Solving Ax = b using a modified conjugate gradient method based on roots of A

P. F. Fischer, S. Gottlieb

Research output: Contribution to journalArticle

Abstract

We consider the modified conjugate gradient procedure for solving Ax = b in which the approximation space is based upon the Krylov space associated with A1/p and b, for any integer p. For the square-root MCG (p = 2) we establish a sharpened bound for the error at each iteration via Chebyshev polynomials in √A. We discuss the implications of the quickly accumulating effect of an error in √A b in the initial stage, and find an error bound even in the presence of such accumulating errors. Although this accumulation of errors may limit the usefulness of this method when √A b is unknown, it may still be successfully applied to a variety of small, "almost-SPD" problems, and can be used to jump-start the conjugate gradient method. Finally, we verify these theoretical results with numerical tests.

Original languageEnglish (US)
Pages (from-to)441-456
Number of pages16
JournalJournal of Scientific Computing
Volume15
Issue number4
DOIs
StatePublished - Dec 1 2001
Externally publishedYes

Keywords

  • Conjugate gradient method
  • Convergence rate
  • Krylov space
  • Modified conjugate gradient method
  • Stability

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Engineering(all)
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

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