One of the most important generalized inverses is the Drazin inverse, which is defined for square matrices having an index. The objective of this work is to investigate and present a computational tool in the form of an iterative method for computing this task. This scheme reaches the seventh rate of convergence as long as a suitable initial matrix is chosen and by employing only five matrix products per cycle. After some analytical discussions, several tests are provided to show the efficiency of the presented formulation.

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In this paper, an improvement to the mid-point method is contributed for finding the square root of a matrix as well as its inverse. To this aim, an iteration scheme to find this matrix function is constructed, and its error and stability estimates are provided to show the theoretical rate of convergence. Our higher-order method can compete with the existing iterative methods of a similar nature. This is illustrated in numerical simulations of various sizes.