## 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 $$\{z: | z-c^p| <c^p\}$$ 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 (MSC2010) 15A18 Eigenvalues, singular values, and eigenvectors 15B51 Stochastic matrices 15A24 Matrix equations and identities

Zbl 1040.65038

ARPREC; MPFUN
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