Volume 50, pp. 144-163, 2018.
The extended global Lanczos method for matrix function approximation
A. H. Bentbib, M. El Ghomari, C. Jagels, K. Jbilou, and L. Reichel
Abstract
The need to compute the trace of a large matrix that is not explicitly known, such as the
matrix , where 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 , can give faster convergence than the standard Lanczos method,
which uses subspaces with nonnegative powers of 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.
Full Text (PDF) [322 KB],
BibTeX
, DOI: 10.1553/etna_vol50s144
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
Links to the cited ETNA articles
