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

Abstract

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].