BIT Numerical Mathematics

, Volume 28, Issue 1, pp 163–178 | Cite as

An optimum iterative method for solving any linear system with a square matrix

  • Dennis C. Smolarski
  • Paul E. Saylor
Part II Numerical Mathematics

Abstract

A method is presented to solveAx=b by computing optimum iteration parameters for Richardson's method. It requires some information on the location of the eigenvalues ofA. The algorithm yields parameters well-suited for matrices for which Chebyshev parameters are not appropriate. It therefore supplements the Manteuffel algorithm, developed for the Chebyshev case. Numerical examples are described.

AMS(MOS) Subject Classifications

65F10 15A06 65N20 33A65 

Keywords

Richardson's method iterative solution Chebyshev method Manteuffel algorithm optimum parameters least squares nonsymmetric matrices nonhermitian matrices orthogonal polynomials eigenvalues 

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References

  1. 1.
    R. S. Anderssen and G. H. Golub,Richardson's non-stationary matrix iterative procedure, Rep. STAN-CS-72-304, Computer Science Department, Stanford University, August, 1972.Google Scholar
  2. 2.
    S. F. Ashby,CHEBYCODE: A FORTRAN implementation of Manteuffel's adaptive Chebyshev algorithm, Rep. UIUCDS-R-85-1203, University of Illinois, Urbana, IL, May, 1985.Google Scholar
  3. 3.
    B. Atlestam,Tschebycheff Polynomials for Sets Consisting of Two Disjoint Intervals with Application to Convergence Estimates for the Conjugate Gradient Method, Rep. 77.06R, Dept. of Computer Science, Chalmers Institute of Technology and the University of Göteborg, Sweden, 1977.Google Scholar
  4. 4.
    C. de Boor and J. R. Rice,Extremal polynomials with application to Richardson's iteration for indefinite linear systems, SIAM J. Scient. Stat. Comput., 3:1 (March 1982).Google Scholar
  5. 5.
    M. Eiermann, W. Niethammer and R. S. Varga,A study of semiiterative methods for nonsymmetric systems of linear equations, Numer. Math., 47 (1985), pp. 505–533.Google Scholar
  6. 6.
    M. Eiermann, R. S. Varga and W. Niethammer, Interationsverfahren für nichtsymmetrische Gleichungssysteme und Approximationsmethoden im Komplexen, Jber. d. Dt. Math.-Verein. 89 (1987), pp. 1–32.Google Scholar
  7. 7.
    S. C. Eisenstat,Efficient implementation of a class of preconditioned conjugate gradient methods, SIAM J. Scient. Stat. Comput., 2:1 (March 1981), pp. 1–4.Google Scholar
  8. 8.
    H. Elman,Iterative methods for large, sparse, nonsymmetric systems of linear equations, Res. Report 229, Dept. of Computer Science, Yale University, New Haven, Conn., 1982.Google Scholar
  9. 9.
    H. Elman, Y. Saad and P. Saylor,A hybrid Chebyshev-Krylov subspace algorithm for solving nonsymmetric systems of linear equations, SIAM J. Scient. Stat. Comput., 7:3 (July 1986), pp. 840–869.Google Scholar
  10. 10.
    H. Elman and R. Streit,Polynomial iteration for nonsymmetric indefinite linear systems, Res. Report 380, Dept. of Computer Science, Yale University, New Haven, Conn., 1985.Google Scholar
  11. 11.
    G. E. Forsythe and W. R. Wasow,Finite Difference Methods for Partial Differential Equations, Wiley, New York, 1960.Google Scholar
  12. 12.
    W. Gautschi,On generating orthogonal polynomials. SIAM J. Sci. Stat. Comput., 3:3 (Sept. 1982), pp. 289–317.Google Scholar
  13. 13.
    G. Golub and J. Welsch,Calculation of Gauss quadrature rules, Math. of Computation. 23:106 (1969), pp. 221-230.Google Scholar
  14. 14.
    W. B. Gragg and L. Reichel,On the application of orthogonal polynomials to the iterative solution of linear systems of equations with indefinite or non-hermitian matrices, Linear Alg. Applic. 88-89 (April, 1987), pp. 349–371.Google Scholar
