Volume 2, Issue 2
Note on Finding an Optimal Deflation for Quadratic Matrix Polynomials

CSIAM Trans. Appl. Math., 2 (2021), pp. 336-356.

Published online: 2021-05

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• Abstract

This paper is concerned with the way to find an optimal deflation for the eigenvalue problem associated with quadratic matrix polynomials. This work is a response of the work by Tisseur et al., $Linear$ $Algebra$ $Appl$., $435:464-479, 2011$, and solves one of open problems raised by them. We build an equivalent unconstrained optimization problem on eigenvalues of a hyperbolic quadratic matrix polynomial of order 2, and develop a technique that transforms the quadratic matrix polynomial to an equivalent one that is easy to solve. Numerical tests are given to illustrate several properties of the problem.

• Keywords

deflation, quadratic matrix polynomials, hyperbolic, eigenvalue optimization.

65F35, 65F15

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@Article{CSIAM-AM-2-336, author = {Xin Liang , }, title = {Note on Finding an Optimal Deflation for Quadratic Matrix Polynomials}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2021}, volume = {2}, number = {2}, pages = {336--356}, abstract = {

This paper is concerned with the way to find an optimal deflation for the eigenvalue problem associated with quadratic matrix polynomials. This work is a response of the work by Tisseur et al., $Linear$ $Algebra$ $Appl$., $435:464-479, 2011$, and solves one of open problems raised by them. We build an equivalent unconstrained optimization problem on eigenvalues of a hyperbolic quadratic matrix polynomial of order 2, and develop a technique that transforms the quadratic matrix polynomial to an equivalent one that is easy to solve. Numerical tests are given to illustrate several properties of the problem.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.2021.nla.05}, url = {http://global-sci.org/intro/article_detail/csiam-am/18888.html} }
TY - JOUR T1 - Note on Finding an Optimal Deflation for Quadratic Matrix Polynomials AU - Xin Liang , JO - CSIAM Transactions on Applied Mathematics VL - 2 SP - 336 EP - 356 PY - 2021 DA - 2021/05 SN - 2 DO - http://doi.org/10.4208/csiam-am.2021.nla.05 UR - https://global-sci.org/intro/article_detail/csiam-am/18888.html KW - deflation, quadratic matrix polynomials, hyperbolic, eigenvalue optimization. AB -

This paper is concerned with the way to find an optimal deflation for the eigenvalue problem associated with quadratic matrix polynomials. This work is a response of the work by Tisseur et al., $Linear$ $Algebra$ $Appl$., $435:464-479, 2011$, and solves one of open problems raised by them. We build an equivalent unconstrained optimization problem on eigenvalues of a hyperbolic quadratic matrix polynomial of order 2, and develop a technique that transforms the quadratic matrix polynomial to an equivalent one that is easy to solve. Numerical tests are given to illustrate several properties of the problem.

Xin Liang. (2021). Note on Finding an Optimal Deflation for Quadratic Matrix Polynomials. CSIAM Transactions on Applied Mathematics. 2 (2). 336-356. doi:10.4208/csiam-am.2021.nla.05
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