Simultaneous Similarity Reductions for a Pair of Matrices to Condensed Forms

Ren Cang Li; Qiang Ye

doi:10.1007/s40304-014-0033-y

Simultaneous Similarity Reductions for a Pair of Matrices to Condensed Forms

Ren Cang Li, Qiang Ye

Mathematics

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

3 Scopus citations

Abstract

We present simultaneous reduction algorithms for two (nonsymmetric) matrices A and B to upper Hessenberg and lower Hessenberg forms, respectively. One is through the simultaneous similarity reduction and the other is through a Lanczos–Arnoldi-type iteration. The algorithm that uses the Lanczos–Arnoldi-type iteration can be considered as a generalization of both the nonsymmetric Lanczos algorithm and the standard Arnoldi algorithm. We shall also apply our reduction to construct a model reduction for certain kind second-order single-input single-output system. It is proved that the model reduction has the desirable moment matching property.

Original language	English
Pages (from-to)	139-153
Number of pages	15
Journal	Communications in Mathematics and Statistics
Volume	2
Issue number	2
DOIs	https://doi.org/10.1007/s40304-014-0033-y
State	Published - Jun 1 2014

Bibliographical note

Funding Information:
Acknowledgments Research of R. Li is supported in part by NSF grants DMS-1115834 and DMS-1317330, and a Research Gift Grant from Intel Corporation. Research of Q. Ye is supported in part by NSF Grants DMS–1318633 and DMS-1317424.

Publisher Copyright:
© 2014, School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg.

Keywords

Krylov subspace method
Lanczos–Arnoldi iteration
Model reduction
Simultaneous reductions

ASJC Scopus subject areas

Statistics and Probability
Computational Mathematics
Applied Mathematics

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10.1007/s40304-014-0033-y

Cite this

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abstract = "We present simultaneous reduction algorithms for two (nonsymmetric) matrices A and B to upper Hessenberg and lower Hessenberg forms, respectively. One is through the simultaneous similarity reduction and the other is through a Lanczos–Arnoldi-type iteration. The algorithm that uses the Lanczos–Arnoldi-type iteration can be considered as a generalization of both the nonsymmetric Lanczos algorithm and the standard Arnoldi algorithm. We shall also apply our reduction to construct a model reduction for certain kind second-order single-input single-output system. It is proved that the model reduction has the desirable moment matching property.",

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N2 - We present simultaneous reduction algorithms for two (nonsymmetric) matrices A and B to upper Hessenberg and lower Hessenberg forms, respectively. One is through the simultaneous similarity reduction and the other is through a Lanczos–Arnoldi-type iteration. The algorithm that uses the Lanczos–Arnoldi-type iteration can be considered as a generalization of both the nonsymmetric Lanczos algorithm and the standard Arnoldi algorithm. We shall also apply our reduction to construct a model reduction for certain kind second-order single-input single-output system. It is proved that the model reduction has the desirable moment matching property.

AB - We present simultaneous reduction algorithms for two (nonsymmetric) matrices A and B to upper Hessenberg and lower Hessenberg forms, respectively. One is through the simultaneous similarity reduction and the other is through a Lanczos–Arnoldi-type iteration. The algorithm that uses the Lanczos–Arnoldi-type iteration can be considered as a generalization of both the nonsymmetric Lanczos algorithm and the standard Arnoldi algorithm. We shall also apply our reduction to construct a model reduction for certain kind second-order single-input single-output system. It is proved that the model reduction has the desirable moment matching property.

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KW - Lanczos–Arnoldi iteration

KW - Model reduction

KW - Simultaneous reductions

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Simultaneous Similarity Reductions for a Pair of Matrices to Condensed Forms

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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