Technical Report CS0823

TR#:CS0823
Class:CS
Title: EXTENSION AND COMPLETION OF WYNN'S THEORY ON CONVERGENCE OF COLUMNS OF THE EPSILON TABLE.
Authors: A. Sidi
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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.
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