An extended Ulm-like method for inverse singular value problems with multiple and/or zero singular values

Abstract

In this paper, we propose an extended Ulm-like method for solving the inverse singular value problem (ISVP for short) with multiple and/or zero singular values. Compared with the Ulm-like method, the proposed method reduces the amount of calculations and is well-defined even when multiple and/or zero singular values appear. Under the nonsingularity assumption for the relative generalized Jacobian at a solution and by using a new technique, a convergence analysis is provided and the quadratic convergence is proved. Our results in the present paper improve and extend significantly the corresponding ones of Vong et al. (2011) and Shen et al. (2018) for the ISVP with distinct and positive singular values and/or with square matrices. In particular, we solve the interesting problem raised in Vong et al. (2011): whether the Ulm-like method and its convergence result can be extended to the cases of multiple singular values and of zero singular values. Numerical results reveal that the extended Ulm-like method is an efficient algorithm for the ISVPs even when multiple and/or zero singular values appear.