  Fast convergent, accurate, computationally efficient, parallelizable, and
robust matrix inversion and parameter estimation algorithms are required in
many time-critical and accuracy-critical applications such as system
identification, signal and image processing, network and big data analysis,
machine learning and in many others. This paper introduces new composite power
series expansion with optionally chosen rates (which can be calculated
simultaneously on parallel units with different computational capacities) for
further convergence rate improvement of high order Newton-Schulz iteration. New
expansion was integrated into the Richardson iteration and resulted in
significant convergence rate improvement. The improvement is quantified via
explicit transient models for estimation errors and by simulations. In
addition, the recursive and computationally efficient version of the
combination of Richardson iteration and Newton-Schulz iteration with composite
expansion is developed for simultaneous calculations. Moreover, unified
factorization is developed in this paper in the form of tool-kit for power
series expansion, which results in a new family of computationally efficient
Newton-Schulz algorithms.
