Computing with Formal Power Series

Abstract

A set of facilities for an algebra system that makes it possible to mampulate formal power series is described. Unlike many previous power series packages, the one presented here does not enforce any predetermined or arbitrary limits on the order of expansions generated. The system can deal with the elementary functions and copes gracefully with removable singularities of the form exemplified by t/sin(t) at t = 0. Examples of the use of the power series package are given; they were run on the SCRATCHPAD algebra system.

Equality in Computer Algebra and Beyond

“…The two main examples of this are "lazy power series" (Norman, 1975;Harrington, 1980) and various forms of "lazy generators" for elements of R (Gosper, 1972;Boehm et al, 1986;Vuillemin, 1990;Ménissier-Morain, 1994;Edalat et al, 1997), though lazy p-adics are also possible. The problem in both cases is fundamentally the same: unless one has some extrinsic information, one can develop the series/numbers to as many terms as one wants, and the fact that all the corresponding terms are equal † does not prove that the two underlying objects are mathematically equal.…”

Symbolic Computation of Padé Approximants

“…In the sample problems presented in this paper, the Taylor series have been generated from initial-value problems by the (expensive) method of repeated differentiation (see [7]). Efficient methods for generating power series expansions are discussed by Norman [11] and by Zippel [14].…”

“… (11). may not have been applied (in case a~-~)k_l --0) to compute a~k -1~, but in any case the required divisor is given by eq.…”

Algorithmic development of power series

“…series objects are given by a finite number of initial terms, and an (internally used) formula to calculate further coefficients, see e.g. [12]. Infinite series representations, however, are not supported in these systems, either.…”