PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1994 | 29 | 1 | 19-33
Tytuł artykułu

Orthogonal polynomials and the Lanczos method

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Lanczos method for solving a system of linear equations is well known. It is derived from a generalization of the method of moments and one of its main interests is that it provides the exact answer in at most n steps where n is the dimension of the system. Lanczos method can be implemented via several recursive algorithms known as Orthodir, Orthomin, Orthores, Biconjugate gradient,... In this paper, we show that all these procedures can be explained within the framework of formal orthogonal polynomials. This theory also provides a natural basis for curing breakdown and near-breakdown in these algorithms. The case of the conjugate gradient squared method can be treated similarly.
Rocznik
Tom
29
Numer
1
Strony
19-33
Opis fizyczny
Daty
wydano
1994
Twórcy
autor
  • Laboratoire d'Analyse Numérique et d'Optimisation, UFR IEEA-M3, Université des Sciences et Technologies de Lille, F-59655 Villeneuve d'Ascq Cedex, France
autor
  • Laboratoire d'Analyse Numérique et d'Optimisation, UFR IEEA-M3, Université des Sciences et Technologies de Lille, F-59655 Villeneuve d'Ascq Cedex, France
  • Dipartimento di Elettronica e Informatica, Università degli Studi di Padova, Via Gradenigo 6/a, I-35131 Padova, Italy
Bibliografia
  • [1] C. Brezinski, Padé-type Approximation and General Orthogonal Polynomials, Internat. Ser. Number. Math. 50, Birkhäuser, Basel 1980.
  • [2] C. Brezinski, The methods of Vorobyev and Lanczos, to appear.
  • [3] C. Brezinski, Biorthogonality and its Applications to Numerical Analysis, Marcel Dekker, New York 1992.
  • [4] C. Brezinski and M. Redivo Zaglia, A new presentation of orthogonal polynomials with application to their computation, Numer. Algorithms 1 (1991), 207-221.
  • [5] C. Brezinski and M. Redivo Zaglia, Treatment of near-breakdown in the CGS algorithm, to appear.
  • [6] C. Brezinski, M. Redivo Zaglia and H. Sadok, A breakdown-free Lanczos type algorithm for solving linear systems, Numer. Math. 63 (1992), 29-38.
  • [7] C. Brezinski, M. Redivo Zaglia and H. Sadok, Avoiding breakdown and near-breakdown in Lanczos type algorithms, Numer. Algorithms 1 (1991), 261-284.
  • [8] C. Brezinski and H. Sadok, Lanczos type methods for solving systems of linear equations, Appl. Numer. Math., to appear.
  • [9] C. Brezinski and H. Sadok, Avoiding breakdown in the CGS algorithm, Numer. Algorithms 1 (1991), 199-206.
  • [10] A. Draux, Polynômes Orthogonaux Formels - Applications, Lecture Notes in Math. 974, Springer, Berlin 1983.
  • [11] R. Fletcher, Conjugate gradient methods for indefinite systems, in: Numerical Analysis, G. A. Watson (ed.), Lecture Notes in Math. 506, Springer, Berlin 1976, 73-89.
  • [12] M. H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms. Part I, SIAM J. Matrix Anal. Appl. 13 (1992), 594-639.
  • [13] M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards 49 (1952), 409-439.
  • [14] C. Lanczos, Solution of systems of linear equations by minimized iterations, ibid., 33-53.
  • [15] P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 10 (1989), 36-52.
  • [16] Yu. V. Vorobyev, Method of Moments in Applied Mathematics, Gordon and Breach, New York 1965.
  • [17] D. M. Young and K. C. Jea, Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods, Linear Algebra Appl. 34 (1980), 159-194.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv29z1p19bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.