Extended nonsymmetric global Lanczos method for matrix function approximation

Abstract

Extended Krylov subspace methods are attractive methods for computing approximations of matrix functions and other problems producing large-scale matrices. In this work, we propose the extended nonsymmetric global Lanczos method for solving some matrix approximation problems. The derived algorithm uses short recursive relations to generate bi-orthonormal bases, with respect to the Frobenius inner product, of the corresponding extended Krylov subspaces ${K^{e}_{m}}(A,V)$ and ${K^{e}_{m}}(A^{T},W)$ . Here, A is a large nonsymmetric matrix; V and $W\in \mathbb {R}^{n\times s}$ are two blocks. New algebraic properties of the proposed method are developed and applications to approximation of both WTf(A)V and trace(WTf(A)V ) are given. Numerical examples are presented to show the performance of the extended nonsymmetric global Lanczos for these problems.