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Approximating dominant singular triplets of large sparse matrices via modified moments (1996)

Varadhan, Sowmini ; Berry, Michael W. ; Golub, Gene H.

Springer

Numerical algorithms 13 (1996), S. 123-152

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ISSN: 1572-9265

Keywords: Chebyshev semi-iterative method ; modified moments ; singular value decomposition ; sparse matrices ; 65K15 ; 65K50 ; 68P20

Source: Springer Online Journal Archives 1860-2000

Topics: Computer Science , Mathematics

Notes: Abstract A procedure for determining a few of the largest singular values and corresponding singular vectors of large sparse matrices is presented. Equivalent eigensystems are solved using a technique originally proposed by Golub and Kent based on the computation of modified moments. The asynchronicity in the computations of moments and eigenvalues makes this method attractive for parallel implementations on a network of workstations. Although no obvious relationship between modified moments and the corresponding eigenvectors is known to exist, a scheme to approximate both eigenvalues and eigenvectors (and subsequently singular values and singular vectors) has been produced. This scheme exploits both modified moments in conjunction with the Chebyshev semi-iterative method and deflation techniques to produce approximate eigenpairs of the equivalent sparse eigensystems. The performance of an ANSI-C implementation of this scheme on a network of UNIX workstations and a 256-processor Cray T3D is presented.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF02143130

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Matrix shapes invariant under the symmetric QR algorithm (1995)

Arbenz, Peter ; Golub, Gene H.

New York, NY [u.a.] : Wiley-Blackwell

Numerical Linear Algebra with Applications 2 (1995), S. 87-93

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ISSN: 1070-5325

Keywords: QR algorithm ; zero pattern ; Engineering ; Engineering General

Source: Wiley InterScience Backfile Collection 1832-2000

Topics: Mathematics

Notes: The QR algorithm is a basic algorithm for computing the eigenvalues of dense matrices. For efficiency reasons it is prerequisite that the algorithm is applied only after the original matrix has been reduced to a matrix of a particular shape, most notably Hessenberg and tridiagonal, which is preserved during the iterative process. In certain circumstances a reduction to another matrix shape may be advantageous. In this paper, we identify which zero patterns of symmetric matrices are preserved under the QR algorithm.