Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems

被引：16

作者：

Dopico, Froilan M. ^{[1
]}

Marcaida, Silvia ^{[2
]}

Quintana, Maria C. ^{[1
]}

Van Dooren, Paul ^{[3
]}

机构：

[1] Univ Carlos III Madrid, Dept Matemat, Avda Univ 30, Leganes 28911, Spain

[2] Univ Pais Vasco UPV EHU, Dept Matemat Aplicada & Estadist & Invest Operat, Apdo Correos 644, Bilbao 48080, Spain

[3] Catholic Univ Louvain, Dept Engn Math, Ave Georges Lemaitre 4, B-1348 Louvain La Neuve, Belgium

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2020年 / 604卷

关键词：

Rational matrix; Rational eigenvalue problem; Nonlinear eigenvalue problem; Linearization; Polynomial system matrix; Rational approximation; Block full rank pencilssolution; SYSTEM MATRIX; MINIMAL BASES; EQUIVALENCE; POLYNOMIALS; RECOVERY; SPACES; FORM;

D O I：

10.1016/j.laa.2020.07.004

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

This paper presents a definition for local linearizations of rational matrices and studies their properties. This definition allows to introduce matrix pencils associated to a rational matrix that preserve its structure of zeros and poles in subsets of any algebraically closed field and also at infinity. This new theory of local linearizations captures and explains rigorously the properties of all the different pencils that have been used from the 1970's until 2020 for computing zeros, poles and eigenvalues of rational matrices. Particular attention is paid to those pencils that have appeared recently in the numerical solution of nonlinear eigenvalue problems through rational approximation. (C) 2020 Elsevier Inc. All rights reserved.

引用

页码：441 / 475

页数：35

共 37 条

[1] LINEARIZATIONS FOR RATIONAL MATRIX FUNCTIONS AND ROSENBROCK SYSTEM POLYNOMIALS [J].

Alam, Rafikul ;

Behera, Namita .

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, 2016, 37 (01) :354-380

[2] Linearization of matrix polynomials expressed in polynomial bases [J].

Amiraslani, A. ;

Corless, R. M. ;

Lancaster, P. .

IMA JOURNAL OF NUMERICAL ANALYSIS, 2009, 29 (01) :141-157

[3] STRONG LINEARIZATIONS OF RATIONAL MATRICES [J].

Amparan, A. ;

Dopico, F. M. ;

Marcaida, S. ;

Zaballa, I .

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, 2018, 39 (04) :1670-1700

[4]

Amparan A, 2015, ELECTRON J LINEAR AL, V30, P196

[5] On the structure invariants of proper rational matrices with prescribed finite poles [J].

Amparan, A. ;

Marcaida, S. ;

Zaballa, I. .

LINEAR & MULTILINEAR ALGEBRA, 2013, 61 (11) :1464-1486

[6]

[Anonymous], 1959, THEORY MATRICES

[7] LOCAL SYSTEM EQUIVALENCE [J].

CULLEN, DJ .

MATHEMATICAL SYSTEMS THEORY, 1986, 19 (01) :67-78

[8] Affine spaces of strong linearizations for rational matrices and the recovery of eigenvectors and minimal bases [J].

Das, Ranjan Kumar ;

Alam, Rafikul .

LINEAR ALGEBRA AND ITS APPLICATIONS, 2019, 569 :335-368

[9] Recovery of minimal bases and minimal indices of rational matrices from Fiedler-like pencils [J].

Das, Ranjan Kumar ;

Alam, Rafikul .

LINEAR ALGEBRA AND ITS APPLICATIONS, 2019, 566 :34-60

[10] Spectral equivalence of matrix polynomials and the Index Sum Theorem [J].

De Teran, Fernando ;

Dopico, Froilan. M. ;

Mackey, D. Steven .

LINEAR ALGEBRA AND ITS APPLICATIONS, 2014, 459 :264-333

← 1 2 3 4 →