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A Schur-Newton method for the matrix $p$th root and its inverse. (English) Zbl 1128.65030

The author presents a convergence analysis of the Newton method for the inverse of the matrix $p$-th root, i.e., ${A}^{-1/p}$ for $p>2$. The $p$-th root arises in the computation of the matrix logarithm by the inverse scaling and squaring method. A quadratic convergence to ${A}^{-1/p}$ is proved provided the eigenvalues of $A$ lie in a wedge-shaped convex set within the disc $\left\{z:|z-{c}^{p}|<{c}^{p}\right\}$ with the initial matrix ${c}^{-1}I$, where $c$ is a positive scalar. It includes an optimal choice of $c$ for $A$ having real positive eigenvalues.

The analysis leads to a hybrid algorithm for general matrices employing a Schur decomposition, matrix square roots of the upper (quasi-) triangular roots, and two coupled versions of the Newton iteration. The algorithm is stable and computes either ${A}^{1/p}$ or ${A}^{-1/p}$. It is more efficient than the Schur method of M. I. Smith [ibid. 24, No. 4, 971–989 (2003; Zbl 1040.65038)] for large $p$ that are not highly composite. An application is supplied to roots of transition matrices for a time-homogeneous continuous-time Markov process. Numerically illustrative experiments are supplied.

##### MSC:
 65F30 Other matrix algorithms 15A18 Eigenvalues, singular values, and eigenvectors 15A51 Stochastic matrices (MSC2000) 15A24 Matrix equations and identities