Specialized and hybrid Newton schemes for the matrix pth root

We discuss different variants of Newton’s method for computing the pth root of a given matrix. A suitable implementation is presented for solving the Sylvester equa-tion, that appears at every Newton’s iteration, via Kronecker products. This approach is quadratically convergent and stable, but too expensive in computational cost. In contrast we propose and analyze some specialized versions that exploit the commuta-tion of the iterates with the given matrix. These versions are relatively inexpensive but have either stability problems or stagnation problems when good precision is re-quired. Hybrid versions are presented to take advantage of the best features in both approaches. Preliminary and encouraging numerical results are presented for p = 3 and p = 5