Computing primary solutions of equations involving primary matrix functions

The matrix equation f(X)=A, where f is an analytic function and A is a square matrix, is considered. Some results on the classification of solutions are provided. When f is rational, a numerical algorithm is proposed to compute all solutions that can be written as a polynomial of A. For real data, the algorithm yields the real solutions using only real arithmetic. Numerical experiments show that the algorithm performs in a stable fashion when run in finite precision arithmetic.