  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.
