Truncation Errors in Padé Approximations to Certain Functions: An Alternative Approach

1. Introduction. The problem of finding rational approximations to functions has received a considerable amount of attention recently, and many methods exist for finding such approximations. A brief survey of the most widely used methods has been given by Cheney and Southard [1]. In all problems of approximation, it is essential to know the truncation error which arises when the function is replaced by its approximation. Ideally, one would also like to have a realistic a priori estimate of this error. However such estimates are not in general easy to obtain. For example , in the case of polynomial approximations to a function, the truncation error can frequently be expressed in terms of a higher derivative of the function at some indeterminate point. Even in cases where the nth derivative of a function may be readily obtained, an estimate of an upper bound of the truncation error obtained by considering upper bounds of the higher-order derivatives is frequently much larger than the actual truncation error. Consequently such an estimate is not very useful for making an a priori estimate of the degree of the approximation to be used. For polynomial approximations, realistic estimates for the truncation error can sometimes be found by first of all expressing the truncation error as a contour integral, and then making use of asymptotic methods to evaluate this integral. It has been observed that if we approximate a function f(x) by a polynomial pn(x) of degree n, then an asymptotic estimate of this contour integral form of the trunca-tion error for large n, frequently gives a good estimate of the error even for small values of n. In this paper we shall show that a similar approach can also be used for certain rational approximations, to give excellent a priori estimates. Suppose that we wish to approximate a given function f(x) by means of a rational approximation pm(x)/qn(x), where pAx) and qn(x) are polynomials of degree m and n respectively. In this paper we shall consider only the so-called Padé approximations to f(x). These approximations are such that if we consider the expansions of f(x), pm(x) and qn(x) about x = 0, then