Representation of Power Series in Terms of Polynomials, Rational Approximations and Continued Fractions

The first part of this paper is devoted to a discussion of a digital computer program, developed for the IBM 704 computer, which furnishes polynomial approximations with accuracy up to 16 digits for power, series or other polynomials. The method used is essentially the economization procedure proposed by C. Lanczos [1, 2], R. C. Minnick [3] and others. In the second part of the paper a method of obtaining efficient rational approximations is developed. I t presupposes the existence of the program described in the first part and constitutes, we believe, a novel approach to this problem, well suited for computer programming. Finally, a computer program based on the ideas of part two is described and an application to the function log2x is given. The program computes double precision polynomial approximations, and rational and continued fraction approximations; gives error estimates for each approximation; and if so specified compares each approximation with the given function at up to 106 points in the interval (--1 _-< x ~ 1).