Advertisement

Iterationsverfahren zur Lösung großer linearer Gleichungssysteme, einige weitere Verfahren

  • Josef Stoer
  • Roland Bulirsch
Chapter
  • 197 Downloads
Part of the Springer-Lehrbuch book series (SLB)

Zusammenfassung

Viele praktische Probleme führen zu der Aufgabe, sehr große lineare Gleichungssyteme Ax = b zu lösen, bei denen glücklicherweise die Matrix A nur schwach besetzt ist, d. h. nur relativ wenige nicht verschwindende Komponenten besitzt. Solche Gleichungssysteme erhält man z. B. bei der Anwendung von Differenzenverfahren oder finite-element Methoden zur näherungsweisen Lösung von Randwertaufgaben bei partiellen Differentialgleichungen. Die üblichen Eliminationsverfahren [s. Kapitel 4] können hier nicht ohne weiteres zur Lösung verwandt werden, weil sie ohne besondere Maßnahmen gewöhnlich zur Bildung von mehr oder weniger voll besetzten Zwischenmatrizen führen und deshalb die Zahl der zur Lösung erforderlichen Rechenoperationen auch für die heutigen Rechner zu groß wird, abgesehen davon, daß die Zwischenmatrizen nicht mehr in die üblicherweise verfügbaren Maschinenspeicher passen.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literatur zu Kapitel 8

