A novel method to compute all eigenvalues of the polynomial eigenvalue problems in an open half plane
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A novel method to compute all eigenvalues of the polynomial eigenvalue problems in an open half plane Xiaoping Chen1,2 · Xi Yang3 · Wei Wei3 · Jinde Cao4 · Shuai Tang2 Received: 26 March 2018 / Revised: 16 October 2018 / Accepted: 19 October 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract In this paper, we introduce the linear fractional mapping and the contour integral method. Based on them, we develop a new numerical method to find all eigenvalues of the polynomial eigenvalue problems in an open half plane. Numerical examples are shown to illustrate the effectiveness of the proposed method. Keywords Polynomial eigenvalue problem · Linear fractional mapping · Open half plane · Contour integral method Mathematics Subject Classification 15A18 · 65F15
1 Introduction In this paper, we consider the polynomial eigenvalue problem (PEP) P(λ)x = 0,
(1)
Communicated by Jinyun Yuan.
B
Xiaoping Chen [email protected] Xi Yang [email protected]; [email protected] Wei Wei [email protected] Jinde Cao [email protected] Shuai Tang [email protected]
1
Department of Mathematics, Southeast University, Nanjing 210096, China
2
Department of Mathematics, Taizhou University, Taizhou 225300, China
3
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
4
School of Mathematics and Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing 210096, China
123
X. Chen et al.
where P(λ) = λd A0 + λd−1 A1 + · · · + λAd−1 + Ad , Ak ∈ C n×n , 0 ≤ k ≤ m. λ and x, known as the eigenvalue and the corresponding eigenvector, respectively, are the variables to be determined. The problem is very general and includes the standard eigenvalue problem P(λ) = λI − A (see, e.g., Wilkinson 1965), the generalized eigenvalue problem P(λ) = λA − B (see, e.g., Dai et al. 2015), the quadratic eigenvalue problem P(λ) = λ2 A + λB + C (see, e.g., Tisseur and Meerbergen 2001) and the cubic eigenvalue problem P(λ) = λ3 A0 + λ2 A1 + λA2 + A3 (see, e.g., Hwang et al. 2005). The PEP arises in a number of various applications, and has received considerable attention in the last few decades. For example, it is ubiquitous in a wide range of problems, such as vibration analysis of viscoelastic systems (Adhikari and Pascual 2009), structural dynamic analysis (Gupta 1976), stability analysis of control systems (Higham and Tisseur 2002), numerical simulation of quantum dots (Hwang et al. 2004) and so on. Considerable efforts have been devoted to the polynomial eigenvalue problem in the literature. Gohberg et al. (1982) developed the mathematical theory concerning matrix polynomials. Gohberg et al. (1979), Dedieu and Tisseur (2003), Higham and Tisseur (2003) and Chu (2003) gave the perturbation theory for the polynomial eigenvalue problem. Higham et al. (2007), Tisseur (2000) and Lawrence and Corless (2015) analyzed backward error of the polynomial eigenvalue problem. The classical approach for solving the PEP is linearizing the problem (1)
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