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

Abstract

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 c*, 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.

Formulations and theorems of quadratically convergent methods for inverse symmetric eigenvalue problems

“…in the same manner as [24,[40][41][42]. It is easy to generalize the following discussion to an arbitrary set of prescribed eigenvalues.…”

“…are always assumed for the proof of the quadratic convergence also in recent papers [40][41][42]. The basic theorem is as follows.…”

“…From the mathematical view points, a geometric interpretation is introduced in [12,14,17]. To reduce the computational cost, [7,8,40] introduce some inexact solvers in line 4.…”

“…Since they are a kind of Newton's methods, a linear system for the Jacobi matrix must be solved per iteration. Thus, inexact linear system solvers are successfully applied as in [7,8,10,40]. The Ulm-like method can also be applied as in [39].…”

An extension of the Cayley transform method for a parameterized generalized inverse eigenvalue problem

“…The case of multiple eigenvalues is more subtle, but was already addressed in Reference 11 with appropriate modifications of the three methods. Some of the above methods have also been adapted for the case of multiple eigenvalues: the inexact Cayley transform method in Reference 17, the Ulm‐like Cayley transform method in Reference 18, and Aishima's method in Reference 19.…”