A note on the space complexity of fast D-finite function evaluation

Abstract

We state and analyze a generalization of the “truncation trick” suggested by Gourdon and Sebah to improve the performance of power series evaluation by binary splitting. It follows from our analysis that the values of D-finite functions (i.e., functions described as solutions of linear differential equations with polynomial coefficients) may be computed with error bounded by 2−p in time O(p(lg p)3+o(1)) and space O(p). The standard fast algorithm for this task, due to Chudnovsky and Chudnovsky, achieves the same time complexity bound but requires Θ(p lg p) bits of memory.