Numerical Algorithms

, Volume 65, Issue 4, pp 759–782 | Cite as

Composite convergence bounds based on Chebyshev polynomials and finite precision conjugate gradient computations

Original Paper

Abstract

The conjugate gradient method (CG) for solving linear systems of algebraic equations represents a highly nonlinear finite process. Since the original paper of Hestenes and Stiefel published in 1952, it has been linked with the Gauss-Christoffel quadrature approximation of Riemann-Stieltjes distribution functions determined by the data, i.e., with a simplified form of the Stieltjes moment problem. This link, developed further by Vorobyev, Brezinski, Golub, Meurant and others, indicates that a general description of the CG rate of convergence using an asymptotic convergence factor has principal limitations. Moreover, CG is computationally based on short recurrences. In finite precision arithmetic its behaviour is therefore affected by a possible loss of orthogonality among the computed direction vectors. Consequently, any consideration concerning the CG rate of convergence relevant to practical computations must include analysis of effects of rounding errors. Through the example of composite convergence bounds based on Chebyshev polynomials, this paper argues that the facts mentioned above should become a part of common considerations on the CG rate of convergence. It also explains that the spectrum composed of small number of well separated tight clusters of eigenvalues does not necessarily imply a fast convergence of CG or other Krylov subspace methods.

Keywords

Conjugate gradient method Stieltjes moment problem Chebyshev semi-iterative method Composite polynomial convergence bounds Finite precision computations Clusters of eigenvalues 

Mathematics Subject Classifications (2010)

