Verified eigenvalue and eigenvector computations using complex moments and the Rayleigh–Ritz procedure for generalized Hermitian eigenvalue problems
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TL;DR: In this paper , the authors proposed a verified computation method for eigenvalues in a region and the corresponding eigenvectors of generalized Hermitian eigenvalue problems, which uses complex moments to extract the eigencomponents of interest from a random matrix and uses the Rayleigh$\unicode{x2013}$Ritz procedure to project a given eigen value problem into a reduced Eigenvalue problem.
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About: This article is published in Journal of Computational and Applied Mathematics. The article was published on 01 May 2023. and is currently open access. The article focuses on the topics: Eigenvalues and eigenvectors & Hermitian matrix.
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Citations
Fast computation of error bounds for all eigenpairs of a Hermitian and all singular pairs of a rectangular matrix with emphasis on eigen- and singular value clusters
Siegfried M. Rump,Marko Lange +1 more
TL;DR: In this article , error bounds for all eigenvectors of a Hermitian matrix as well as for all singular vectors of a rectangular real or complex matrix have been computed, and the computed bounds do contain the true result with mathematical certainty.
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References
A projection method for generalized eigenvalue problems using numerical integration
Tetsuya Sakurai,Hiroshi Sugiura +1 more
TL;DR: In this article, a method for finding certain eigenvalues of a generalized eigenvalue problem that lie in a given domain of the complex plane is proposed, which projects the matrix pencil onto a subspace associated with the eigen values that are located in the domain via numerical integration.
402
Verification methods: Rigorous results using floating-point arithmetic
TL;DR: Verification methods are introduced and it is discussed how floating-point arithmetic is used and how they can assist in achieving a mathematically rigorous result.
322
A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura projection method
TL;DR: The Sakurai-Sugiura projection method, which solves generalized eigenvalue problems to find certain eigenvalues in a given domain, was reformulated by using the resolvent theory.
159
CIRR: a Rayleigh-Ritz type method with contour integral for generalized eigenvalue problems
Tetsuya Sakurai,Hiroto Tadano +1 more
TL;DR: In this paper, a Rayleigh-Ritz type eigensolver for finding a limited set of eigenvalues and their corresponding eigenvectors in a certain region of generalized eigen-value problems is considered.