Extension and Completion of Wynn's Theory on Convergence of Columns of the Epsilon Table

Let [Sn] n=0 be such that Sn tS+ j=1 aj* n j as n , with 1>|*1 |> |*2 |> } } } , such that limj *j=0. A well-known result by Wynn states that when the Shanks transformation or its equivalent =-algorithm is applied to [Sn] n=0 , then = (n) 2k &Stak+1[> k i=1 (*k+1&*i) (1&*i)] 2 *k+1 as n . In the present work we extend this result (i) by allowing some of the *j to have the same modulus and (ii) by replacing the constants aj by some polynomials Pj (n) in n. Sequences [Sn] 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 endpoint singularities and their multidimensional analogues. The results of this work are obtained by exploiting the connection between the Shanks transformation and Pade approximants and by using some recent results of the author on Pade approximants for meromorphic functions.