Residual methods for the large-scale matrix pth root and some related problems

The problem of finding the pth root of a matrix has received special attention in the last few years. Standard approaches for this problem include and combine some variations of Newton’s method, which in turn involve matrix factorizations that, in general, are not suitable for large-scale problems. Motivated by some recently developed low-cost iterative schemes for nonlinear problems, we consider and analyze specialized residual methods that only require a few matrix-matrix products per iteration, and hence are suitable for the large-scale case. As a by-product we also discuss the advantages of residual methods for general nonlinear problems whose variables separate. Preliminary and encouraging numerical results are presented for computing pth roots of large-scale symmetric and positive definite matrices, for different values of p.