Rounding error and perturbation bounds for the symplectic QR factorization

To compute the eigenvalues of a skew-symmetric matrix A, we can use a one-sided Jacobi-like algorithm to enhance accuracy. This algorithm begins by a suitable Cholesky-like factorization of A, A=G T JG . In some applications, A is given implicitly in that form and its natural Cholesky-like factor G...

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Bibliographic Details
Published in:Linear algebra and its applications Vol. 358; no. 1; pp. 255 - 279
Main Authors: Singer, Sanja, Singer, Saša
Format: Journal Article
Language:English
Published: Elsevier Inc 2003
Subjects:
Perturbation bounds
Rounding error bounds
Skew-symmetric eigenproblem
Symplectic QR factorization
Skew-symmetric eigenproblem
Symplectic QR factorization
Rounding error bounds
Perturbation bounds
ISSN:0024-3795, 1873-1856
Online Access:Get full text
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Summary:To compute the eigenvalues of a skew-symmetric matrix A, we can use a one-sided Jacobi-like algorithm to enhance accuracy. This algorithm begins by a suitable Cholesky-like factorization of A, A=G T JG . In some applications, A is given implicitly in that form and its natural Cholesky-like factor G is immediately available, but “tall”, i.e., not of full row rank. This factor G is unsuitable for the Jacobi-like process. To avoid explicit computation of A, and possible loss of accuracy, the factor has to be preprocessed by a QR-like factorization. In this paper we present the symplectic QR algorithm to achieve such a factorization, together with the corresponding rounding error and perturbation bounds. These bounds fit well into the relative perturbation theory for skew-symmetric matrices given in factorized form.
ISSN:0024-3795
1873-1856
DOI:10.1016/S0024-3795(02)00263-X