2,391 research outputs found
An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices
The nonsymmetric Lanczos method can be used to compute eigenvalues of large sparse non-Hermitian matrices or to solve large sparse non-Hermitian linear systems. However, the original Lanczos algorithm is susceptible to possible breakdowns and potential instabilities. An implementation is presented of a look-ahead version of the Lanczos algorithm that, except for the very special situation of an incurable breakdown, overcomes these problems by skipping over those steps in which a breakdown or near-breakdown would occur in the standard process. The proposed algorithm can handle look-ahead steps of any length and requires the same number of matrix-vector products and inner products as the standard Lanczos process without look-ahead
Computation of generalized matrix functions
We develop numerical algorithms for the efficient evaluation of quantities
associated with generalized matrix functions [J. B. Hawkins and A. Ben-Israel,
Linear and Multilinear Algebra 1(2), 1973, pp. 163-171]. Our algorithms are
based on Gaussian quadrature and Golub--Kahan bidiagonalization. Block variants
are also investigated. Numerical experiments are performed to illustrate the
effectiveness and efficiency of our techniques in computing generalized matrix
functions arising in the analysis of networks.Comment: 25 paged, 2 figure
- …