  15. 15.
    I. Gustafsson,A class of first order factorizations, BIT 18 (1978), pp. 142–156.Google Scholar
  16. 16.
    L. Hageman and D. Young,Applied Iterative Methods, Academic Press, New York, 1981.Google Scholar
  17. 17.
    V. I. Lebedev and S. A. Fingenov,On the order of choice of the iteration parameters in the Chebyshev cyclic iteration method, Zhur. Vych. Mat. I Mat. Fiz. 11:2 (1972), pp. 425–438 (in Russian). An English translation appears in R. S. Anderssen and G. H. Golub,Richardson's non-stationary matrix iterative procedure, Rep. STAN-CS-72-304, Computer Science Department, Stanford University, August, 1972.Google Scholar
  18. 18.
    T. A. Manteuffel,The Tchebychev iteration for non-symmetric linear systems, Numer. Math., 28 (1977), pp. 307–327.CrossRefGoogle Scholar
  19. 19.
    T. A. Manteuffel,Adaptive procedure for estimating parameters for the nonsymmetric Tchebychev iteration, Numer. Math., 31 (1978), pp. 183–208.Google Scholar
  20. 20.
    J. A. Meijerink and H. A. van der Vorst,An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. of Comput., 31:137 (Jan., 1977), pp. 148–162.Google Scholar
  21. 21.
    G. Opfer and G. Schober,Richardson's iteration for nonsymmetric matrices, Linear Alg. Applic., 58 (1984), pp. 343–361.Google Scholar
  22. 22.
    B. N. Parsons,General k-part stationary iterative solutions to linear systems, SIAM J. Numer. Anal., 24:1 (Feb., 1987), pp. 188–198.Google Scholar
  23. 23.
    R. R. Roloff,Iterative solution of matrix equations for symmetric matrices possessing positive and negative eigenvalues, Rep. UIUCDCS-R-79-1018, University of Illinois, Urbana, IL, Oct., 1979.Google Scholar
  24. 24.
    Y. Saad,Least squares polynomials in the complex plane with applications to solving sparse nonsymmetric matrix problems, Tech. Rep. 276, Dept. of Computer Science, Yale University, New Haven, Conn., June 1983.Google Scholar
  25. 25.
    Y. Saad,Least squares polynomials in the complex plane and their use for solving nonsymmetric linear systems, SIAM J. of Numerical Analysis, 24:1 (Feb., 1987), pp. 155–169.Google Scholar
  26. 26.
    P. E. Saylor and D. C. Smolarski, S.J.Computing the roots of complex orthogonal and kernel polynomials, in SIAM J. on Scientific and Statistical Computing, 9:1 (Jan. 1988).Google Scholar
  27. 27.
    D. C. Smolarski,Optimum semi-iterative methods for the solution of any linear algebraic system with a square matrix, Rep. UIUCDCS-R-81-1077, University of Illinois, Urbana, IL, Dec., 1981.Google Scholar
  28. 28.
    D. C. Smolarski and P. E. Saylor,An optimum semi-iterative method for solving any linear set with a square matrix, Rep. UIUCDCS-R-85-1218, University of Illinois, Urbana, IL, July, 1985.Google Scholar
  29. 29.
    E. L. Stiefel,Kernel polynomials in linear algebra and their numerical applications, U.S. National Bureau of Standards, Applied Mathematics Series, 49 (Jan., 1958), pp. 1–22.Google Scholar
  30. 30.
    R. L. Streit and A. H. Nuttall,A note on the semi-infinite programming approach to complex approximation, Math. of Comp., 40 (April, 1983), 162, pp. 599–605.Google Scholar
  31. 31.
    G. Szegö,Orthogonal Polynomials, 4th Ed., AMS, Providence, RI, 1939, 1975.Google Scholar
  32. 32.
    H. Tal'ezer,Polynomial approximation of functions of matrices and applications. ICASE Report No. 87-63, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23665, 1987.Google Scholar

Copyright information

© BIT Foundations 1988

Authors and Affiliations

  • Dennis C. Smolarski
    • 1
  • Paul E. Saylor
    • 1
  1. 1.Department of Computer ScienceUniversity of Illinois at Urbana-ChampaignUrbana

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