A fast algorithm for reversion of power series

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 ) + M M ( n 1 / 2 ) ) ) O(n^{1/2}(M(n) + M\!M(n^{1/2}))) operations where M ( n ) M(n) and M M ( n ) M\!M(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.