Extrapolation methods for vector sequences
Received: 16 October 1990 DOI:
Cite this article as: Graves-Morris, P.R. Numer. Math. (1992) 61: 475. doi:10.1007/BF01385521 Summary
An analogue of Aitken's Δ
2 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 References
aitken, A.C. (1926): On Bernoulli's numerical solution of algebraic equations. Proc. Roy. Soc. Edin.
Baker, G.A. Jr., Graves-Morris, P.R. (1981): Padé approximants. Addison Wesley, Cambridge
Brezinski, C. (1975): Généralisations de la transformation de Shanks, de la table de Wynn et de l'ε-algorithme. Calcolo
Cordellier, F. (1989): Thesis, Univ. Lille
Graves-Morris, P.R. (1983): Vector-valued rational interpolants I. Numer. Math.
Graves-Morris, P.R. (1990): Solution of integral equations using generalised inverse, function-velued Padé approximants I. J. Comput. Appl. Math.
Graves-Morris, P.R., Jenkins, C.D. (1986): Vector-valued rational interpolants III. Constr. Approx.
Graves-Morris, P.R., Jenkins, C.D. (1989): Degeneracies of generalised inverse, vector-valued Padé approximants, Constr. Approx.
Graves-Morris, P.R., Saff, E.B. (1988): Row convergence theorems for generalised inverse vector-valued Padé approximants. J. Comput. Appl. Math.
Macleod, A.J. (1986): Acceleration of vector sequence by multidimensional Δ
methods. Comm. Appl. Numer. Meth.
McLeod, J.B. (1971): A note on the ε-algorithm. Computing
Smith, D.A., Ford, W.F., Sidi, A. (1987): Extrapolation methods for vector sequences. SIAM Rev.
Varga, R.S. (1962): Matrix iterative analysis. Prentice-Hall, Englewood Cliffs, N.J.
Weniger, E.J. (1989): Non-linear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep.
Wilkinson, J.H. (1965): The algebraic eigenvalue problem. Oxford, Oxford University Press
Wynn, P. (1962): Acceleration techniques for iterated vector and matrix problems. Math. Comput.
Wynn, P. (1963): Continued fractions whose coefficients obey a non-commutative law of multiplication. Arch. Rat. Mech. Anal.
Zienkiewicz, O.C., Löhner, R. (1985): Accelerated ‘relaxation’ or direct solution? Future prospects for FEM. Int. J. Numer. Meth. Eng.