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On the Superlinear Convergence of MINRES

In: Numerical Mathematics and Advanced Applications 2011

Author

Listed:
  • V. Simoncini

    (Università di Bologna, Dipartimento di Matematica
    CIRSA)

  • D. B. Szyld

    (Temple University (038-16), Department of Mathematics)

Abstract

Quantitative bounds are presented for the superlinear convergence of the MINRES method of Paige and Saunders (SIAM J Numer Anal 12:617–629, 1975) for the solution of sparse linear systems Ax=b, with A symmetric and indefinite. It is shown that the superlinear convergence is observed as soon as the harmonic Ritz values approximate well the eigenvalues of A that are either closest to zero or farthest from zero. This generalizes a well-known corresponding result obtained by van der Sluis and van der Vorst with respect to the Conjugate Gradients method, for A symmetric and positive definite.

Suggested Citation

  • V. Simoncini & D. B. Szyld, 2013. "On the Superlinear Convergence of MINRES," Springer Books, in: Andrea Cangiani & Ruslan L. Davidchack & Emmanuil Georgoulis & Alexander N. Gorban & Jeremy Levesley (ed.), Numerical Mathematics and Advanced Applications 2011, edition 127, pages 733-740, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-33134-3_77
    DOI: 10.1007/978-3-642-33134-3_77
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