A Lanczos-type method for multiple starting vectors

Aliaga, José I.; Boley, Daniel; Freund, Robert M.; Hernández, Vicente

doi:10.1090/s0025-5718-99-01163-1

Cited by 92 publications

(121 citation statements)

References 40 publications

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“…Note that by construction, the matrix (53) has full column rank. The n-th block-Krylov subspace (induced by M and R) K n M, R is defined as the subspace of C N1 spanned by the first n columns of the matrix (53); see, [1] for more details of this construction.…”

Section: Block-krylov Subspacesmentioning

confidence: 99%

Padé-Type Model Reduction of Second-Order and Higher-Order Linear Dynamical Systems

Freund

Lecture Notes in Computational Science and Engineering

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Summary.A standard approach to reduced-order modeling of higher-order linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for reduced-order modeling of first-order systems. While this approach results in reduced-order models that are characterized as Padé-type or even true Padé approximants of the system's transfer function, in general, these models do not preserve the form of the original higher-order system. In this paper, we present a new approach to reduced-order modeling of higher-order systems based on projections onto suitably partitioned Krylov basis matrices that are obtained by applying Krylov-subspace techniques to an equivalent first-order system. We show that the resulting reduced-order models preserve the form of the original higher-order system. While the resulting reduced-order models are no longer optimal in the Padé sense, we show that they still satisfy a Padé-type approximation property. We also introduce the notion of Hermitian higher-order linear dynamical systems, and we establish an enhanced Padé-type approximation property in the Hermitian case.

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Section: Block-krylov Subspacesmentioning

confidence: 99%

Padé-Type Model Reduction of Second-Order and Higher-Order Linear Dynamical Systems

Freund

Lecture Notes in Computational Science and Engineering

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“…Next, we briefly review the notion of block Krylov subspaces; see [1] for a more detailed discussion. The matrix sequence…”

Section: Block Krylov Subspacesmentioning

confidence: 99%

Structure-Preserving Model Order Reduction of RCL Circuit Equations

Freund

2008

Mathematics in Industry

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Summary. In recent years, order-reduction techniques based on Krylov subspaces have become the methods of choice for generating macromodels of large multi-port RLC circuits. Despite the success of these techniques and the extensive research efforts in this area, for general RCL circuits, the existing Krylov subspace-based reduction algorithms do not fully preserve all essential structures of the given large RCL circuit. In this paper, we describe the problem of structure-preserving model order reduction of general RCL circuits, and we discuss two state-of-the-art algorithms, PRIMA and SPRIM, for the solution of this problem. Numerical results are reported that illustrate the higher accuracy of SPRIM vs. PRIMA. We also mention some open problems.

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“…There exist various generalizations of the Lanczos process, like block variants, e.g., [36,12,4], and variants with different numbers of left-hand and right-hand side starting vectors, e.g., [1]. Recently, Lanczos processes with s ∈ N left-hand side starting vectors and one right-hand side starting vector are enjoying a renaissance: in disguise, namely, as the workhorses of the IDR [39] and IDR(s) [34,10] based methods, see, e.g., [15,33,32].…”

Section: Introductionmentioning

confidence: 99%

An augmented analysis of the perturbed two-sided Lanczos tridiagonalization process

Paige

Panayotov

Zemke

2014

Linear Algebra and its Applications

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We generalize an augmented rounding error result that was proven for the symmetric Lanczos process in [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2347-2359, to the two-sided (usually unsymmetric) Lanczos process for tridiagonalizing a square matrix. We extend the analysis to more general perturbations than rounding errors in order to provide tools for the analysis of inexact and related methods. The aim is to develop a deeper understanding of the behavior of all these methods. Our results take the same form as those for the symmetric Lanczos process, except for the bounds on the backward perturbation terms (the generalizations of backward rounding errors for the augmented system). In general we cannot derive tight a priori bounds for these terms as was done for the symmetric process, but a posteriori bounds are feasible, while bounds related to certain properties of matrices would be theoretically desirable.

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A Lanczos-type method for multiple starting vectors

Cited by 92 publications

References 40 publications

Padé-Type Model Reduction of Second-Order and Higher-Order Linear Dynamical Systems

Padé-Type Model Reduction of Second-Order and Higher-Order Linear Dynamical Systems

Structure-Preserving Model Order Reduction of RCL Circuit Equations

An augmented analysis of the perturbed two-sided Lanczos tridiagonalization process

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