Refining estimates of invariant and deflating subspaces for large and sparse matrices and pencils

Hung Yuan Fan; Peter Chang Yi Weng; Eric King wah Chu

doi:10.1007/s10543-014-0469-1

Refining estimates of invariant and deflating subspaces for large and sparse matrices and pencils

Hung Yuan Fan, Peter Chang Yi Weng, Eric King wah Chu

Department of Mathematics

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

4 Citations (Scopus)

Abstract

We consider the refinement of estimates of invariant (or deflating) subspaces for a large and sparse real matrix (or pencil) in ℝ^n×n, through some (generalized) nonsymmetric algebraic Riccati equations or their associated (generalized) Sylvester equations via Newton's method. The crux of the method is the inversion of some well-conditioned unstructured matrices via the efficient and stable inversion of the associated structured but near-singular matrices.All computations have complexity proportional to n, under appropriate assumptions, as illustrated by several numerical examples.

Original language	English
Pages (from-to)	147-169
Number of pages	23
Journal	BIT Numerical Mathematics
Volume	54
Issue number	1
DOIs	https://doi.org/10.1007/s10543-014-0469-1
Publication status	Published - 2014 Mar

Keywords

Deflating subspace
Invariant subspace
Large-scale problem
Newton's method
Nonsymmetric algebraic Riccati equation
Sparse matrix
Sylvester equation

ASJC Scopus subject areas

Software
Computer Networks and Communications
Computational Mathematics
Applied Mathematics

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10.1007/s10543-014-0469-1

Cite this

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title = "Refining estimates of invariant and deflating subspaces for large and sparse matrices and pencils",

abstract = "We consider the refinement of estimates of invariant (or deflating) subspaces for a large and sparse real matrix (or pencil) in ℝn×n, through some (generalized) nonsymmetric algebraic Riccati equations or their associated (generalized) Sylvester equations via Newton's method. The crux of the method is the inversion of some well-conditioned unstructured matrices via the efficient and stable inversion of the associated structured but near-singular matrices.All computations have complexity proportional to n, under appropriate assumptions, as illustrated by several numerical examples.",

keywords = "Deflating subspace, Invariant subspace, Large-scale problem, Newton's method, Nonsymmetric algebraic Riccati equation, Sparse matrix, Sylvester equation",

author = "Fan, {Hung Yuan} and Weng, {Peter Chang Yi} and Chu, {Eric King wah}",

note = "Funding Information: Acknowledgments The first author has been supported by the NSC, Taiwan Grant Number NSC 102-2115-M-003-009, and the second author by a Monash Graduate Scholarship and a Monash International Postgraduate Research Scholarship. Part of the work was completed when the third author visited the Shanghai Key Laboratory of Contemporary Applied Mathematics at FuDan University and National Taiwan Normal University.",

year = "2014",

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doi = "10.1007/s10543-014-0469-1",

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AU - Weng, Peter Chang Yi

AU - Chu, Eric King wah

N1 - Funding Information: Acknowledgments The first author has been supported by the NSC, Taiwan Grant Number NSC 102-2115-M-003-009, and the second author by a Monash Graduate Scholarship and a Monash International Postgraduate Research Scholarship. Part of the work was completed when the third author visited the Shanghai Key Laboratory of Contemporary Applied Mathematics at FuDan University and National Taiwan Normal University.

PY - 2014/3

Y1 - 2014/3

N2 - We consider the refinement of estimates of invariant (or deflating) subspaces for a large and sparse real matrix (or pencil) in ℝn×n, through some (generalized) nonsymmetric algebraic Riccati equations or their associated (generalized) Sylvester equations via Newton's method. The crux of the method is the inversion of some well-conditioned unstructured matrices via the efficient and stable inversion of the associated structured but near-singular matrices.All computations have complexity proportional to n, under appropriate assumptions, as illustrated by several numerical examples.

AB - We consider the refinement of estimates of invariant (or deflating) subspaces for a large and sparse real matrix (or pencil) in ℝn×n, through some (generalized) nonsymmetric algebraic Riccati equations or their associated (generalized) Sylvester equations via Newton's method. The crux of the method is the inversion of some well-conditioned unstructured matrices via the efficient and stable inversion of the associated structured but near-singular matrices.All computations have complexity proportional to n, under appropriate assumptions, as illustrated by several numerical examples.

KW - Deflating subspace

KW - Invariant subspace

KW - Large-scale problem

KW - Newton's method

KW - Nonsymmetric algebraic Riccati equation

KW - Sparse matrix

KW - Sylvester equation

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JO - BIT Numerical Mathematics

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Refining estimates of invariant and deflating subspaces for large and sparse matrices and pencils

Abstract

Keywords

ASJC Scopus subject areas

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