Divide-and-conquer eigenvalue algorithm
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The nonlinear eigenvalue problem Open
Nonlinear eigenvalue problems arise in a variety of science and engineering applications, and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue p…
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Eigenvalue and Generalized Eigenvalue Problems: Tutorial Open
This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problem…
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Quantum phase estimation for a class of generalized eigenvalue problems Open
Quantum phase estimation provides a path to quantum computation of solutions to Hermitian eigenvalue problems Hv = λv , such as those occurring in quantum chemistry. It is natural to ask whether the same technique can be applied to general…
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An Explicit Formula for the Splitting of Multiple Eigenvalues for Nonlinear Eigenvalue Problems and Connections with the Linearization for the Delay Eigenvalue Problem Open
© 2017 Society for Industrial and Applied Mathematics. We contribute to the perturbation theory of nonlinear eigenvalue problems in three ways. First, we extend the formula for the sensitivity of a simple eigenvalue with respect to a varia…
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Iterative refinement for symmetric eigenvalue decomposition Open
An efficient refinement algorithm is proposed for symmetric eigenvalue problems. The structure of the algorithm is straightforward, primarily comprising matrix multiplications. We show that the proposed algorithm converges quadratically if…
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Solving Singular Generalized Eigenvalue Problems by a Rank-Completing Perturbation Open
Generalized eigenvalue problems involving a singular pencil are very\nchallenging to solve, both with respect to accuracy and efficiency. The\nexisting package Guptri is very elegant but may sometimes be time-demanding,\neven for small and…
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Iterative refinement for symmetric eigenvalue decomposition II: clustered eigenvalues Open
We are concerned with accurate eigenvalue decomposition of a real symmetric matrix A. In the previous paper (Ogita and Aishima in Jpn J Ind Appl Math 35(3): 1007–1035, 2018), we proposed an efficient refinement algorithm for improving the …
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An optimization problem for the first eigenvalue of the ‐fractional Laplacian Open
In this paper we analyze an eigenvalue problem related to the nonlocal p ‐Laplace operator plus a potential. After reviewing some elementary properties of the first eigenvalue of these operators (existence, positivity of associated eigenfu…
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The Waveguide Eigenvalue Problem and the Tensor Infinite Arnoldi Method Open
We present a new computational approach for a class of large-scale nonlinear eigenvalue problems (NEPs) that are nonlinear in the eigenvalue. The contribution of this paper is two fold. We derive a new iterative algorithm for NEPs, the ten…
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The eigenvalue problem with interaction conditions at one interior singular point Open
Some physical processes, both classical physics and quantum physics reduced to eigenvalue problems for Sturm-Liouville equations. In the recent years there has been an increasing interest in discontinuous eigenvalue problems for various St…
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Automatic rational approximation and linearization of nonlinear eigenvalue problems Open
We present a method for solving nonlinear eigenvalue problems (NEPs) using rational approximation. The method uses the Antoulas–Anderson algorithm (AAA) of Nakatsukasa, Sète and Trefethen to approximate the NEP via a rational eigenvalue pr…
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Low-Rank Solution Methods for Stochastic Eigenvalue Problems Open
In this report, we study efficient solution methods for stochastic eigenvalue problems arising from discretization of self-adjoint partial differential equations with random data, where the underlying operators depend linearly on the rando…
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Eigenvalue problem for fractional differential equations with nonlinear integral and disturbance parameter in boundary conditions Open
This paper is concerned with the existence, nonexistence, uniqueness, and multiplicity of positive solutions for a class of eigenvalue problems of nonlinear fractional differential equations with a nonlinear integral term and a disturbance…
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Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems Open
Summary This paper discusses techniques for computing a few selected eigenvalue–eigenvector pairs of large and sparse symmetric matrices. A recently developed class of techniques to solve this type of problems is based on integrating the m…
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Approximation of operator eigenvalue problems in a Hilbert space Open
The eigenvalue problem for a compact symmetric positive definite operator in an infinite-dimensional Hilbert space is approximated by an operator eigenvalue problem in finitedimensional subspace. Error estimates for the approximate eigenva…
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Comparison of Numerical Methods and Open-Source Libraries for Eigenvalue Analysis of Large-Scale Power Systems Open
