On the Newton Method for the Matrix P th Root
Bruno Iannazzo · SIAM Journal on Matrix Analysis and Applications · 2006
Stable versions of Newton's iteration for computing the principal matrix pth root A 1/p of an n x n matrix A are provided. In the case in which X 0 is the identity matrix, it is proved that the method converges for any matrix A having eigenvalues with modulus less than 1 and with positive real parts. Based on these results we provide a general algorithm for computing the principal pth root of any matrix A having no nonpositive real eigenvalues. The algorithm has quadratic convergence, is stable in a neighborhood of the solution, and has a cost of O(n 3 log p) operations per step. Numerical experiments and comparisons are performed.