Automation of the manipulation of multivariate power series

Abstract

This paper reports on the development of compact and remarkably general algorithms for the manipulation of multivariate power series. The problem of efficiently storing the coefficients of such series is solved in a way which admits weighted truncation and yields simple algorithms for (i) algebraic operations, (ii) composition of special functions with power series and (iii) composition and reversion of multivariate power series. The algorithms, which are expressed in a form that can readily be translated into any standard computer language, can manipulate power series in an arbitrary number of variables while retaining all terms up to an arbitrary weighted order with respect to an arbitrary set of weights. The size of the power series which can be manipulated is limited only by memory capacity. For most purposes, a conventional microcomputer is adequate.