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A Schur-Newton method for the matrix pth 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.

65F30Other matrix algorithms
15A18Eigenvalues, singular values, and eigenvectors
15A51Stochastic matrices (MSC2000)
15A24Matrix equations and identities