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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| Vydáno v: | Linear algebra and its applications Ročník 358; číslo 1; s. 255 - 279 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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2003
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| ISSN: | 0024-3795, 1873-1856 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. |
| Author | Singer, Sanja Singer, Saša |
| Author_xml | – sequence: 1 givenname: Sanja surname: Singer fullname: Singer, Sanja email: ssinger@math.hr, singer@math.hr organization: Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lučića 5, 10000 Zagreb, Croatia – sequence: 2 givenname: Saša surname: Singer fullname: Singer, Saša organization: Department of Mathematics, University of Zagreb, P.O. Box 335, 10002 Zagreb, Croatia |
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| Cites_doi | 10.1007/978-94-015-8196-7_52 10.1016/0024-3795(86)90265-X 10.1016/S0024-3795(99)00156-1 |
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| Keywords | Skew-symmetric eigenproblem Symplectic QR factorization Rounding error bounds Perturbation bounds |
| Language | English |
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| References | S. Singer, Indefinite QR factorization and its applications, Ph.D. thesis, Department of Mathematics, University of Zagreb, 1997 (in Croatian) Singer, Singer (BIB8) 2000; 309 E. Pietzsch, Genaue Eigenwertberechnung Nichtsingulärer Schiefsymmetrischer Matrizen, Ph.D. thesis, FernUniversität–Gesamthochschule, Hagen, 1993 Stewart, Sun (BIB10) 1990 Bunch (BIB2) 1982; 38 Higham (BIB5) 1996 I. Slapničar, Accurate symmetric eigenreduction by a Jacobi method, Ph.D. thesis, FernUniversität–Gesamthochschule, Hagen, 1992 Z. Drmač, Computing the singular and the generalized singular values, Ph.D. thesis, FernUniversität–Gesamthochschule, Hagen, 1994 Benner, Byers, Fassbender, Mehrmann, Watkins (BIB1) 2000; 11 Bunse-Gerstner (BIB3) 1986; 83 10.1016/S0024-3795(02)00263-X_BIB9 Higham (10.1016/S0024-3795(02)00263-X_BIB5) 1996 10.1016/S0024-3795(02)00263-X_BIB7 Singer (10.1016/S0024-3795(02)00263-X_BIB8) 2000; 309 Benner (10.1016/S0024-3795(02)00263-X_BIB1) 2000; 11 Bunch (10.1016/S0024-3795(02)00263-X_BIB2) 1982; 38 Bunse-Gerstner (10.1016/S0024-3795(02)00263-X_BIB3) 1986; 83 10.1016/S0024-3795(02)00263-X_BIB6 Stewart (10.1016/S0024-3795(02)00263-X_BIB10) 1990 10.1016/S0024-3795(02)00263-X_BIB4 |
| References_xml | – volume: 11 start-page: 85 year: 2000 end-page: 93 ident: BIB1 article-title: Cholesky-like factorizations of skew-symmetric matrices publication-title: Electron. Trans. Numer. Anal. – volume: 38 start-page: 475 year: 1982 end-page: 479 ident: BIB2 article-title: A note on the stable decomposition of skew-symmetric matrices publication-title: Math. Comp. – volume: 83 start-page: 49 year: 1986 end-page: 77 ident: BIB3 article-title: Matrix factorizations for symplectic QR-like methods publication-title: Linear Algebra Appl. – reference: Z. Drmač, Computing the singular and the generalized singular values, Ph.D. thesis, FernUniversität–Gesamthochschule, Hagen, 1994 – reference: S. Singer, Indefinite QR factorization and its applications, Ph.D. thesis, Department of Mathematics, University of Zagreb, 1997 (in Croatian) – reference: I. Slapničar, Accurate symmetric eigenreduction by a Jacobi method, Ph.D. thesis, FernUniversität–Gesamthochschule, Hagen, 1992 – year: 1996 ident: BIB5 publication-title: Accuracy and Stability of Numerical Algorithms – year: 1990 ident: BIB10 article-title: Matrix Perturbation Theory – reference: E. Pietzsch, Genaue Eigenwertberechnung Nichtsingulärer Schiefsymmetrischer Matrizen, Ph.D. thesis, FernUniversität–Gesamthochschule, Hagen, 1993 – volume: 309 start-page: 103 year: 2000 end-page: 119 ident: BIB8 article-title: Rounding-error and perturbation bounds for the indefinite QR factorization publication-title: Linear Algebra Appl. – volume: 11 start-page: 85 year: 2000 ident: 10.1016/S0024-3795(02)00263-X_BIB1 article-title: Cholesky-like factorizations of skew-symmetric matrices publication-title: Electron. Trans. Numer. Anal. – ident: 10.1016/S0024-3795(02)00263-X_BIB9 doi: 10.1007/978-94-015-8196-7_52 – ident: 10.1016/S0024-3795(02)00263-X_BIB4 – volume: 83 start-page: 49 year: 1986 ident: 10.1016/S0024-3795(02)00263-X_BIB3 article-title: Matrix factorizations for symplectic QR-like methods publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(86)90265-X – volume: 309 start-page: 103 year: 2000 ident: 10.1016/S0024-3795(02)00263-X_BIB8 article-title: Rounding-error and perturbation bounds for the indefinite QR factorization publication-title: Linear Algebra Appl. doi: 10.1016/S0024-3795(99)00156-1 – year: 1990 ident: 10.1016/S0024-3795(02)00263-X_BIB10 – ident: 10.1016/S0024-3795(02)00263-X_BIB7 – year: 1996 ident: 10.1016/S0024-3795(02)00263-X_BIB5 – ident: 10.1016/S0024-3795(02)00263-X_BIB6 – volume: 38 start-page: 475 year: 1982 ident: 10.1016/S0024-3795(02)00263-X_BIB2 article-title: A note on the stable decomposition of skew-symmetric matrices publication-title: Math. Comp. |
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| Snippet | 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... |
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| SubjectTerms | Perturbation bounds Rounding error bounds Skew-symmetric eigenproblem Symplectic QR factorization |
| Title | Rounding error and perturbation bounds for the symplectic QR factorization |
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