A Note on the Space Complexity of Fast D-Finite Function Evaluation

TL;DR

This paper analyzes the space complexity of evaluating D-finite functions, proposing a method that achieves near-optimal time bounds with significantly reduced memory requirements compared to traditional algorithms.

Contribution

It generalizes the truncation trick for power series evaluation, providing a space-efficient algorithm with near-optimal time complexity for D-finite functions.

Findings

Achieves error bounds of 2^(-p) in time O(p*(log p)^(3+o(1)))

Uses space O(p), significantly less than the standard (p*log p) bits

Provides a practical approach for efficient 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 \Theta(p*lg p) bits of memory.