A Survey of Truncation Error Analysis for Padé and Continued Fraction Approximants

Abstract

Many important functions\(f\left( z \right)\) of mathematical physics, chemistry, engineering, and statistics are represented by convergent sequences \(\left\{ {fn\left( z \right)} \right\} \) of rational functions that are entries of a (1-point or multipoint) Pade table for \(f\left( z \right)\) In most cases of practicalinterest \(\left\{ {{{f}_{n}}\left( z \right)} \right\} \) is the sequence of approximants of a continued fraction (see, e.g., [1],[37], [45] and references contained therein). One reason for the importance of Pade tables and related continued fractions is that sequences of their approximants may converge in larger regions of the complex plane C than the power series expansion, which may not converge at all. Also the algorithmic character of continued fractions and Pade approximants provides efficient methods for the computation of special functions.