s-step iterative methods for symmetric linear systems

A. T. Chronopoulos, C. W. Gear

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140 Scopus citations


In this paper we introduce s-step Conjugate Gradient Method for Symmetric and Positive Definite (SPD) linear systems of equations and discuss its convergence. In the s-step Conjugate Gradient Method iteration s new directions are formed simultaneously from {geometrically equivalent to}ri, Ari,...,As-1ri{geometrically equivalent to} and the preceding s directions. All s directions are chosen to be A-orthogonal to the preceding s directions. The approximation to the solution is then advanced by minimizing an error functional simultaneously in all s directions. This intuitively means that the progress towards the solution in one iteration of the s-step method equals the progress made over s consecutive steps of the one-step method. This is proven to be true.

Original languageEnglish (US)
Pages (from-to)153-168
Number of pages16
JournalJournal of Computational and Applied Mathematics
Issue number2
StatePublished - Feb 1989

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics


  • Iterative methods
  • conjugate gradient
  • convergence
  • s-step


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