65F10 65B99 65G50 65N15 

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References

  1. 1.
    Arioli, M., Liesen, J., Miedlar, A., Strakoš, Z.: Interplay between discretization and algebraic computation in adaptive numerical solution of elliptic PDE problems. GAMM Mitt. Ges. Angew. Math. Mech. (2013)Google Scholar
  2. 2.
    Axelsson, O.: A class of iterative methods for finite element equations. Comput. Methods Appl. Mech. Engrg. 9, 123–127 (1976)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Axelsson, O.: Iterative solution methods. Cambridge University Press, Cambridge (1994)CrossRefMATHGoogle Scholar
  4. 4.
    Axelsson, O.: A generalized conjugate gradient minimum residual method with variable preconditioners In: Advanced mathematics: computations and applications (Novosibirsk, 1995), pp. 14–25. NCC Publ.Google Scholar
  5. 5.
    Axelsson, O.: Optimal preconditioners based on rate of convergence estimates for the conjugate gradient method. Numer. Funct. Anal. Optim. 22, 277–302 (2001)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Axelsson, O., Kaporin, I.: On the sublinear and superlinear rate of convergence of conjugate gradient methods. Numer. Algorithms 25, 1–22 (2000)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Axelsson, O., Karátson, J.: Equivalent operator preconditioning for elliptic problems. Numer. Algorithms 50, 297–380 (2009)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Axelsson, O., Lindskog, G.: On the eigenvalue distribution of a class of preconditioning methods. Numer. Math. 48, 479–498 (1986)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Beckermann, B., Kuijlaars, A.B.J.: On the sharpness of an asymptotic error estimate for conjugate gradients. BIT 41, 856–867 (2001)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Beckermann, B., Kuijlaars, A.B.J.: Superlinear convergence of conjugate gradients. SIAM J. Numer. Anal. 39, 300–329 (2001)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Beckermann, B., Kuijlaars, A.B.J.: Superlinear CG convergence for special right-hand sides. Electron. Trans. Numer. Anal. 14, 1–19 (2002)MATHMathSciNetGoogle Scholar
  12. 12.
    Brezinski, C.: Projection Methods for Systems of Equations. Vol. 7 of Studies in Computational Mathematics. North-Holland, Amsterdam (1997)Google Scholar
  13. 13.
    Brezinski, C.: Error estimates for the solution of linear systems. SIAM J. Sci. Comput. 21, 764–781 (1999)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Computable error bounds and estimates for the conjugate gradient method. Numer. Algorithms 25, 75–88 (2000)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: An iterative method with error estimators. J. Comput. Appl. Math. 127, 93–119 (2001)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Dahlquist, G., Björck, Å.: Numerical Methods in Scientific Computing, vol. 1. SIAM, Philadelphia (2008)CrossRefMATHGoogle Scholar
  17. 17.
    Dahlquist, G., Eisenstat, S.C., Golub, G.H.: Bounds for the error of linear systems of equations using the theory of moments. J. Math. Anal. Appl. 37, 151–166 (1972)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Dahlquist, G., Golub, G.H., Nash, S.G.: Bounds for the error in linear systems. In: Semi-infinite programming (Proc. Workshop, Bad Honnef, 1978). Lecture Notes in Control and Information Sci., vol. 15, pp. 154–172. Springer, Berlin (1979)Google Scholar
  19. 19.
    Daniel, J.W.: The conjugate gradient method for linear and nonlinear operator equations. SIAM J. Numer. Anal. 4, 10–26 (1967)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Deuflhard, P.: Cascadic conjugate gradient methods for elliptic partial differential equations, algorithm and numerical results In: Domain decomposition methods in scientific and engineering computing (University Park, PA , 1993), vol. 180 of Contemp. Math American Mathematical Society, pp 29–42. Providence (1994)Google Scholar
  21. 21.
    Engeli, M., Ginsburg, T., Rutishauser, H., Stiefel, E.: Refined iterative methods for computation of the solution and the eigenvalues of self-adjoint boundary value problems. Mitt. Inst. Angew. Math. Zürich. 8, 107p. (1959)Google Scholar
  22. 22.
    Faber, V., Manteuffel, T.A., Parter, S.V.: On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differential equations. Adv. Appl. Math. 11, 109–163 (1990)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Flanders, D.A., Shortley, G.: Numerical determination of fundamental modes. J. Appl. Phys. 21, 1326–1332 (1950)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Golub, G.H., Meurant, G.: Matrices, moments and quadrature with applications. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2010)Google Scholar
  25. 25.
    Golub, G.H., Strakoš, Z.: Estimates in quadratic formulas. Numer. Algorithms 8, 241–268 (1994)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Gratton, S., Titley-Peloquin, D., Toint, P., Tshimanga, J.: Linearizing the method of conjugate gradients. Technical Report naXys-15-2012, Namur Centre for Complex Systems. FUNDP–University of Namur, Belgium (2012)Google Scholar
  27. 27.
    Greenbaum, A.: Behaviour of slightly perturbed Lanczos and conjugate-gradient recurrences. Linear Algebra Appl. 113, 7–63 (1989)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Greenbaum, A.: Iterative Methods for Solving Linear Systems, vol. 17 of Frontiers in Applied Mathematics. SIAM, Philadelphia (1997)CrossRefGoogle Scholar
  29. 29.
    Greenbaum, A., Strakoš, Z.: Predicting the behavior of finite precision Lanczos and conjugate gradient computations. SIAM J. Matrix Anal. Appl. 13, 121–137 (1992)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Günnel, A., Herzog, R., Sachs, E.: A note on preconditioners and scalar products for Krylov methods in Hilbert space. Preprint (2013)Google Scholar
  31. 31.
