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  • ddc:000  (2)
  • 42C05  (1)
  • 65F30  (1)
  • 65F35  (1)
  • 1
    Digitale Medien
    Digitale Medien
    A look-ahead algorithm for the solution of general Hankel systems (1993)
    Springer
    Numerische Mathematik 64 (1993), S. 295-321 
    Details
    ISSN: 0945-3245
    Schlagwort(e): 65F05 ; 15A57 ; 42C05
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The solution of systems of linear equations with Hankel coefficient matrices can be computed with onlyO(n 2) arithmetic operations, as compared toO(n 3) operations for the general cases. However, the classical Hankel solvers require the nonsingularity of all leading principal submatrices of the Hankel matrix. The known extensions of these algorithms to general Hankel systems can handle only exactly singular submatrices, but not ill-conditioned ones, and hence they are numerically unstable. In this paper, a stable procedure for solving general nonsingular Hankel systems is presented, using a look-ahead technique to skip over singular or ill-conditioned submatrices. The proposed approach is based on a look-ahead variant of the nonsymmetric Lanczos process that was recently developed by Freund, Gutknecht, and Nachtigal. We first derive a somewhat more general formulation of this look-ahead Lanczos algorithm in terms of formally orthogonal polynomials, which then yields the look-ahead Hankel solver as a special case. We prove some general properties of the resulting look-ahead algorithm for formally orthogonal polynomials. These results are then utilized in the implementation of the Hankel solver. We report some numerical experiments for Hankel systems with ill-conditioned submatrices.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01388691
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  • 2
    Digitale Medien
    Digitale Medien
    The product-product singular value decomposition of matrix triplets (1991)
    Springer
    BIT 31 (1991), S. 711-726 
    Details
    ISSN: 1572-9125
    Schlagwort(e): 65F25 ; 65F30 ; 65F35 ; singular value decomposition ; matrix product ; implicit Kogbetliantz technique
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract A new decomposition of a matrix triplet (A, B, C) corresponding to the singular value decomposition of the matrix productABC is developed in this paper, which will be termed theProduct-Product Singular Value Decomposition (PPSVD). An orthogonal variant of the decomposition which is more suitable for the purpose of numerical computation is also proposed. Some geometric and algebraic issues of the PPSVD, such as the variational and geometric interpretations, and uniqueness properties are discussed. A numerical algorithm for stably computing the PPSVD is given based on the implicit Kogbetliantz technique. A numerical example is outlined to demonstrate the accuracy of the proposed algorithm.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01933183
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  • 3
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    A Numerical Algorithm for Computing the Restricted Singular Value Decomposition of Matrix Triplets. (1989)
    Details
    Publikationsdatum: 2014-02-26
    Beschreibung: This paper presents a numerical algorithm for computing the restricted singular value decomposition of matrix triplets (RSVD). It is shown that one can use unitary transformations to separate the regular part from a general matrix triplet. After preprocessing on the regular part, one obtains a matrix triplet consisting of three upper triangular matrices of the same dimensions. The RSVD of this special matrix triplet is computed using the implicit Kogbetliantz technique. The algorithm is well suited for parallel computation. {\bf Keywords:} Restricted singular values, matrix triplets, unitary transformations, implicit Kogbetliantz technique.
    Schlagwort(e): ddc:000
    Sprache: Englisch
    Materialart: reportzib , doc-type:preprint
    Format: application/pdf
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  • 4
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    Restricted Singular Value Decomposition of Matrix Triplets. (1989)
    Details
    Publikationsdatum: 2014-02-26
    Beschreibung: In this paper we introduce the concept of restricted singular values (RSV's) of matrix triplets. A theorem concerning the RSV's of a general matrix triplet $ (A,B,C) $, where $ A \in C^{m\times n} $, $B\in C^{m\times p} $ and $ C\in C^{q\times n} $, which is called restricted singular value decomposition (RSVD) of matrix triplets, is derived. This result generalizes the wellknown SVD, GSVD and the recently proposed product induced SVD (PSVD). Connection of RSV's with the problem of determination of matrix rank under restricted perturbation is also discussed. {\bf Keywords:} Matrix rank, singular values, generalized singular values, product induced singular values, restricted singular values, matrix decompositions.
    Schlagwort(e): ddc:000
    Sprache: Englisch
    Materialart: reportzib , doc-type:preprint
    Format: application/pdf
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    OPUS
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