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Perturbation theory for the LDU factorization and accurate computations for diagonally dominant matrices

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Abstract

We present a structured perturbation theory for the LDU factorization of (row) diagonally dominant matrices and we use this theory to prove that a recent algorithm of Ye (Math Comp 77(264):2195–2230, 2008) computes the L, D and U factors of these matrices with relative errors less than 14n 3 u, where u is the unit roundoff and n × n is the size of the matrix. The relative errors for D are componentwise and for L and U are normwise with respect the “max norm” \({\|A\|_M = \max_{ij} |a_{ij}|}\). These error bounds guarantee that for any diagonally dominant matrix A we can compute accurately its singular value decomposition and the solution of the linear system Ax = b for most vectors b, independently of the magnitude of the traditional condition number of A and in O(n 3) flops.

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Authors and Affiliations

  1. Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM and Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Spain

    Froilán M. Dopico

  2. Department of Mathematics, San Jose State University, One Washington Square, San Jose, CA, 95192, USA

    Plamen Koev

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  1. Froilán M. Dopico
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  2. Plamen Koev
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Correspondence to Froilán M. Dopico.

Additional information

F. M. Dopico was partially supported by the Ministerio de Ciencia e Innovación of Spain through grant MTM-2009-09281.

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Dopico, F.M., Koev, P. Perturbation theory for the LDU factorization and accurate computations for diagonally dominant matrices. Numer. Math. 119, 337–371 (2011). https://doi.org/10.1007/s00211-011-0382-3

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  • DOI: https://doi.org/10.1007/s00211-011-0382-3

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