An Introduction to Padé Approximations with an Application
by
J. C. Butcher
University of Auckland

Padé approximations are generalizations to power series approximations. If N and D are polynomials of degrees n and d respectively then ``N/D is a Padé approximation to a function f'' means that

f(x) = \fracN(x)D(x) + O(xn+d+1).

In this introduction to this subject we will consider conditions on f for these approximations to exist for all choices of the non-negative integers n and d. We will also establish various ``three-term recurrence relations'' relating nearby members of the Padé table. In the case of f(x) = exp(x), explicit values of coefficients occurring in these formulae will be found.

The specific application that will be considered, is the question of which (n, d) pairs satisfy the computationally important property that the corresponding Padé approximation to the exponential function is such that |N(z)| <= |D(z)|, whenever z is a complex number with non-positive real part. The sufficient condition (which happens also to be necessary), d-2 <= n <= d will be proved using the three-term recurrence relations.

Date received: June 9, 1998