Convergence and preconditioning of inexact inverse subspace iteration for generalized eigenvalue problems (Q1985904)

Let \(A-\lambda B\) be regular matrix pencil (\(\det (A-\lambda B)\neq 0\)), where \(A\) and \(B\) are large and sparse \(n \times n\) matrices, and let \(X\in C^{n \times p}\), \(p\ll n\). The subspace range\((X)\) is a deflating subspace for \(A-\lambda B\) if there exists \(L\in C^{p \times p}\) such that \(AX-BXL=0\). This paper focuses on the inner iteration that arises in the inexact inverse subspace iteration for computing a small deflating subspace of a large matrix pencil. First, it is shown that the method achieves linear rate of convergence if the inner iteration is performed with increasing accuracy (see, Theorem 1 and Algorithm 1). In Section 3 the authors discussed the use of block-GMRES as inner iteration of Algorithm 1. The investigations of this paper generalize well-known results by \textit{M. Robbé} et al. [SIAM J. Matrix Anal. Appl. 31, No. 1, 92--113 (2009; Zbl 1269.65036)]. In particular, it is shown that the preconditioners help to maintain the number of iterations needed by block-GMRES to approximately small constant. Some numerical tests are presented in Section 4.