A Stable Coupled Newton's Iteration for the Matrix Inverse $p$-th Root

The computation of the inverse roots of matrices arises in evaluating non-symmetric eigenvalue problems, solving nonlinear matrix equations, computing some matrix functions, control theory and several other areas of applications. It is possible to approximate the matrix inverse pth roots by exploiting a specialized version of New- ton''s method, but previous researchers have mentioned that some iterations have poor convergence and stability properties. In this work, a stable recursive technique to evaluate an inverse pth root of a given matrix is presented. The scheme is analyzed and its properties are investigated. Computational experiments are also performed to illustrate the strengths and weaknesses of the proposed method.