Homotopy method for the numerical solution of the eigenvalue problem of self-adjoint partial differential operators
type: Item
instance of: scholarly article
title: Homotopy method for the numerical solution of the eigenvalue problem of self-adjoint partial differential operators
zbMATH Open document ID: 0836.65115
DOI: 10.1007/BF02140775
author: Yanyan Li
published in: Numerical Algorithms
publication date: 1996-02-01T00:00:00Z
review text: The homotopy method is used to compute the smallest few eigenvalues and their corresponding eigenfunctions of a linear self-adjoint partial differential operator. By this method the homotopy \(H(t) = (1 - t)A_0 + tA_1\) is formed (\(A_0\) is a matrix whose smallest few eigenpairs are known). The eigenpairs at \(t + dt\) are obtained using the Rayleigh quotient iterations, \(dt\) is chosen so that \(H(t)\) and \(H(t + dt)\) do not differ too much. This step is repeated until \(t\) reaches 1, when the solution to the original problem is found. The issues relating to the linear solver and the low-rank perturbations and the case when several eigenvalues are clustered together are discussed. Numerical examples for the Schrödinger eigenvalue problem in two space dimensions are given. The algorithm is used to locate the bifurcation point of a parametrized partial differential equation.
Mathematics Subject Classification ID: 65N25
Mathematics Subject Classification ID: 35P15
zbMATH DE Number: 826916
zbMATH Keywords: smallest eigenvalues
zbMATH Keywords: numerical examples
zbMATH Keywords: eigenfunctions
zbMATH Keywords: Rayleigh quotient iterations
zbMATH Keywords: Schrödinger eigenvalue problem
zbMATH Keywords: bifurcation point
MaRDI profile type: Publication
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cites work: Homotopy-Determinant Algorithm for Solving Nonsymmetric Eigenvalue Problems
cites work: Solving Eigenvalue Problems of Real Nonsymmetric Matrices with Real Homotopies
cites work: Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue Problem
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cites work: The method of conjugate gradients used in inverse iteration
cites work: Criteria for Combining Inverse and Rayleigh Quotient Iteration
full work available at URL: https://doi.org/10.1007/bf02140775
OpenAlex ID: W2006918424
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description: scientific article; zbMATH DE number 826916