  We address the general mathematical problem of computing the inverse $p$-th
root of a given matrix in an efficient way. A new method to construct iteration
functions that allow calculating arbitrary $p$-th roots and their inverses of
symmetric positive definite matrices is presented. We show that the order of
convergence is at least quadratic and that adaptively adjusting a parameter $q$
always leads to an even faster convergence. In this way, a better performance
than with previously known iteration schemes is achieved. The efficiency of the
iterative functions is demonstrated for various matrices with different
densities, condition numbers and spectral radii.
