Numerische Mathematik

, Volume 40, Issue 1, pp 39–46

# The ɛ-algorithm and multivariate Padé-approximants

• Annie A. M. Cuyt
Article

## Summary

In the univariate case the ɛ-algorithm of Wynn is closely related to the Padé-table in the following sense: if we apply the ɛ-algorithm to the partial sums of the power series$$f(x) = \sum\limits_{i = 0}^\infty {c_i x^i }$$ then ε 2m lm is the (l, m) Padé-approximant tof(x) wherel is the degree of the numerator andm is the degree of the denominator [1 pp. 66–68].

Several generalizations of the ɛ-algorithm exist but without any connection with a theory of Padé-approximants.

Also several definitions of the Padé-approximant to a multivariate function exist, but up till now without any connection with the ɛ-algorithm.

In this paper, we see that the multivariate Padé-approximants introduced in [3], satisfy the same property as the univariate Padé-approximants: if we apply the ɛ-algorithm to the partial sums of the power series
$$f\left( {x_1 ,...,x_n } \right) = \sum\limits_{i_1 + ... + i_n = 0}^\infty {c_{i_1 ...i_n } x_1^{i_1 } ...x_n^{i_n } }$$
then ε 2m (lm) is the (l, m) multivariate Padé-approximant tof(x1, ...,x n ).

## Subject classification

AMS(MOS): 65D15 CR: 5.13

## References

1. 1.
Brezinski, C.: Algoritmes d'accélération de la convergence. Editions Technip, Paris 1978Google Scholar
2. 2.
Brezinski, C.: Accélération de la convergence en analyse numérique. LNM 584, Berlin: Springer 1977Google Scholar
3. 3.
Cuyt, A.: Abstract Padé Approximants in operator theory, pp-61–87. LNM 765, Berlin: Springer 1979Google Scholar
4. 4.
Perron, O.: Die Lehre von den Kettenbrüchen II. Stuttgart: Teubner 1977Google Scholar