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Numerische Mathematik

, Volume 75, Issue 3, pp 267–292 | Cite as

On multisplitting methods for band matrices

  • Götz Alefeld
  • Ingrid Lenhardt
  • Günter Mayer

Summary.

We present new theoretical results on two classes of multisplitting methods for solving linear systems iteratively. These classes are based on overlapping blocks of the underlying coefficient matrix \( A \) which is assumed to be a band matrix. We show that under suitable conditions the spectral radius \( \rho(H) \) of the iteration matrix \( H \) does not depend on the weights of the method even if these weights are allowed to be negative. For a certain class of splittings we prove an optimality result for \( \rho(H) \) with respect to the weights provided that \( A \) is an M–matrix. This result is based on the fact that the multisplitting method can be represented by a single splitting \( A = M - N \) which in our situation surprisingly turns out to be a regular splitting. Furthermore we show by numerical examples that weighting factors \( \alpha \not \in [0,1] \) may considerably improve the convergence.

Mathematics Subject Classification (1991):65F10 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Götz Alefeld
    • 1
  • Ingrid Lenhardt
    • 1
  • Günter Mayer
    • 2
  1. 1.Institut für Angewandte Mathematik, Universität Karlsruhe, D-76128 Karlsruhe, GermanyDE
  2. 2.Fachbereich Mathematik, Universität Rostock, D-18051 Rostock, GermanyDE

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