…”

“…Any n-dimensional vector c * satisfying (1.4) is called a solution of the ISVPs. In particular, when s = 1, the ISVPs mentioned above has been simplified to the distinct case, which considered in [1,16,[18][19][20][21][22]. In this section, we first start with the inexact Newton-type method for solving the ISVP (1.4) satisfying the singular values given in (2.1).…”