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2014•SIAM Journal on Matrix Analysis and ApplicationsRequires access

Fast Enclosure for All Eigenvalues and Invariant Subspaces in Generalized Eigenvalue Problems

Shinya Miyajima

Open publisher page 26 citations

Abstract

Two fast algorithms for enclosing all eigenvalues and invariant subspaces in generalized eigenvalue problems are proposed. In these algorithms, individual eigenvectors and invariant subspaces are enclosed when eigenvalues are well separated and closely clustered, respectively. The first algorithm involves only cubic complexity and automatically determines eigenvalue clusters. The second algorithm is applicable even for defective eigenvalues. Numerical results show the properties of the proposed algorithms.

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What this paper is about

Two fast algorithms for enclosing all eigenvalues and invariant subspaces in generalized eigenvalue problems are proposed. In these algorithms, individual eigenvectors and invariant subspaces are enclosed when eigenvalues are well separated and closely clustered, respectively. The first algorithm involves only cubic complexity and automatically determines eigenvalue clusters. The second algorithm is applicable even for defective eigenvalues. Numerical results show the properties of the proposed algorithms.

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Available abstract

Two fast algorithms for enclosing all eigenvalues and invariant subspaces in generalized eigenvalue problems are proposed. In these algorithms, individual eigenvectors and invariant subspaces are enclosed when eigenvalues are well separated and closely clustered, respectively. The first algorithm involves only cubic complexity and automatically determines eigenvalue clusters. The second algorithm is applicable even for defective eigenvalues. Numerical results show the properties of the proposed algorithms.

Key concepts: Eigenvalues and eigenvectors, Linear subspace, Mathematics, Invariant (physics), Divide-and-conquer eigenvalue algorithm, Eigenvalue perturbation, Invariant subspace, Pure mathematics

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