  1. Arnoldi, W.E. (1951): The principle of minimized iteration in the solution of the matrix eigenvalue problem Quart. Appl. Math. 9 17–29MathSciNetzbMATHGoogle Scholar
  2. Axelsson, O. (1977): Solution of linear systems of equations: Iterative methods. In: Barker (1977).Google Scholar
  3. Axelsson, O. (1994): Iterative Solution Methods Cambridge, UK: Cambridge University Press.zbMATHCrossRefGoogle Scholar
  4. Barker, V.A. (Ed.) (1977): Sparse Matrix techniques. Lecture Notes in Mathematics Vol. 572, Berlin-Heidelberg-New York: Springer.zbMATHCrossRefGoogle Scholar
  5. Braess, D. (1997): Finite Elemente. Berlin-Heidelberg-New York: SpringerzbMATHGoogle Scholar
  6. Bramble, J.H. (1993): Multigrid Methods. Harlow: Longman.zbMATHGoogle Scholar
  7. Brandt, A. (1977): Multi-level adaptive solutions to boundary value problems Math. of Comput. 31 333–390zbMATHCrossRefGoogle Scholar
  8. Briggs, W.L. (1987) A Multigrid Tutorial. Philadelphia: SIAMzbMATHGoogle Scholar
  9. Buneman, O. (1969): A compact non-iterative Poisson solver. Stanford University, Institute for Plasma Research Report Nr. 294, Stanford, CA.Google Scholar
  10. Buzbee, B.L., Dorr, F.W. (1974): The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular regions. SIAM J. Numer. Anal 11, 753–763.MathSciNetzbMATHCrossRefGoogle Scholar
  11. F.W., George, J.A., Golub, G.H. (1971): The direct solution of the discrete Poisson equation on irregular regions SIAM J. Numer. Anal. 8 722–736.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Golub, G.H., Nielson, C.W. (1970): On direct methods for solving Poisson’s equations SIAM J. Numer. Anal. 7 627–656MathSciNetzbMATHCrossRefGoogle Scholar
  13. Chan, T.F., Glowinski, R., Periaux, J., Widlund, O. (Eds.) (1989): Proceedings of the Second International Symposium on Domain Decomposition Methods. Philadelphia: SIAM.Google Scholar
  14. Fletcher, R. (1974). Conjugate gradient methods for indefinite systems. In: G.A. Watson (ed.), Proceedings of the Dundee Biennial Conference on Numerical Analysis 1974, p. 73–89. New York: Springer-Verlag 1975.Google Scholar
  15. Forsythe, G.E., Moler, C.B. (1967): Computer Solution of Linear Algebraic Systems. Series in Automatic Computation. Englewood Cliffs, N.J.: Prentice Hall.Google Scholar
  16. Freund, R.W., Nachtigal, N.M. (1991): QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numerische Mathematik 60, 315–339.MathSciNetzbMATHCrossRefGoogle Scholar
  17. George, A. (1973): Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal 10, 345–363.zbMATHCrossRefGoogle Scholar
  18. Glowinski, R., Golub, G.H., Meurant, G.A., Periaux, J. (Eds.) (1988): Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations. Philadelphia: SIAM.Google Scholar
  19. Hackbusch, W. (1985): Multigrid Methods and Applications. Berlin-Heidelberg-New York: Springer-Verlag.CrossRefGoogle Scholar
  20. Trottenberg, U. (Eds.) (1982): Multigrid Methods. Lecture Notes in Mathematics. Vol. 960. Berlin-Heidelberg-New York: Springer-Verlag.zbMATHGoogle Scholar
  21. Hestenes, M.R., Stiefel, E. (1952): Methods of conjugate gradients for solving linear systems. Nat. Bur. Standards, J. of Res. 49, 409–436.MathSciNetzbMATHCrossRefGoogle Scholar
  22. Hockney, R.W. (1969): The potential calculation and some applications. Methods of Computational Physics 9 136–211. New York, London: Academic Press.Google Scholar
  23. Householder, A.S. (1964): The Theory of Matrices in Numerical Analysis. New York: Blaisdell Publ. Comp.zbMATHGoogle Scholar
  24. Keyes, D.E., Gropp, W.D. (1987): A comparison of domain decomposition techniques for elliptic partial differential equations. SIAM J. Sci. Statist. Comput 8, s166 – s202.MathSciNetCrossRefGoogle Scholar
  25. Lanczos, C. (1950): An iteration method for the solution of the eigenvalue problem of linear differential and integral equations. J. Res. Nat. Bur. Standards 45, 255-282.MathSciNetCrossRefGoogle Scholar
  26. Lanczos, C. (1952): Solution of systems of linear equations by minimized iterations. J. Res. Nat. Bur. Standards 49, 33–53.MathSciNetCrossRefGoogle Scholar
  27. McCormick, S. (1987): Multigrid Methods. Philadelphia: SIAM.zbMATHCrossRefGoogle Scholar
  28. Meijerink, J.A., van der Vorst, H.A. (1977): An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comp 31, 148–162.MathSciNetzbMATHGoogle Scholar
  29. O’Leary, D.P., Widlund, O. (1979): Capacitance matrix methods for the Helmholtz equation on general three-dimensional regions. Math. Comp 33, 849–879.Google Scholar
  30. Paige, C.C., Saunders, M.A. (1975): Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Analysis 12, 617–624.MathSciNetzbMATHCrossRefGoogle Scholar
  31. Proskurowski, W., Widlund, O. (1976): On the numerical solution of Helmholtz’s equation by the capacitance matrix method. Math. Comp 30, 433–468.MathSciNetzbMATHGoogle Scholar
  32. Quarteroni, A., Valli, A. (1997): Numerical Approximation of Partial Differential Equations. 2nd Ed., Berlin-Heidelberg-New York: Springer.Google Scholar
  33. Reid, J.K. (Ed.) (1971 a): Large Sparse Sets of Linear Equations London, New York: Academic Press. (1971 b): On the method of conjugate gradients for the solution of large sparse systems of linear equations. In: Reid (1971 a), 231–252.Google Scholar
  34. Rice, J.R., Boisvert, R.F. (1984): Solving Elliptic Problems Using ELLPACK. BerlinHeidelberg-New York: Springer.Google Scholar
  35. Saad, Y. (1996): Iterative Methods for Sparse Linear Systems. Boston: PWS Publishing Company.zbMATHGoogle Scholar
  36. Schultz, M.H (1986): GMRES: a generalized minimal residual algorithmGoogle Scholar
  37. for solving nonsymmetric linear systems. SIAM J. Scientific and Statistical Computing 7 856–869.Google Scholar
  38. Schröder, J., Trottenberg, U. (1973): Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben I. Numer.Math 22, 37–68.MathSciNetzbMATHCrossRefGoogle Scholar
  39. Reutersberg, H. (1976): Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben II. Numer. Math 26, 429–459.Google Scholar
  40. Sonneveldt, P. (1989): CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Scientific and Statistical Computing 10, 36–52.CrossRefGoogle Scholar
  41. Swarztrauber, P.N. (1977): The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson’s equation on a rectangle. SIAM Review 19, 490–501.MathSciNetzbMATHCrossRefGoogle Scholar
  42. van der Vorst (1992): Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J Scientific and Statistical Computing 12, 631–644.Google Scholar
  43. Varga, R.S. (1962): Matrix Iterative Analysis. Series in Automatic Computation. Englewood Cliffs: Prentice Hall.Google Scholar
  44. Wachspress, E.L. (1966): Iterative Solution of Elliptic Systems and Application to the Neutron Diffusion Equations of Reactor Physics. Englewood Cliffs, N.J.: Prentice-Hall.Google Scholar
  45. Wilkinson, J.H., Reinsch, C. (1971): Linear Algebra. Handbook for Automatic Computation, Vol. II. Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 186. Berlin-Heidelberg-New York: Springer.Google Scholar
  46. Young, D.M. (1971): Iterative Solution of Large Linear Systems. Computer Science and Applied Mathematics. New York: Academic Press.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Josef Stoer
    • 1
  • Roland Bulirsch
    • 2
  1. 1.Institut für Angewandte Mathematik und StatistikUniversität WürzburgWürzburgDeutschland
  2. 2.Zentrum MathematikTechnische Universität MünchenMünchenDeutschland

Personalised recommendations