A Symmetric Band Lanczos Process Based on Coupled Recurrences and Some Applications

The symmetric band Lanczos process is an extension of the classical Lanczos algorithm for symmetric matrices and single starting vectors to multiple starting vectors. After n iterations, the symmetric band Lanczos process has generated an n× n projection, Tn, of the given symmetric matrix onto the n-dimensional subspace spanned by the first n Lanczos vectors. This subspace is closely related to the nth block-Krylov subspace induced by the given symmetric matrix and the given block of multiple starting vectors. The standard algorithm produces the entries of Tn directly. In this paper, we propose a variant of the symmetric band Lanczos process that employs coupled recurrences to generate the factors of an LDLT factorization of a closely related n× n projection, rather than Tn. This is done in such a way that the factors of the LDL T factorization inherit the “fish-bone” structure of Tn. Numerical examples from reduced-order modeling of large electronic circuits and algebraic eigenvalue problems demonstrate that the proposed variant of the band Lanczos process based on coupled recurrences is more robust and accurate than the standard algorithm.