×

On the quadratic convergence of Kogbetliantz’s algorithm for computing the singular value decomposition. (English) Zbl 0621.65031

The paper is concerned with Kogbetliantz’s algorithm for computing the singular value decomposition of matrices. Under some conditions, and for certain variants of the method, the authors prove its quadratic convergence. The proof is inspired by the paper of J. H. Wilkinson [Numer. Math. 4, 296-300 (1962; Zbl 0104.345)] concerning the quadratic convergence of the cyclic Jacobi method.
Reviewer: T.Reginska

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A18 Eigenvalues, singular values, and eigenvectors

Citations:

Zbl 0104.345
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brent, P.; Luk, F.; Van Loan, C., Computation of the singular value decomposition using mesh-connected processors, (Tech. Rep. CS-528 (1983), Dept. of Computer Science, Cornell Univ: Dept. of Computer Science, Cornell Univ Ithaca, N.Y) · Zbl 0597.65029
[2] Forsythe, G.; Henrici, P., The cyclic Jacobi method for computing the principal values of a complex matrix, Trans. Amer. Math. Soc., 94, 1-23 (1960) · Zbl 0092.32504
[3] M. Heath, A. Laub, C. C. Paige, and R. Ward, Computing the SDV of a product of matrices, in preparation.; M. Heath, A. Laub, C. C. Paige, and R. Ward, Computing the SDV of a product of matrices, in preparation. · Zbl 0607.65013
[4] Henrici, P., On the speed of convergence of cyclic and quasi cyclic Jacobi methods for computing eigenvalues of Hermitian matrices, J. SIAM, 6, 144-162 (1958) · Zbl 0097.32601
[5] Kogbetliantz, E., Diagonalization of general complex matrices as a new method for solution of linear equations, Amsterdam. Amsterdam, Proceedings of the International Congress on Mathematics, Vol. 2, 356-357 (1954)
[6] Kogbetliantz, E., Solution of linear equations by diagonalization of coefficient matrices, Quart. Appl. Math., 13, 123-132 (1955) · Zbl 0066.10101
[7] Lawson, C.; Hanson, R., Solving Least Squares Problems (1974), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J · Zbl 0860.65028
[8] C. C. Paige, Computing the generalized singular value decomposition, SIAM J. Sci. Statist. Comput.; C. C. Paige, Computing the generalized singular value decomposition, SIAM J. Sci. Statist. Comput. · Zbl 0621.65030
[9] Schönhage, A., Zur Konvergenz des Jacobi-Verfahrens, Numer. Math., 3, 374-380 (1961) · Zbl 0100.33105
[10] Stewart, G. W., Error and perturbation bounds for subspaces associated with certain eigenvalue problems, SIAM Rev., 15, 727-764 (1973) · Zbl 0297.65030
[11] van Kempen, H., On the convergence of the classical Jacobi method for real symmetric matrices with non-distinct eigenvalues, Numer. Math., 9, 11-18 (1966) · Zbl 0229.65037
[12] van Kempen, H., On the quadratic convergence of the special Jacobi method, Numer. Math., 9, 19-22 (1966) · Zbl 0229.65038
[13] Wilkinson, J. H., Note on the quadratic convergence of the cyclic Jacobi process, Numer. Math., 4, 296-300 (1962) · Zbl 0104.34501
[14] Wilkinson, J. H., Almost diagonal matrices with multiple or close eigenvalues, Linear Algebra Appl., 1, 1-12 (1968) · Zbl 0167.30303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.