Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems
Applied Numerical Mathematics
Accurate conjugate gradient methods for families of shifted systems
Applied Numerical Mathematics - Numerical algorithms, parallelism and applications
Deflated block Krylov subspace methods for large scale eigenvalue problems
Journal of Computational and Applied Mathematics
A model-order reduction technique for low rank rational perturbations of linear eigenproblems
PARA'04 Proceedings of the 7th international conference on Applied Parallel Computing: state of the Art in Scientific Computing
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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 $n\times n$ projection, ${\mathbf{T}}_n^{\rm s}$, of the given symmetric matrix onto the $n$-dimensional subspace spanned by the first $n$ Lanczos vectors. This subspace is closely related to the $n$th block-Krylov subspace induced by the given symmetric matrix and the given block of multiple starting vectors. The standard algorithm produces the entries of ${\mathbf{T}}_n^{\rm s}$ directly. In this paper, we propose a variant of the symmetric band Lanczos process that employs coupled recurrences to generate the factors of an LDL$^{\mathrm{T}}$ factorization of a closely related $n\times n$ projection, rather than ${\mathbf{T}}_n^{\rm s}$. This is done in such a way that the factors of the LDL$^{\mathrm{T}}$ factorization inherit the ``fish-bone'' structure of ${\mathbf{T}}_n^{\rm s}$. 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.