| TR#: | CS0823 |
| Class: | CS |
| Title: | EXTENSION AND COMPLETION OF WYNN'S THEORY ON
CONVERGENCE OF COLUMNS OF THE EPSILON TABLE. |
| Authors: | A. Sidi |
| Not Available | |
| Abstract: | Let \{S_n\}^{\infty}_{n=0} be such that S_n \sim S + \sum^{\infty}_{j=1} a_j \lambda^n_j as n \rightarrow \infty, with 1 > |\lambda_1| > |\lambda_2| > \cdots, such that \lim_{j \rightarrow \infty} \lambda_j = 0 . A well known result by Wynn states that when the Shanks transformation or its equivalent \varepsilon-algorithm is applied to \{S_n\}^{\infty}_{n=0}, then \varepsilon^{(n)}_{2k} - S \sim a_{k+1} \left [ \prod^k_{i=1} \left (\lambda_{k+1}-\lambda_i\right )/(1-\lambda_i) \right]^2 \lambda^n_{k+1} as n \rightarrow \infty. In the present work we extend this result (i) by allowing some of the \lambda_j to have the same modulus and (ii) by replacing the constants a_j by some polynomials P_j(n) in n. Sequences \{S_n\}^{\infty}_{n=0} with these characteristics arise frequently, e.g., in fixed point iterative solution of linear systems and in trapezoidal rule approximation of finite range integrals with logarithmic end point singularities and their multidimensional analogues. The results of this work are obtained by exploiting the connection between the Shanks transformation and Pad\'{e} approximants and by using some recent results of the author on Pad\'{e} approximants for meromorphic functions. |
| Copyright | The above paper is copyright by the Technion, Author(s), or others. Please contact the author(s) for more information |
Remark: Any link to this technical report should be to this page (http://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-info.cgi/1994/CS/CS0823), rather than to the URL of the PDF or PS files directly. The latter URLs may change without notice.
To the list of the CS technical reports of 1994
To the main CS technical reports page