Exploiting a higher-order scheme for matrix square root and its inverse simultaneously

In this work, an investigation on an iterative scheme to calculate the matrix square root and its inversion simultaneously is performed and further discussed via the concept of matrix sign function. Convergence properties are discussed under some conditions on the choice of the initial matrix as well as the input matrix A$$ A $$. It is then attempted to propose an iterative method possessing higher convergence order, which is also stable. Extension of the proposed scheme to the p$$ p $$th root of a matrix is also given. Ultimately, several tests including an application of the proposed iterative method to solve matrix differential equations are brought forward.