A cost-efficient variant of the incremental Newton iteration for the   matrix $p$th root
  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. 
Fuminori Tatsuoka, Tomohiro Sogabe, Yuto Miyatake, Shao-Liang Zhang
math.NA