Convergence Rate Improvement of Richardson and Newton-Schulz Iterations

TL;DR

This paper proposes a new composite power series expansion to enhance the convergence rate of Richardson and Newton-Schulz iterations, enabling faster and more efficient matrix inversion and parameter estimation.

Contribution

It introduces a novel composite power series expansion integrated into existing iterative methods, improving convergence rates and computational efficiency.

Findings

Significant convergence rate improvements demonstrated through simulations.

Development of a recursive, efficient combined iteration algorithm.

Creation of a unified factorization toolkit for power series expansion.

Abstract

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.