A linear system solver based on a modified Krylov subspace method for breakdown recovery

Charles H. Tong, Qiang Ye

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


Despite its usefulness in solving eigenvalue problems and linear systems of equations, the nonsymmetric Lanczos method is known to suffer from a potential breakdown problem. Previous and recent approaches for handling the Lanczos exact and near-breakdowns include, for example, the look-ahead schemes by Parlett-Taylor-Liu [23], Freund-Gutknecht-Nachtigal [9], and Brezinski-Redivo Zaglia-Sadok [4]; the combined look-ahead and restart scheme by Joubert [18]; and the low-rank modified Lanczos scheme by Huckle [17]. In this paper, we present yet another scheme based on a modified Krylov subspace approach for the solution of nonsymmetric linear systems. When a breakdown occurs, our approach seeks a modified dual Krylov subspace, which is the sum of the original subspace and a new Krylov subspace Km(wj, AT) where wj is a newstart vector (this approach has been studied by Ye [26] for eigenvalue computations). Based on this strategy, we have developed a practical algorithm for linear systems called the MLAN/QM algorithm, which also incorporates the residual quasi-minimization as proposed in [12]. We present a few convergence bounds for the method as well as numerical results to show its effectiveness.

Original languageEnglish
Pages (from-to)233-251
Number of pages19
JournalNumerical Algorithms
Issue number1
StatePublished - Apr 1996


  • Lanczos breakdown
  • Nonsymmetric linear systems
  • Residual quasi-minimization

ASJC Scopus subject areas

  • Applied Mathematics


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