A fast algorithm for reversion of power series

We give an algorithm for reversion of formal power series, based on an efficient way to evaluate the Lagrange inversion formula. Our algorithm requires O(n1/2(M(n)+MM(n1/2))) operations where M(n) and MM(n) are the costs of polynomial and matrix multiplication respectively. This matches 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.