Power Series Arithmetic with the BPAS Library

Abstract

We discuss the design and implementation of multivariate power series, univariate polynomials over power series, and their associated arithmetic operations within the Basic Polynomial Algebra Subprograms (BPAS) Library. This implementation employs lazy variations of Weierstrass preparation and the factorization of univariate polynomials over power series following Hensel’s lemma. Our implementation is lazy in that power series terms are only computed when explicitly requested. The precision of a power series is dynamically extended upon request, without requiring any re-computation of existing terms. This design extends into an “ancestry” of power series whereby power series created from the result of arithmetic or Weierstrass preparation automatically hold on to enough information to dynamically update themselves to higher precision using information from their “parents”.