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The simplified topological ε-algorithms: software and applications

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Abstract

In this paper, we describe the Matlab toolbox EPSfun for implementing and using the simplified topological ε-algorithms for accelerating the convergence of sequences of elements of a vector space. The functions for other similar algorithms are also provided. We give applications to the solution of linear and nonlinear systems of equations and to the computation of matrix functions.

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References

  1. Amat, S., Bermúdez, C., Busquier, S., Mestiri, D.: A family of Halley–Chebyshev iterative schemes for non–Fréchet differentiable operators. J. Comput. Appl. Math. 228, 486–493 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bader, B.W., Kolda, T.G., et al.: matlab Tensor Toolbox Version 2.6. http://www.sandia.gov/∼tgkolda/TensorToolbox/ (2015)

  3. Bai, Z. -Z.: A class of iteration methods based on the Moser formula for nonlinear equations in Markov chains. Linear Algebra Appl. 266, 219–241 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brezinski, C.: Application de l’ ε–algorithme à la résolution des systèmes non linéaires. C. R. Acad. Sci. Paris 271 A, 1174–1177 (1970)

    MATH  Google Scholar 

  5. Brezinski, C.: Sur un algorithme de résolution des systèmes non linéaires. C. R. Acad. Sci. Paris 272 A, 145–148 (1971)

    MATH  Google Scholar 

  6. Brezinski, C.: Some results in the theory of the vector ε-algorithm. Linear Alg. Appl. 8, 77–86 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brezinski, C.: Some results and applications about the vector ε-algorithm. Rocky Mt. J Math. 4, 335–338 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brezinski, C.: Numerical stability of a quadratic method for solving systems of nonlinear equations. Computing 14, 205–211 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brezinski, C.: Généralisation de la transformation de Shanks, de la table de Padé et de l’ ε–algorithme. Calcolo 12, 317–360 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brezinski, C., Redivo-Zaglia, M.: Extrapolation Methods. Theory and Practice. North–Holland, Amsterdam (1991)

    MATH  Google Scholar 

  11. Brezinski, C., Redivo–Zaglia, M.: Fortran 77 Library, supplement to the book [10]. http://www.math.unipd.it/∼michela/extracode/Extrapolation_Library.html

  12. Brezinski, C., Redivo-Zaglia, M.: The simplified topological ε-algorithms for accelerating sequences in a vector space. SIAM J. Sci. Comput. 36, A2227–A2247 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Brezinski, C., Redivo-Zaglia, M.: Convergence acceleration of Kaczmarz’s method. J. Engrg. Math. 93, 3–19 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brown, K.M.: Computer oriented algorithms for solving systems of simultaneous nonlinear algebraic equations. In: Byrne, G.D., Hall, C.A. (eds.) Numerical Solution of Systems of Nonlinear Algebraic Equations, pp 281–348. Academic Press, New York (1973)

  15. Cai, X. -C., Keyes, D. E.: Nonlinearly preconditioned inexact Newton algorithms. SIAM J. Sci. Comput. 24, 183–200 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cordellier, F.: Démonstration algébrique de l’extension de l’identité de Wynn aux tables de Padé non normales. In: Wuytack, L. (ed.) Padé Approximation and its Applications, LNM, vol. 765, pp 36–60. Springer, Berlin (1979)

  17. El-Sayed, S.M., Al-Dbiban, A.M.: A new inversion free iteration for solving the equation X + A ∗ X −1 A = Q. J. Comput. Appl. Math. 181, 148–156 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gekeler, E.: On the solution of systems of equations by the epsilon algorithm of Wynn. Math. Comput. 26, 427–436 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  19. Grau–Sánchez, M., Noguera, M., Diaz–Barrero, J. L.: Note on the efficiency of the iterative methods for solving nonlinear equations. SeMA J. 71, 15–22 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Graves-Morris, P. R., Roberts, D. E., Salam, A.: The epsilon algorithm and related topics. J. Comput. Appl. Math. 122, 51–80 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guo, C. -H.: On the numerical solution of a nonlinear matrix equation in Markov chains. Linear Algebra Appl. 288, 175–186 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Guo, C. -H.: Convergence rate of an iterative method for a nonlinear matrix equation. SIAM J. Matrix Anal. Appl. 23, 295–302 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Higham, N. J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)

  24. Higham, N.J.: The Matrix Computation Toolbox. http://www.ma.man.ac.uk/∼higham/mctoolbox

  25. Jbilou, K., Sadok, H.: Some results about vector extrapolation methods and related fixed point iteration. J. Comp. Appl. Math. 36, 385–398 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jbilou, K., Sadok, H.: Matrix polynomial and epsilon–type extrapolation methods with applications. Numer. Algorithms 68, 107–119 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Le Ferrand, H.: The quadratic convergence of the topological epsilon algorithm for systems of nonlinear equations. Numer. Algorithms 3, 273–284 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. Monsalve, M., Raydan, M.: A new inversion free method for a rational matrix equation. Lin. Alg. Appl. 43, 64–71 (2010). Corrigendum in Lin. Alg. Appl., 448 (2014) 343–344

    Article  MathSciNet  MATH  Google Scholar 

  29. Shanks, D.: Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, 1–42 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  30. Steffensen, J.F.: Remarks on iteration. Skand. Aktuarietidskr. 16, 64–72 (1933)

    MATH  Google Scholar 

  31. Wynn, P.: On a device for computing the e m (S n ) transformation. MTAC 10, 91–96 (1956)

    MATH  Google Scholar 

  32. Wynn, P.: On the propagation of error in certain non-linear algorithms. Numer. Math. 1, 142–149 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wynn, P.: Singular rules for certain nonlinear algorithms. BIT 3, 175–195 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wynn, P.: General purpose vector epsilon algorithm Algol procedures. Numer Math. 6, 22–36 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wynn, P.: An arsenal of Algol procedures for the evaluation of continued fractions and for effecting the epsilon algorithm. Chiffres 4, 327–362 (1966)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire Paul Painlevé, UMR CNRS 8524, UFR de Mathématiques, Université de Lille - Sciences et Technologies, 59655, Villeneuve d’Ascq Cedex, France

    Claude Brezinski

  2. Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121, Padova, Italy

    Michela Redivo-Zaglia

Authors
  1. Claude Brezinski
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  2. Michela Redivo-Zaglia
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Correspondence to Michela Redivo-Zaglia.

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Brezinski, C., Redivo-Zaglia, M. The simplified topological ε-algorithms: software and applications. Numer Algor 74, 1237–1260 (2017). https://doi.org/10.1007/s11075-016-0238-0

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  • DOI: https://doi.org/10.1007/s11075-016-0238-0

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