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

Guo, Chun-Hua; Higham, Nicholas J.

SIAM J. Matrix Anal. Appl. 28, No. 3, 788-804 (2006).

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.

Reviewer: Lubomír Bakule (Praha)

MSC:

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

Keywords:

matrix \(p\)-th root; principal \(p\)-th root; matrix logarithm; Newton’s method; preprocessing; Schur decomposition; numerical stability; convergence; Markov model; transition matrix; inverse scaling and squaring method; eigenvalues; algorithm; numerical experiments

Citations:

Zbl 1040.65038

Software:

ARPREC; MPFUN