Orthogonal polynomials of the R-linear generalized minimal residual method

Marko Huhtanen*, Allan Perämäki

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

5 Citations (Scopus)


The speed of convergence of the R-linear GMRES method is bounded in terms of a polynomial approximation problem on a finite subset of the spectrum. This result resembles the classical GMRES convergence estimate except that the matrix involved is assumed to be condiagonalizable. The bounds obtained are applicable to the CSYM method, in which case they are sharp. Then a new three term recurrence for generating a family of orthogonal polynomials is shown to exist, yielding a natural link with complex symmetric Jacobi matrices. This shows that a mathematical framework analogous to the one appearing with the Hermitian Lanczos method exists in the complex symmetric case. The probability of being condiagonalizable is estimated with random matrices.

Original languageEnglish
Pages (from-to)220-239
Number of pages20
JournalJournal of Approximation Theory
Publication statusPublished - Mar 2013
MoE publication typeA1 Journal article-refereed


  • Condiagonalizable
  • CSYM
  • Jacobi matrix
  • Orthogonal polynomial
  • Polynomial approximation
  • R-linear GMRES
  • Random matrix
  • Spectrum
  • Three term recurrence


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