Weiping Shen

An inexact Cayley transform method for inverse eigenvalue problems with multiple eigenvalues

We consider the convergence problem of an inexact Cayley transform method for solving inverse eigenvalue problems with multiple eigenvalues. Under the nonsingularity assumption of the relative generalized Jacobian matrix at the solution , a convergence analysis covering both the distinct and multiple eigenvalues cases is provided and the superlinear convergence is proved. Moreover, numerical experiments are given in the last section and comparisons with the Cayley transform method are made.

Convergence Analysis of Newton-Like Methods for Inverse Eigenvalue Problems with Multiple Eigenvalues

We provide in the present paper a corrected proof for the classical quadratical convergence theorem (i.e., Theorem 3.3 in Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634--667]) of the Newton-like method for solving inverse eigenvalue problems with possible multiple eigenvalues. Moreover, as a by-product, our approach developed here can be extended to establish a similar convergence result for an inexact version of the Newton-like method with possible multiple eigenvalues, which is an extension of the corresponding inexact Newton-like method for the distinct case in Chan, Chung, and Xu [BIT Numer. Math., 43 (2003), pp. 7--20].

Convergence of a Ulm-like method for square inverse singular value problems with multiple and zero singular values

An interesting problem was raised in Vong et al. (SIAM J. Matrix Anal. Appl. 32:412–429, 2011): whether the Ulm-like method and its convergence result can be extended to the cases of multiple and zero singular values. In this paper, we study the convergence of a Ulm-like method for solving the square inverse singular value problem with multiple and zero singular values. Under the nonsingularity assumption in terms of the relative generalized Jacobian matrices, a convergence analysis for the multiple and zero case is provided and the quadratical convergence property is proved. Moreover, numerical experiments are given in the last section to demonstrate our theoretic results.