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

• Carmo P. Brás ^[1] ; William W. Hager ^[2] ; Joaquim J. Júdice ^[3]
  1. [1] Universidade Nova de Lisboa

     Universidade Nova de Lisboa

     Socorro, Portugal

     
  2. [2] University of Florida

     University of Florida

     Estados Unidos

     
  3. [3] Universidade de Coimbra

     Universidade de Coimbra

     Coimbra (Sé Nova), Portugal

     

  Mostrar afiliaciones +
• Localización: Top, ISSN-e 1863-8279, ISSN 1134-5764, Vol. 20, Nº. 3, 2012, págs. 791-809
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • 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.