This paper discusses the numerical solution of the generalized non-Hermitian eigenvalue problem. It provides a comprehensive comparison of existing algorithms, as well as of available free and open-source software tools, which are suitable…
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Nonlinear eigenvalue problems for generalized Painlevé equations Open
Eigenvalue problems for linear differential equations, such as\ntime-independent Schr\\"odinger equations, can be generalized to eigenvalue\nproblems for nonlinear differential equations. In the nonlinear context a\nseparatrix plays the ro…
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On Rayleigh Quotient Iteration for the Dual Quaternion Hermitian Eigenvalue Problem Open
The application of eigenvalue theory to dual quaternion Hermitian matrices holds significance in the realm of multi-agent formation control. In this paper, we study the use of Rayleigh quotient iteration (RQI) for solving the right eigenpa…
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Multiparameter Eigenvalue Problems and Shift-invariance Open
sponsorship: This work was supported by (1) KU Leuven: Research Fund (projects C16/15/059, C3/19/053, C24/18/022, C3/20/117), Industrial Research Fund (Fellowships 13-0260, IOF/16/004) and several Leuven Research and Development bilateral …
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Multi-Eigenvalue Demodulation Using Complex Moment-Based Eigensolver and Neural Network Open
Optical eigenvalues originating in optical solitons are the potential for becoming information carriers not affected by chromatic dispersion and nonlinear effects in optical fibers. They are obtained by attributing the associated eigenvalu…
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Finite difference approximation of electron balance problem in the stationary high-frequency induction discharges Open
\nThe problem of finding the minimal eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem for the ordinary differential equation with coefficients depending on a spectral parameter is investigated. This …
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Analytical solutions to some generalized and polynomial eigenvalue problems Open
It is well-known that the finite difference discretization of the Laplacian eigenvalue problem − Δu = λu leads to a matrix eigenvalue problem (EVP) Ax = λx where the matrix A is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal mat…
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Fractional eigenvalue problems that approximate Steklov eigenvalue problems Open
In this paper we analyse possible extensions of the classical Steklov eigenvalue problem to the fractional setting. In particular, we find a non-local eigenvalue problem of fractional type that approximates, when taking a suitable limit, t…
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A spectral Newton-Schur algorithm for the solution of symmetric generalized eigenvalue problems Open
This paper proposes a numerical algorithm based on spectral Schur complements to compute a few eigenvalues and the associated eigenvectors of symmetric matrix pencils. The proposed scheme follows an algebraic domain decomposition viewpoint…
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On pole-swapping algorithms for the eigenvalue problem Open
Pole-swapping algorithms, which are generalizations of the QZ algorithm for the generalized eigenvalue problem, are studied. A new modular (and therefore more flexible) convergence theory that applies to all pole-swapping algorithms is dev…
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Performance of the parallel block Jacobi method with dynamic ordering for the symmetric eigenvalue problem Open
We investigate the performance of the parallel block Jacobi method for the symmetric eigenvalue problem with dynamic ordering both theoretically and experimentally. First, we present an improved global convergence theorem of the method tha…
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Iterative Algorithms for Solving the Partial Eigenvalue Problem for Symmetric Interval Matrixes Open
In this paper, we consider iterative methods for solving a partial eigenvalue problem for real symmetric interval matrices.Such matrices have applications in modeling many technical problems where a lot of data suffers from limited variati…
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Stability of heterogeneous beams with three supports through Green functions Open
The present paper is devoted to the issue how the critical load of some heterogeneous beams with three supports can be determined by using Green functions. The stability problems of these beams are equivalent to three-point boundary value …
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An extension of the Cayley transform method for a parameterized generalized inverse eigenvalue problem Open
Summary Since recent studies have shown that the Cayley transform method can be an effective iterative method for solving the inverse eigenvalue problem, in this work, we consider using an extension of it for solving a type of parameterize…
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An optimization problem for the first eigenvalue of the $p-$fractional laplacian Open
In this paper we analyze an eigenvalue problem related to the nonlocal $p-$laplace operator plus a potential. After reviewing some elementary properties of the first eigenvalue of these operators (existence, positivity of associated eigenf…