Faster algorithms for the square root and reciprocal of power series

TL;DR

This paper introduces faster algorithms for computing square roots and reciprocals of power series, reducing the computational complexity compared to previous methods by optimizing polynomial multiplication costs.

Contribution

The paper presents new algorithms that improve the asymptotic efficiency of calculating power series square roots and reciprocals, achieving lower constants in complexity.

Findings

Square root computation costs approximately 1.333 M(n)

Reciprocal computation costs approximately 1.444 M(n)

Both algorithms outperform previous best results

Abstract

We give new algorithms for the computation of square roots and reciprocals of power series in C[[x]]. If M(n) denotes the cost of multiplying polynomials of degree n, the square root to order n costs (1.333... + o(1)) M(n) and the reciprocal costs (1.444... + o(1)) M(n). These improve on the previous best results, respectively (1.8333... + o(1)) M(n) and (1.5 + o(1)) M(n).