Library

X
feed icon rss

Your email was sent successfully. Check your inbox.

An error occurred while sending the email. Please try again.

Proceed reservation?

Export
  • 1
    Electronic Resource
    Electronic Resource
    A look-ahead Bareiss algorithm for general Toeplitz matrices (1994)
    Springer
    Numerische Mathematik 68 (1994), S. 35-69 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65F05, 15A57, 42C05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. The Bareiss algorithm is one of the classical fast solvers for systems of linear equations with Toeplitz coefficient matrices. The method takes advantage of the special structure, and it computes the solution of a Toeplitz system of order~ $N$ with only~ $O(N^2)$ arithmetic operations, instead of~ $O(N^3)$ operations. However, the original Bareiss algorithm requires that all leading principal submatrices be nonsingular, and the algorithm is numerically unstable if singular or ill-conditioned submatrices occur. In this paper, an extension of the Bareiss algorithm to general Toeplitz systems is presented. Using look-ahead techniques, the proposed algorithm can skip over arbitrary blocks of singular or ill-conditioned submatrices, and at the same time, it still fully exploits the Toeplitz structure. Implementation details and operations counts are given, and numerical experiments are reported. We also discuss special versions of the proposed look-ahead Bareiss algorithm for Hermitian indefinite Toeplitz systems and banded Toeplitz systems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050047
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 2
    Electronic Resource
    Electronic Resource
    QMR: a quasi-minimal residual method for non-Hermitian linear systems (1991)
    Springer
    Numerische Mathematik 60 (1991), S. 315-339 
    Details
    ISSN: 0945-3245
    Keywords: 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The biconjugate gradient (BCG) method is the “natural” generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. In this paper, we present a novel BCG-like approach, the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385726
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 3
    Electronic Resource
    Electronic Resource
    A look-ahead algorithm for the solution of general Hankel systems (1993)
    Springer
    Numerische Mathematik 64 (1993), S. 295-321 
    Details
    ISSN: 0945-3245
    Keywords: 65F05 ; 15A57 ; 42C05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01388691
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 4
    Electronic Resource
    Electronic Resource
    Congratulations (1994)
    Springer
    Numerische Mathematik 68 (1994), S. 1-1 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050044
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 5
    Electronic Resource
    Electronic Resource
    An interior-point method for fractional programs with convex constraints (1994)
    Springer
    Mathematical programming 67 (1994), S. 407-440 
    Details
    ISSN: 1436-4646
    Keywords: Interior-point method ; Fractional programs ; Convex constraints ; Polynomial-rate convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We present an interior-point method for a class of fractional programs with convex constraints. The proposed algorithm converges at a polynomial rate, similarly as in the case of a convex problem, even though fractional programs are only pseudo-convex. Here, the rate of convergence is measured in terms of the area of two-dimensional convex setsC k containing the origin and certain projections of the optimal points, and the area ofC k is reduced by a constant factorc 〈 1 at each iteration. The factorc depends only on the self-concordance parameter of a barrier function associated with the feasible set. We present an outline of a practical implementation of the proposed method, and we report results of some preliminary numerical experiments.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582229
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 6
    Electronic Resource
    Electronic Resource
    Editorial (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 335-335 
    Details
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680010402
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 7
    Electronic Resource
    Electronic Resource
    On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 403-420 
    Details
    ISSN: 1070-5325
    Keywords: Quasi-minimal residual iteration ; Non-Hermitian matrix ; Singular linear system ; Markov chain modeling ; Krylov-subspace method ; Convergence ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Recently, Freund and Nachtigal proposed the quasi-minimal residual algorithm (QMR) for solving general nonsingular non-Hermitian linear systems. The method is based on the Lanczos process, and thus it involves matrix - vector products with both the coefficient matrix of the linear system and its transpose. Freund developed a variant of QMR, the transpose-free QMR algorithm (TFQMR), that only requires products with the coefficient matrix. In this paper, the use of QMR and TFQMR for solving singular systems is explored. First, a convergence result for the general class of Krylov-subspace methods applied to singular systems is presented. Then, it is shown that QMR and TFQMR both converge for consistent singular linear systems with coefficient matrices of index 1. Singular systems of this type arise in Markov chain modeling. For this particular application, numerical experiments are reported.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680010406
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
Close ⊗
This website uses cookies and the analysis tool Matomo. More information can be found here...