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.


65F30 Other matrix algorithms (MSC2010)
15A18 Eigenvalues, singular values, and eigenvectors
15B51 Stochastic matrices
15A24 Matrix equations and identities


Zbl 1040.65038


Full Text: DOI