We derive inclusion regions for the eigenvalues of matrix polynomials expressed in a general polynomial basis, which can lead to significantly better results than traditional bounds. We present several applications to engineering problems.
A real square matrix is Perron-like if it has a real eigenvalue $s$, called the principal eigenvalue of the matrix, and $\mbox{Re}\,μ<s$ for any other eigenvalue $μ$. Nonnegative matrices and symmetric ones are typical examples of this class of matrices. The main purpose of this paper is to develop a set of new schemes to compute the principal eigenvalues of Perron-like matrices and the associated generalized eigenspaces by using polynomial approximations of matrix exponentials. Numerical examples show that these schemes are effective in practice.
In this paper, we propose a unified approach for solving structure-preserving eigenvalue embedding problem (SEEP) for quadratic regular matrix polynomials with symmetry structures. First, we determine perturbations of a quadratic matrix polynomial, unstructured or structured, such that the perturbed polynomials reproduce a desired invariant pair while maintaining the invariance of another invariant pair of the unperturbed polynomial. If the latter is unknown, it is referred to as no spillover perturbation. Then we use these results for solving the SEEP for structured quadratic matrix polynomia…
A Thick-Restart Lanczos algorithm with polynomial filtering for Hermitian eigenvalue problems
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；University of Minnesota, Twin Cities（明尼苏达大学双城分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
In this work we present a rational Krylov subspace method for solving real large-scale polynomial eigenvalue problems with T-even (that is, symmetric/skew-symmetric) structure. Our method is based on the Even-IRA algorithm. To preserve the structure, a sparse T-even linearization from the class of block minimal bases pencils is applied. Due to this linearization, the Krylov basis vectors can be computed in a cheap way. A rational decomposition is derived so that our method explicitly allows for changes of the shift during the iteration. This leads to a method that is able to compute parts of t…
In this paper, we introduce a new family of orthogonal systems, termed as the Müntz ball polynomials (MBPs), which are orthogonal with respect to the weight function: $\|x\|^{2θ+2μ-2} (1-\|x\|^{2θ})^α$ with the parameters $α>-1, μ>- 1/2$ and $θ>0$ in the $d$-dimensional unit ball $x\in {\mathbb B}^d=\big\{x\in\mathbb{R}^d: r=\|x\|\leq1\big\}$. We then develop efficient and spectrally accurate MBP spectral-Galerkin methods for singular eigenvalue problems including degenerating elliptic problems with perturbed ellipticity and Schrödinger's operators with fractional potentials. We demonstrate th…
A well known method to solve the Polynomial Eigenvalue Problem (PEP) is via linearization. That is, transforming the PEP into a generalized linear eigenvalue problem with the same spectral information and solving such linear problem with some of the eigenvalue algorithms available in the literature. Linearizations of matrix polynomials are usually defined using unimodular transformations. In this paper we establish a connection between the standard definition of linearization for matrix polynomials introduced by Gohberg, Lancaster and Rodman and the notion of polynomial system matrix introduce…
Sharp Error Bounds for Piecewise Polynomial Approximation: Revisit and Application to Elliptic PDE Eigenvalue Computation
LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）；Beijing Computational Science Research Center（北京计算科学研究中心）；Department of Mathematics, Wayne State University（韦恩州立大学数学系）
We discuss a fast approximate solution to the associated classical -- classical orthogonal polynomial connection problem. We first show that associated classical orthogonal polynomials are solutions to a fourth-order quadratic eigenvalue problem with polynomial coefficients such that the differential operator is degree-preserving. Upon linearization, the discretization of this quadratic eigenvalue problem is block upper-triangular and banded. After a perfect shuffle, we extend a divide-and-conquer approach to the upper-triangular and banded generalized eigenvalue problem to the blocked case, w…
Improved Inclusion-Exclusion Eigenvalue Localization Sets for Matrices and Matrix Polynomials
Faculty of Sciences, University of Novi Sad（诺维萨德大学理学院）；National Technical University of Athens（雅典国立技术大学）；Faculty of Technical Sciences, University of Novi Sad（诺维萨德大学技术科学学院）
In latest years, several advancements have been made in symbolic-numerical eigenvalue techniques for solving polynomial systems. In this article, we add to this list. We design an algorithm which solves systems with isolated solutions reliably and efficiently. In overdetermined cases, it reduces the task to an eigenvalue problem in a simpler and considerably faster way than in previous methods, and it can outperform the homotopy continuation approach. We provide many examples and an implementation in the proof-of-concept Julia package EigenvalueSolver.jl.
