Numerische Mathematik

, Volume 61, Issue 1, pp 475–487 | Cite as

Extrapolation methods for vector sequences

  • P. R. Graves-Morris
Article

Summary

An analogue of Aitken's Δ2 method, suitable for vector sequences, is proposed. Aspects of the numerical performance of the vector ε-algorithm, based on using the Moore-Penrose inverse, are investigated. The fact that the denominator polynomial associated with a vector Padé approximant is the square of its equivalent in the scalar case is shown to be a source of approximation error. In cases where the convergence of the vector sequence is dominated by real eigenvalues, a hybrid form of the vector Padé approximant, having a denominator polynomial of minimal degree, is proposed and its effectiveness is demonstrated on several standard examples.

Mathematics Subject Classification (1991)

65305 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. aitken, A.C. (1926): On Bernoulli's numerical solution of algebraic equations. Proc. Roy. Soc. Edin.46, 289–305Google Scholar
  2. Baker, G.A. Jr., Graves-Morris, P.R. (1981): Padé approximants. Addison Wesley, CambridgeGoogle Scholar
  3. Brezinski, C. (1975): Généralisations de la transformation de Shanks, de la table de Wynn et de l'ε-algorithme. Calcolo12, 317–360Google Scholar
  4. Cordellier, F. (1989): Thesis, Univ. LilleGoogle Scholar
  5. Graves-Morris, P.R. (1983): Vector-valued rational interpolants I. Numer. Math.42, 331–348Google Scholar
  6. Graves-Morris, P.R. (1990): Solution of integral equations using generalised inverse, function-velued Padé approximants I. J. Comput. Appl. Math.32, 117–124Google Scholar
  7. Graves-Morris, P.R., Jenkins, C.D. (1986): Vector-valued rational interpolants III. Constr. Approx.2, 263–289Google Scholar
  8. Graves-Morris, P.R., Jenkins, C.D. (1989): Degeneracies of generalised inverse, vector-valued Padé approximants, Constr. Approx.5, 463–485Google Scholar
  9. Graves-Morris, P.R., Saff, E.B. (1988): Row convergence theorems for generalised inverse vector-valued Padé approximants. J. Comput. Appl. Math.23, 63–85Google Scholar
  10. Macleod, A.J. (1986): Acceleration of vector sequence by multidimensional Δ2 methods. Comm. Appl. Numer. Meth.2, 385–392Google Scholar
  11. McLeod, J.B. (1971): A note on the ε-algorithm. Computing7, 17–24Google Scholar
  12. Smith, D.A., Ford, W.F., Sidi, A. (1987): Extrapolation methods for vector sequences. SIAM Rev.29, 199–233Google Scholar
  13. Varga, R.S. (1962): Matrix iterative analysis. Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
  14. Weniger, E.J. (1989): Non-linear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep.10, 189–371Google Scholar
  15. Wilkinson, J.H. (1965): The algebraic eigenvalue problem. Oxford, Oxford University PressGoogle Scholar
  16. Wynn, P. (1962): Acceleration techniques for iterated vector and matrix problems. Math. Comput.16, 301–322Google Scholar
  17. Wynn, P. (1963): Continued fractions whose coefficients obey a non-commutative law of multiplication. Arch. Rat. Mech. Anal.12, 273–312Google Scholar
  18. Zienkiewicz, O.C., Löhner, R. (1985): Accelerated ‘relaxation’ or direct solution? Future prospects for FEM. Int. J. Numer. Meth. Eng.21, 1–11Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • P. R. Graves-Morris
    • 1
  1. 1.Department of MathematicsUniversity of BradfordBradfordUK

Personalised recommendations