Computation of the c-Table Related to the Padé Approximation

Abstract

The aim of this paper is to give a complete and practical method for numerical application of Padé approximation with the help of the $c$-table analysis. We present an exhaustive list of useful formulas to compute a $c$-table related to a formal power series $C(z)={\sum }_{n=0}^{\infty }{c}_n{z}^n$. Some of these formulas are not widely known, because they were presented in publications of limited circulation. Some others were never published, as three symmetric Paszkowski-like formulas to overcome the blocks in a $c$-table or an extension of local error formula for Padé approximants in the blocks. All formulas are given in two versions: in terms of Toeplitz determinants ($c$-table) and in the version of Hankel determinants ($c$-table). We compare the theory with numerical observations by reproducing different computational aspects of software producing the $c$-tables with the presence of blocks and their evolution following the evolution of computer environment.