A Schur Algorithm for Computing Matrix pth Roots

Any nonsingular matrix has pth roots. One way to compute matrix pth roots is via a specialized version of Newton’s method, but this iteration has poor convergence and stability properties in general. A Schur algorithm for computing a matrix pth root that generalizes methods of Björck and Hammarling [Linear Algebra Appl., 52/53 (1983), pp. 127–140] and Higham [Linear Algebra Appl., 88/89 (1987), pp. 405–430] for the square root is presented. The algorithm forms a Schur decomposition of A and computes a pth root of the (quasi-)triangular factor by a recursion. The backward error associated with the Schur method is examined, and the method is shown to have excellent numerical stability.