Preserving symmetry in preconditioned krylov subspace methods

T. F. Chan*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


The research of these authors was supported by the NSF under contract We consider the problem of solving a linear system Ax = b when A is nearly symmetric and when the system is preconditioned by a symmetric positive definite matrix M. In the symmetric case, we can recover symmetry by using M-inner products in the conjugate gradient (CG) algorithm. This idea can also be used in the nonsymmetric case, and near symmetry can be preserved similarly. Like CG, the new algorithms are mathematically equivalent to split preconditioning but do not require M to be factored. Better robustness in a specific sense can also be observed. When combined with truncated versions of iterative methods, tests show that this is more effective than the common practice of forfeiting near-symmetry altogether.

Original languageEnglish (US)
Pages (from-to)568-581
Number of pages14
JournalSIAM Journal on Scientific Computing
Issue number2
StatePublished - 1998
Externally publishedYes


  • Incomplete orthogonalization
  • Inner-products
  • Preconditioned iterative methods

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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