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

Abstract

Listen

Translate article

Let {Sn}∞n=0be such thatSn∽S+∑∞j=1ajλnjasn→∞, with 1>|λ1|>|λ2|>…, such that limj→∞λj=0. A well-known result by Wynn states that when the Shanks transformation or its equivalentε-algorithm is applied to {Sn}∞n=0, thenε(n)2k−S∽ak+1[∏ki=1(λk+1−λi)/(1−λi)]2λnk+1asn→∞. In the present work we extend this result (i) by allowing some of theλjto have the same modulus and (ii) by replacing the constantsajby some polynomialsPj(n) inn. Sequences {Sn}∞n=0with 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 Padé approximants and by using some recent results of the author on Padé approximants for meromorphic functions.