Truncation and manipulation of multivariate power series
A new approach to the manipulation of power series in several variables is described. It is based on the new concept of weighted truncation: The series \(f(x)=\sum_{r\in {\mathbb{N}}^ n}f_ rx^ r\quad (x\in {\mathbb{R}}^ n)\) with coefficients \(f_ r\) is truncated by retaining only those terms with \(w\cdot r\leq M\), where \(w\in {\mathbb{N}}^ n\) is a prescribed weight vector and M is a positive constant. Algorithms using this approach are then discussed for various operations with power series: addition, multiplication, division, evaluation of a function of a series, such as the logarithm, the exponential, the sine, etc., and, finally, composition and reversion. In connection with function evaluation, often the coefficients of \(f^*(x)=\sum w_ ix_ i\partial f/\partial x_ i\) are aimed at first, and \(f_ r=f^*_ r/(w\cdot r)\) is then used to obtain those of f.
scientific article; zbMATH DE number 4020291