Approximating the principal matrix square root using some novel third-order iterative methods

It is known that the matrix square root has a significant role in linear algebra computations arisen in engineering and sciences. Any matrix with no eigenvalues in R- has a unique square root for which every eigenvalue lies in the open right half-plane. In this research article, a relationship between a scalar root finding method and matrix computations is exploited to derive new iterations to the matrix square root. First, two algorithms that are cubically convergent with conditional stability will be proposed. Then, for solving the stability issue, we will introduce coupled stable schemes that can compute the square root of a matrix with very acceptable accuracy. Furthermore, the convergence and stability properties of the proposed recursions will be analyzed in details. Numerical experiments are included to illustrate the properties of the modified methods.