Reaching the superlinear convergence phase of the CG method

O. Axelsson, J. Karátson

Research output: Contribution to journalArticle

2 Citations (Scopus)


The rate of convergence of the conjugate gradient method takes place in essentially three phases, with respectively a sublinear, a linear and a superlinear rate. The paper examines when the superlinear phase is reached. To do this, two methods are used. One is based on the K-condition number, thereby separating the eigenvalues in three sets: small and large outliers and intermediate eigenvalues. The other is based on annihilating polynomials for the eigenvalues and, assuming various analytical distributions of them, thereby using certain refined estimates. The results are illustrated for some typical distributions of eigenvalues and with some numerical tests.

Original languageEnglish
Pages (from-to)244-257
Number of pages14
JournalJournal of Computational and Applied Mathematics
Publication statusPublished - Jan 1 2014


  • Conjugate gradient method
  • Superlinear convergence

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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