Krylov type subspace methods for matrix polynomials

Leonard Hoffnung; Ren Cang Li; Qiang Ye

doi:10.1016/j.laa.2005.09.016

Krylov type subspace methods for matrix polynomials

Leonard Hoffnung, Ren Cang Li, Qiang Ye

Research output: Contribution to journal › Article › peer-review

19 Scopus citations

Abstract

We consider solving eigenvalue problems or model reduction problems for a quadratic matrix polynomial Iλ² - Aλ - B with large and sparse A and B. We propose new Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with the coefficient matrices of much smaller size, which is used to approximate the original problem. We shall apply the new processes to solve eigenvalue problems and model reductions of a second order linear input-output system and discuss convergence properties. Our new processes are also extendable to cover a general matrix polynomial of any degree.

Original language	English
Pages (from-to)	52-81
Number of pages	30
Journal	Linear Algebra and Its Applications
Volume	415
Issue number	1
DOIs	https://doi.org/10.1016/j.laa.2005.09.016
State	Published - May 1 2006

Bibliographical note

Funding Information:
Keywords: Quadratic matrix polynomial; Krylov subspace; Quadratic eigenvalue problem; Model reduction ∗ Corresponding author. E-mail addresses: [email protected] (L. Hoffnung), [email protected] (R.-C. Li), [email protected] (Q. Ye). 1 Supported in part by the NSF grant nos. CCR-9875201 and CCR-0098133. 2 Supported in part by NSF CAREER award grant no. CCR-9875201 and by NSF grant no. DMS-0510664. 3 Supported in part by NSF grant no. CCR-0098133 and NSF grant no. DMS-0411502.

Funding

Keywords: Quadratic matrix polynomial; Krylov subspace; Quadratic eigenvalue problem; Model reduction \u2217 Corresponding author. E-mail addresses: [email protected] (L. Hoffnung), [email protected] (R.-C. Li), [email protected] (Q. Ye). 1 Supported in part by the NSF grant nos. CCR-9875201 and CCR-0098133. 2 Supported in part by NSF CAREER award grant no. CCR-9875201 and by NSF grant no. DMS-0510664. 3 Supported in part by NSF grant no. CCR-0098133 and NSF grant no. DMS-0411502.

Funders	Funder number
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	0098133, 0510664, DMS-0411502, CCR-9875201, DMS-0510664

Keywords

Krylov subspace
Model reduction
Quadratic eigenvalue problem
Quadratic matrix polynomial

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/j.laa.2005.09.016

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abstract = "We consider solving eigenvalue problems or model reduction problems for a quadratic matrix polynomial Iλ2 - Aλ - B with large and sparse A and B. We propose new Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with the coefficient matrices of much smaller size, which is used to approximate the original problem. We shall apply the new processes to solve eigenvalue problems and model reductions of a second order linear input-output system and discuss convergence properties. Our new processes are also extendable to cover a general matrix polynomial of any degree.",

keywords = "Krylov subspace, Model reduction, Quadratic eigenvalue problem, Quadratic matrix polynomial",

author = "Leonard Hoffnung and Li, \{Ren Cang\} and Qiang Ye",

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AU - Hoffnung, Leonard

AU - Li, Ren Cang

AU - Ye, Qiang

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N2 - We consider solving eigenvalue problems or model reduction problems for a quadratic matrix polynomial Iλ2 - Aλ - B with large and sparse A and B. We propose new Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with the coefficient matrices of much smaller size, which is used to approximate the original problem. We shall apply the new processes to solve eigenvalue problems and model reductions of a second order linear input-output system and discuss convergence properties. Our new processes are also extendable to cover a general matrix polynomial of any degree.

AB - We consider solving eigenvalue problems or model reduction problems for a quadratic matrix polynomial Iλ2 - Aλ - B with large and sparse A and B. We propose new Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with the coefficient matrices of much smaller size, which is used to approximate the original problem. We shall apply the new processes to solve eigenvalue problems and model reductions of a second order linear input-output system and discuss convergence properties. Our new processes are also extendable to cover a general matrix polynomial of any degree.

KW - Krylov subspace

KW - Model reduction

KW - Quadratic eigenvalue problem

KW - Quadratic matrix polynomial

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Krylov type subspace methods for matrix polynomials

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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