A New Iterative Method to Find Polar Decomposition

Abstract

In this paper, we present a novel and efficient iterative approach for computing the polar decomposition of rectangular (or square) complex (or real) matrices. The method proposed herein entails four matrix multiplications in each iteration, effectively circumventing the need for matrix inversions. We substantiate that this method exhibits fourth-order convergence. To illustrate its efficacy relative to alternative techniques, we conduct numerical experiments using randomly generated matrices of dimensions $n\times n$, where n assumes values of 80, 90, 100, 120, 150, 180, and 200. Through two illustrative examples, we provide numerical results. We gauge the performance of different methods by calculating essential metrics based on ten matrices for each dimension. These metrics include the average iteration count, the average total matrix multiplication count, the average precision, and the average execution time. Through meticulous comparison, our newly devised method emerges as a proficient and rapid solution, boasting a reduced computational overhead.