Faster exponentials of power series

TL;DR

This paper introduces a more efficient algorithm for computing the exponential of a power series, reducing the computational complexity compared to previous methods.

Contribution

The paper presents a novel algorithm that computes exp(f) with lower asymptotic cost, improving the efficiency of power series exponential calculations.

Findings

Reduces the complexity constant from approximately 2.333 to 2.1666.

Provides an asymptotic complexity improvement for power series exponentials.

Enhances computational efficiency in symbolic and numerical analysis involving power series.

Abstract

We describe a new algorithm for computing exp(f) where f is a power series in C[[x]]. If M(n) denotes the cost of multiplying polynomials of degree n, the new algorithm costs (2.1666... + o(1)) M(n) to compute exp(f) to order n. This improves on the previous best result, namely (2.333... + o(1)) M(n).