A Fixed Point Method for Power Series Computation

This paper presents a novel technique for manipulating structures which represent infinite power series. When power series are implemented using lazy evaluation, many operations can be written as simple recursive procedures. For example, the programs to generate the series for the elementary transcendental functions are almost transliterations of the defining integral equations. However, a naive lazy algorithm provides an implementation which may be orders of magnitude slower than a method which manipulates the coefficients explicitly. The technique described here allows a power series to be defined in a very natural but computationally inefficient way and transforms it to an equivalent, efficient form. This is achieved by using a fixed point operator on the delayed part to remove redundant calculations. The paper describes this fixed point method and the class of problems to which it is applicable. It has been used in Scratchpad II to improve the performance of a number of operations on infinite series, including division, reversion, special functions and the solution of linear and non-linear ordinary differential equations. A few examples are given of the method and of the speed up obtained. To illustrate, the computation of the first n terms of exp(u) for a dense, infinite series u is reduced from O(n 4) to O(n 2) coefficient operations, the same as required by the standard on-line algorithms.