A fast algorithm for reversion of power series

TL;DR

This paper introduces a fast algorithm for reversion of formal power series using an optimized implementation of the Lagrange inversion formula, achieving improved practical performance over existing methods.

Contribution

It presents a new algorithm that reduces the computational complexity of power series reversion and offers a constant factor speedup compared to previous algorithms.

Findings

The algorithm matches the asymptotic complexity of Brent and Kung's method.

Benchmarks show the algorithm performs efficiently in practice.

Achieves a speedup depending on polynomial and matrix multiplication algorithms.

Abstract

We give an algorithm for reversion of formal power series, based on an efficient way to implement the Lagrange inversion formula. Our algorithm requires $O(n^{1/2}(M(n) + MM(n^{1/2})))$ operations where $M(n)$ and $MM(n)$ are the costs of polynomial and matrix multiplication respectively. This matches the asymptotic complexity of an algorithm of Brent and Kung, but we achieve a constant factor speedup whose magnitude depends on the polynomial and matrix multiplication algorithms used. Benchmarks confirm that the algorithm performs well in practice.