Extension and completion of Wynn's theory on convergence of columns of the epsilon table (Q1919081)

The sequence \(\{S_n\}^\infty_{n=0}\), \(S_n\sim S+\sum^\infty_{j=1} a_j\lambda^n_j\) as \(n\to\infty\) of complex numbers whose limit or anti-limit is \(S\) and the computation of approximations of \(S\) by the transformation of \textit{D. Shanks} [J. Math. Phys. 34, 1-42 (1955; Zbl 0067.28602)] were considered. This transformation generates an array \(e_k(S_n)\), which is computed by the \(\varepsilon\)-algorithm by \textit{P. Wynn} [Math. Tables Aids Comput. 10, 91-96 (1956; Zbl 0074.04601)]. The connection between the Shanks transformation and Padé approximation was exploited. Generalization and completion were deduced for the procedure by allowing some of the \(\lambda_j\) to have the same modulus and by replacing the constants \(a\) by some polynomials. As examples of problems in which these sequences occur iterative solutions of linear systems, Euler-MacLaurin expansions for integrands with logarithmic end point singularities were studied.