    Gutknecht, M.H., Strakoš, Z.: Accuracy of two three-term and three two-term recurrences for Krylov space solvers. SIAM J. Matrix Anal. Appl. 22, 213–229 (2000)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Hackbusch, W.: Iterative solution of large sparse systems of equations. Vol. 95 of Applied Mathematical Sciences. Translated and Revised from the 1991 German Original. Springer-Verlag, New York (1994)Google Scholar
  33. 33.
    Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Research Nat. Bur. Standards 49, 409–436 (1952)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Hiptmair, R.: Operator preconditioning. Comput. Math. Appl. 52, 699–706 (2006)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Jennings, A.: Influence of the eigenvalue spectrum on the convergence rate of the conjugate gradient method. J. Inst. Math. Appl. 20, 61–72 (1977)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Lanczos, C.: Chebyshev polynomials in the solution of large-scale linear systems In: Proceedings of the Association for Computing Machinery, Toronto, 1952 (1953), pp 124–133. Sauls Lithograph Co. (for the Association for Computing Machinery), Washington, D. C.Google Scholar
  37. 37.
    Liesen, J., Strakoš, Z.: Krylov subspace methods: principles and analysis. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2012)CrossRefGoogle Scholar
  38. 38.
    Mardal, K.-A., Winther, R.: Preconditioning discretizations of systems of partial differential equations. Numer. Linear Algebra Appl. 18, 1–40 (2011)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Markoff, A.: Démonstration de certaines inégalités de M. Tchébychef. Math. Ann. 24, 172–180 (1884)CrossRefMathSciNetGoogle Scholar
  40. 40.
    Meurant, G.: The Lanczos and conjugate gradient algorithms: from theory to finite precision computations. Vol. 19 of Software, Environments, and Tools. Society for Industrial and Applied Mathematics. SIAM, Philadelphia (2006)Google Scholar
  41. 41.
    Meurant, G., Strakoš, Z.: The Lanczos and conjugate gradient algorithms in finite precision arithmetic. Acta. Numer. 15, 471–542 (2006)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Meurant, G., Tichý, P.: On computing quadrature-based bounds for the A-norm of the error in conjugate gradients. Numer. Algorithms 62, 163–191 (2013)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Naiman, A.E., Babuška, I.M., Elman, H.C.: A note on conjugate gradient convergence. Numer. Math. 76, 209–230 (1997)CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Naiman, A.E., Engelberg, S.: A note on conjugate gradient convergence. II, III. Numer. Math. 85, 665–683, 685–696 (2000)CrossRefMathSciNetGoogle Scholar
  45. 45.
    Nevanlinna, O.: Convergence of iterations for linear equations. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1993)CrossRefGoogle Scholar
  46. 46.
    Notay, Y.: On the convergence rate of the conjugate gradients in presence of rounding errors. Numer. Math. 65, 301–317 (1993)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    O’Leary, D.P., Strakoš, Z., Tichý, P.: On sensitivity of Gauss-Christoffel quadrature. Numer. Math. 107, 147–174 (2007)CrossRefMATHMathSciNetGoogle Scholar
  48. 48.
    Paige, C.C.: Error analysis of the Lanczos algorithm for tridiagonalizing a symmetric matrix. J. Inst. Math. Appl. 18, 341–349 (1976)CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    Paige, C.C.: Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem. Linear Algebra Appl. 34, 235–258 (1980)CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Richardson, L.F.: The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Phil. Trans. Roy. Soc. London A 210, 307–357 (1911)CrossRefMATHGoogle Scholar
  51. 51.
    Rivlin, T.J.: Chebyshev Polynomials, 2nd edn. Pure and Applied Mathematics. Wiley, New York (1990)Google Scholar
  52. 52.
    Silvester, D.J., Simoncini, V.: An optimal iterative solver for symmetric indefinite systems stemming from mixed approximation. ACM Trans. Math. Softw. 37, Art. 42, 22 (2011)Google Scholar
  53. 53.
    Spielman, D.A., Woo, J.: A note on preconditioning by low-stretch spanning trees. Computing Research Repository (2009)Google Scholar
  54. 54.
    Strakoš, Z.: On the real convergence rate of the conjugate gradient method. Linear Algebra Appl. 154–156, 535–549 (1991)CrossRefGoogle Scholar
  55. 55.
    Strakoš, Z., Tichý, P.: On error estimation in the conjugate gradient method and why it works in finite precision computations. Electron. Trans. Numer. Anal. 13, 56–80 (2002)MATHMathSciNetGoogle Scholar
  56. 56.
    Strakoš, Z., Tichý, P.: Error estimation in preconditioned conjugate gradients. BIT 45, 789–817 (2005)CrossRefMATHMathSciNetGoogle Scholar
  57. 57.
    Tyrtyshnikov, E.E.: A Brief Introduction to Numerical Analysis. Birkhäuser, Boston (1997)CrossRefMATHGoogle Scholar
  58. 58.
    van der Sluis, A., van der Vorst, H.A.: The rate of convergence of conjugate gradients. Numer. Math. 48, 543–560 (1986)CrossRefMATHMathSciNetGoogle Scholar
  59. 59.
    van der Vorst, H.A.: Iterative solution methods for certain sparse linear systems with a nonsymmetric matrix arising from PDE-problems. J. Comput. Phys. 44, 1–19 (1981)CrossRefMATHMathSciNetGoogle Scholar
  60. 60.
    Varga, R.S.: Matrix iterative analysis, expanded, 2nd edn. Vol. 27 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (2000)Google Scholar
  61. 61.
    Vorobyev, Y.V.: Methods of moments in applied mathematics. Translated from the Russian by Bernard Seckler. Gordon and Breach Science Publishers, New York (1965)Google Scholar
  62. 62.
    Winther, R.: Some superlinear convergence results for the conjugate gradient method. SIAM J. Numer. Anal. 17, 14–17 (1980)CrossRefMATHMathSciNetGoogle Scholar
  63. 63.
    Young, D.: On Richardson’s method for solving linear systems with positive definite matrices. J. Math. Phys. 32, 243–255 (1954)MATHGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic

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