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An investigation of feasible descent algorithms for estimating the condition number of a matrix

Author

Listed:
  • Carmo Brás

    ()

  • William Hager

    ()

  • Joaquim Júdice

    ()

Abstract

Techniques for estimating the condition number of a nonsingular matrix are developed. It is shown that Hager’s 1-norm condition number estimator is equivalent to the conditional gradient algorithm applied to the problem of maximizing the 1-norm of a matrix-vector product over the unit sphere in the 1-norm. By changing the constraint in this optimization problem from the unit sphere to the unit simplex, a new formulation is obtained which is the basis for both conditional gradient and projected gradient algorithms. In the test problems, the spectral projected gradient algorithm yields condition number estimates at least as good as those obtained by the previous approach. Moreover, in some cases, the spectral gradient projection algorithm, with a careful choice of the parameters, yields improved condition number estimates. Copyright Sociedad de Estadística e Investigación Operativa 2012

Suggested Citation

  • Carmo Brás & William Hager & Joaquim Júdice, 2012. "An investigation of feasible descent algorithms for estimating the condition number of a matrix," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(3), pages 791-809, October.
  • Handle: RePEc:spr:topjnl:v:20:y:2012:i:3:p:791-809 DOI: 10.1007/s11750-010-0161-9
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    References listed on IDEAS

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    1. Still, G., 1999. "Generalized semi-infinite programming: Theory and methods," European Journal of Operational Research, Elsevier, vol. 119(2), pages 301-313, December.
    2. Tijs, S.H., 1979. "Semi-infinite linear programs and semi-infinite matrix games," Other publications TiSEM 3e08d5e1-de1d-45b2-81b3-3, Tilburg University, School of Economics and Management.
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