Fixed point iterations for computing square roots and the matrix sign function of complex matrices

The purpose of this work has been the development of new set of rational iterations for computing square roots and the matrix sign function of complex matrices. Given any positive integer r ≥ 2, we presented a systematic way of deriving rth order convergent algorithms for matrix square roots, the matrix sign function, invariant subspaces in different half-planes, and the polar decomposition. We have shown that these iterations are applicable for computing square roots of more general type of matrices than previously reported, such as matrices in which some of its eigenvalues are negative. Also, algorithms for computing square roots and the invariant subspace of a given matrix in any given half-plane are derived.