A cost-efficient variant of the incremental Newton iteration for the matrix $p$th root

TL;DR

This paper introduces a more cost-efficient variant of the incremental Newton iteration for computing the matrix pth root, significantly reducing computational complexity for large p values.

Contribution

A novel variant of the incremental Newton iteration that reduces computational cost from (n^3 p) to (n^3 \u221a p) flops per iteration for p up to 100.

Findings

The new method maintains stability for matrix pth root computation.

Computational cost is reduced to (n^3 log p) flops per iteration.

Effective for p up to at least 100.

Abstract

Incremental Newton (IN) iteration, proposed by Iannazzo, is stable for computing the matrix $p$th root, and its computational cost is $\mathcal{O}(n^3p)$ flops per iteration. In this paper, a cost-efficient variant of IN iteration is presented. The computational cost of the variant well agrees with $\mathcal{O} (n^3 \log p)$ flops per iteration, if $p$ is up to at least 100.