The extended global Lanczos method for matrix function approximation

The need to compute the trace of a large matrix that is not explicitly known, such as the matrix exp(A), where A is a large symmetric matrix, arises in various applications including in network analysis. The global Lanczos method is a block method that can be applied to compute an approximation of the trace. When the block size is one, this method simplifies to the standard Lanczos method. It is known that for some matrix functions and matrices, the extended Lanczos method, which uses subspaces with both positive and negative powers of A, can give faster convergence than the standard Lanczos method, which uses subspaces with nonnegative powers of A only. This suggests that it may be beneficial to use an extended global Lanczos method instead of the (standard) global Lanczos method. This paper describes an extended global Lanczos method and discusses properties of the associated Gauss-Laurent quadrature rules. Computed examples that illustrate the performance of the extended global Lanczos method are presented. Key words. extended Krylov subspace, extended moment matching, Laurent polynomial, global Lanczos method, matrix function, Gauss quadrature rule AMS subject classifications. 65F25, 65F30, 65F60, 33C47