Optimization of Rational Approximations by Continued Fractions

Abstract

To get guaranteed machine enclosures of a special function f(x), an upper bound ε(f) of the relative error is needed, where ε(f) itself depends on the error bounds ε(app); ε(eval) of the approximation and evaluation error respectively. The approximation function g(x) ≈ f(x) is a rational function (Remez algorithm), and with sufficiently high polynomial degrees ε(app) becomes sufficiently small. Evaluating g(x) on the machine produces a rather great ε(eval) because of the division of the two erroneous polynomials. However, ε(eval) can distinctly be decreased, if the rational function g(x) is substituted by an appropriate continued fraction c(x) which in general needs less elementary operations than the original rational function g(x). Numerical examples will illustrate this advantage.