The conjugate gradient squared algorithm can suffer of similar breakdowns as Lanczos type methods for the same reason that is the non-existence of some formal orthogonal polynomials. Thus curing such breakdowns is possible by jumping over these non-existing polynomials and using only those of them which exist. The technique used is similar to that employed for avoiding breakdowns in Lanczos type methods. The implementation of these new methods is discussed. Numerical examples are given.
Subject classificationsAMS (MOS) 65F10 65F25
KeywordsLanczos method conjugate gradient squared algorithm orthogonal polynomials
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- D.L. Boley, S. Elhay, G.H. Golub and M.H. Gutknecht, Nonsymmetric Lanczos and finding orthogonal polynomials associated with indefinite weights, Numer. Algorithms 1 (1991) 21–43.Google Scholar
- C. Brezinski and M. Redivo Zaglia, A new presentation of orthogonal polynomials with applications to their computation, Numer. Algorithms 1 (1991) 207–222.Google Scholar
- C. Brezinski, M. Redivo Zaglia and H. Sadok, A breakdown-free Lanczos type algorithm for solving linear systems, Numer. Math., to appear.Google Scholar
- C. Brezinski, M. Redivo Zaglia and H. Sadok, Avoiding breakdown and near-breakdown in Lanczos type algorithms Numer. Algorithms, 1 (1991), to appear.Google Scholar
- A. Draux,Polynômes Orthogonaux Formels. Applications, LNM 974 (Springer Verlag, Berlin, 1983).Google Scholar
- M.H. Gutknecht, The unsymmetric Lanczos algorithms and their relations to Padé approximation, continued fractions, and the qd-algorithm, to appear.Google Scholar
- N.M. Nachtigal, S.C. Reddy, L.N. Trefethen, How fast are nonsymmetric matrix iteractions?, SIAM J. Sci. Stat. Comp., to appear.Google Scholar
- P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comp. 10 (1989) 36–52.Google Scholar
- Yu. V. Vorobyev,Method of Moments in Applied Mathematics (Gordon and Breach, New York, 1965).Google Scholar