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NLEVP: A Collection of Nonlinear Eigenvalue Problems
, 2010
"... We present a collection of 46 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of reallife applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems acco ..."
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Cited by 51 (12 self)
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We present a collection of 46 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of reallife applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems according to their structural properties. Identifiers based on these and other properties can be used to extract particular types of problems from the collection. A brief description of each problem is given. NLEVP serves both to illustrate the tremendous variety of applications of nonlinear Eigenvalue problems and to provide representative problems for testing, tuning, and benchmarking of algorithms and codes.
Möbius Transformations of Matrix Polynomials
, 2014
"... We discuss Möbius transformations for general matrix polynomials over arbitrary fields, analyzing their influence on regularity, rank, determinant, constructs such as compound matrices, and on structural features including sparsity and symmetry. Results on the preservation of spectral information co ..."
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Cited by 11 (1 self)
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We discuss Möbius transformations for general matrix polynomials over arbitrary fields, analyzing their influence on regularity, rank, determinant, constructs such as compound matrices, and on structural features including sparsity and symmetry. Results on the preservation of spectral information contained in elementary divisors, partial multiplicity sequences, invariant pairs, and minimal indices are presented. The effect on canonical forms such as Smith forms and local Smith forms, on relationships of strict equivalence and spectral equivalence, and on the property of being a linearization or quadratification are investigated. We show that many important transformations are special instances of Möbius transformations, and analyze a Möbius connection between alternating and palindromic matrix polynomials. Finally, the use of Möbius transformations in solving polynomial inverse eigenproblems is illustrated.
Chebyshev interpolation for nonlinear eigenvalue problems,
 BIT,
, 2012
"... This work is concerned with numerical methods for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. In particular, we focus on eigenvalue problems for which the evaluation of the matrixvalued function is computationally expensive. Such problems arise, e.g., from boundary i ..."
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Cited by 9 (1 self)
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This work is concerned with numerical methods for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. In particular, we focus on eigenvalue problems for which the evaluation of the matrixvalued function is computationally expensive. Such problems arise, e.g., from boundary integral formulations of elliptic PDE eigenvalue problems and typically exclude the use of established nonlinear eigenvalue solvers. Instead, we propose the use of polynomial approximation combined with nonmonomial linearizations. Our approach is intended for situations where the eigenvalues of interest are located on the real line or, more generally, on a prespecified curve in the complex plane. A firstorder perturbation analysis for nonlinear eigenvalue problems is performed. Combined with an approximation result for Chebyshev interpolation, this shows exponential convergence of the obtained eigenvalue approximations with respect to the degree of the approximating polynomial. Preliminary numerical experiments demonstrate the viability of the approach in the context of boundary element methods.
Several properties of invariant pairs of nonlinear algebraic eigenvalue problems, Temple Research Report 120209
, 2012
"... Abstract. We analyze several important properties of invariant pairs of nonlinear algebraic eigenvalue problems of the form T (λ)v = 0. Invariant pairs are generalizations of invariant subspaces in association with block Rayleigh quotients of square matrices to a nonlinear matrixvalued function T ( ..."
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Cited by 2 (2 self)
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Abstract. We analyze several important properties of invariant pairs of nonlinear algebraic eigenvalue problems of the form T (λ)v = 0. Invariant pairs are generalizations of invariant subspaces in association with block Rayleigh quotients of square matrices to a nonlinear matrixvalued function T (·). They play an important role in the analysis of nonlinear eigenvalue problems and algorithms. In this paper, we first show that the algebraic, partial, and geometric multiplicities together with the Jordan chains corresponding to an eigenvalue of T (λ)v = 0 are completely represented by the Jordan canonical form of a simple invariant pair that captures this eigenvalue. We then investigate approximation errors and perturbations of a simple invariant pair. We also show that second order accuracy in eigenvalue approximation can be achieved by the twosided block Rayleigh functional for nondefective eigenvalues. Finally, we study the matrix representation of the Fréchet derivative of the eigenproblem, and we discuss the norm estimate of the inverse derivative, which measures the conditioning and sensitivity of simple invariant pairs.
Parallel Iterative Refinement in Polynomial Eigenvalue Problems∗
, 2015
"... Methods for the polynomial eigenvalue problem often require to be followed by an iterative refinement process to improve the accuracy of the computed solutions. This can be accomplished by means of a Newton iteration tailored to matrix polynomials. The computational cost of this step is usually hig ..."
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Cited by 1 (1 self)
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Methods for the polynomial eigenvalue problem often require to be followed by an iterative refinement process to improve the accuracy of the computed solutions. This can be accomplished by means of a Newton iteration tailored to matrix polynomials. The computational cost of this step is usually higher than the cost of computing the initial approximations, due to the need of solving multiple linear systems of equations with a modified system matrix. An effective parallelization is thus important, and we propose different approaches for the messagepassing scenario. Some schemes use a subcommunicator strategy in order to improve the scalability whenever direct linear solvers are used. We show performance results for the various alternatives implemented in the context of SLEPc, the Scalable Library for Eigenvalue Problem Computations. 1
Parallel Krylov solvers for the polynomial eigenvalue problem
, 2015
"... Abstract. Polynomial eigenvalue problems are often found in scientific computing applications. When the coefficient matrices of the polynomial are large and sparse, usually only a few eigenpairs are required and projection methods are the best choice. We focus on Krylov methods that operate on the c ..."
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Abstract. Polynomial eigenvalue problems are often found in scientific computing applications. When the coefficient matrices of the polynomial are large and sparse, usually only a few eigenpairs are required and projection methods are the best choice. We focus on Krylov methods that operate on the companion linearization of the polynomial, but exploit the block structure with the aim of being memoryefficient in the representation of the Krylov subspace basis. The problem may appear in the form of a lowdegree polynomial (quartic or quintic, say) expressed in the monomial basis, or a highdegree polynomial (coming from interpolation of a nonlinear eigenproblem) expressed in a nonmonomial basis. We have implemented a parallel solver in SLEPc covering both cases, that is able to compute exterior as well as interior eigenvalues via spectral transformation. We discuss important issues such as scaling and restart, and illustrate the robustness and performance of the solver with some numerical experiments.
LOCAL CONVERGENCE OF NEWTONLIKE METHODS FOR DEGENERATE EIGENVALUES OF NONLINEAR EIGENPROBLEMS∗
, 2012
"... methods for degenerate eigenvalues of nonlinear eigenproblemsa ..."
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Matrix Polynomials with Specified Eigenvalues
, 2014
"... This work concerns the distance in 2norm from a matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Perturbations are allowed only on the constant coefficient matrix. Singular value optimization formulas are derived for ..."
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This work concerns the distance in 2norm from a matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Perturbations are allowed only on the constant coefficient matrix. Singular value optimization formulas are derived for these distances facilitating their computation. The singular value optimization problems, when the number of specified eigenvalues is small, can be solved numerically by exploiting the Lipschitzness and piecewise analyticity of the singular values with respect to the parameters.