For a Hermitian matrix $H \in \mathbb C^{n,n}$ and symmetric matrices $S_0, S_1,\ldots,S_k \in \mathbb C^{n,n}$, we consider the problem of computing the supremum of $\left\{ \frac{v^*Hv}{v^*v}:~v\in \mathbb C^{n}\setminus \{0\},\,v^TS_iv=0~\text{for}~i=0,\ldots,k\right\}$. For this, we derive an estimation in the form of minimizing the second largest eigenvalue of a parameter depending Hermitian matrix, which is exact when the eigenvalue at the optimal is simple. The results are then applied to compute the eigenvalue backward errors of higher degree matrix polynomials with T-palindromic, T-an…
Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots
INRIA（法国国家信息与自动化研究所）；CMAP, École polytechnique（巴黎综合理工学院应用数学中心）；School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（伊朗基础科学研究院数学学院）
Fast and backward stable computation of the eigenvalues and eigenvectors of matrix polynomials
Instituto de Ciencias Matemáticas, Universidad Autónoma de Madrid（马德里自治大学数学科学研究所）；Nazarbayev University（纳扎尔巴耶夫大学）；ISTI, Area della ricerca CNR（意大利国家研究委员会ISTI研究所）；KU Leuven（荷语鲁汶大学）；Washington State University（华盛顿州立大学）
It is well-known that the finite difference discretization of the Laplacian eigenvalue problem $-Δu = λu$ leads to a matrix eigenvalue problem (EVP) $A x= λx$ where the matrix $A$ is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in Strang and MacNamara \cite{strang2014functions}. We generalize the results and develop analytical solutions to the generalized matrix eigenvalue problems (GEVPs) $A x= λBx$ which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diag…
This paper is concerned with the convergence analysis of an extended variation of the locally optimal preconditioned conjugate gradient method (LOBPCG) for the extreme eigenvalue of a Hermitian matrix polynomial which admits some extended form of Rayleigh quotient. This work is a generalization of the analysis by Ovtchinnikov (SIAM J. Numer. Anal., 46(5):2567-2592, 2008). As instances, the algorithms for definite matrix pairs and hyperbolic quadratic matrix polynomials are shown to be globally convergent and to have an asymptotically local convergence rate. Also, numerical examples are given t…
This paper is concerned with polynomial approximations of the spectral abscissa function (the supremum of the real parts of the eigenvalues) of a parameterized eigenvalue problem, which are closely related to polynomial chaos approximations if the parameters correspond to realizations of random variables. Unlike in existing works, we highlight the major role of the smoothness properties of the spectral abscissa function. Even if the matrices of the eigenvalue problem are analytic functions of the parameters, the spectral abscissa function may not be everywhere differentiable, even not everywhe…
Given a possibly singular matrix polynomial $P(z)$, we study how the eigenvalues, eigenvectors, root polynomials, minimal indices, and minimal bases of the pencils in the vector space $\mathbb{DL}(P)$ introduced in Mackey, Mackey, Mehl, and Mehrmann [SIAM J. Matrix Anal. Appl. 28(4), 971-1004, 2006] are related to those of $P(z)$. If $P(z)$ is regular, it is known that those pencils in $\mathbb{DL}(P)$ satisfying the generic assumptions in the so-called eigenvalue exclusion theorem are strong linearizations for $P(z)$. This property and the block-symmetric structure of the pencils in $\mathbb{…
Solving polynomial eigenvalue problems with eigenvector nonlinearities (PEPv) is an interesting computational challenge, outside the reach of the well-developed methods for nonlinear eigenvalue problems. We present a natural generalization of these methods which leads to a contour integration approach for computing all eigenvalues of a PEPv in a compact region of the complex plane. Our methods can be used to solve any suitably generic system of polynomial or rational function equations.
Generalized eigenvalue methods for Gaussian quadrature rules
Georgia Institute of Technology（佐治亚理工学院）；Technische Universität Berlin（柏林工业大学）；Uit – The Arctic University of Norway（挪威北极大学）；Universität Konstanz（康斯坦茨大学）；North Carolina State University（北卡罗来纳州立大学）
Structure-preserving deflation of critical eigenvalues in quadratic eigenvalue problems associated with damped mass-spring systems
IIT Guwahati（印度理工学院古瓦哈提分校）；TU Berlin（柏林工业大学）；Josip Juraj Strossmayer University of Osijek, School of Applied Mathematics and Informatics（奥西耶克约瑟普尤拉伊斯特罗斯梅亚耶尔大学应用数学与信息学院）
A posteriori error estimates for discontinuous Galerkin methods using non-polynomial basis functions. Part II: Eigenvalue problems
University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；RWTH Aachen University（亚琛工业大学）；Forschungszentrum Jülich（于利希研究中心）
In this paper, we investigate a class of nonlinear eigenvalue problems resulting from quantum physics. We first prove that the eigenfunction cannot be a polynomial on any open set, which may be reviewed as a refinement of the classic unique continuation property. Then we apply the non-polynomial behavior of eigenfunction to show that the adaptive finite element approximations are convergent even if the initial mesh is not fine enough. We finally remark that similar arguments can be applied to a class of linear eigenvalue problems that improve the relevant existing result.
Quantum estimates for classical polynomial optimization
High Energy Physics Research Unit, Faculty of Science, Chulalongkorn University（朱拉隆功大学理学院高能物理研究组）；Vrije Universiteit Brussel (VUB)（布鲁塞尔自由大学）；International Solvay Institutes（国际索尔维研究所）
