Extended Newton-type method for inverse singular value problems with multiple and/or zero singular values

Abstract

We study the issue of numerically solving inverse singular value problems (ISVPs). Motivated by the Newton-type method introduced in [] for solving ISVPs with distinct and positive singular values, we propose an extended Newton-type method working for ISVPs with multiple and/or zero singular values. Because of the absence of some important and crucial properties, the approach/technique used in the case of distinct and positive singular values no longer works for the case of multiple and/or zero singular values, and we develop a new approach/technique to treat the case of multiple and/or zero singular values. Under the standard nonsingularity assumption of the relative generalized Jacobian matrix at a solution, the quadratic convergence result is established for the extended Newton-type method, and numerical experiments are provided to illustrate the convergence performance of the extended method. Our extended method and convergence result in the present paper improve and extend significantly the corresponding ones in [, , ] for the special cases of distinct and positive singular values and/or of square matrices.