University of Guilan Journal of Mathematical Modeling 2345-394X 2026 07 07 A class of inversion-free iterative methods to find polar decomposition 9736 10.22124/jmm.2026.32920.2996 EN Salman Sheikhi Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran Hamid Esmaeili Shahid Mostafa Ahmadi Roshan Street Hamedan Iran 0000-0001-5650-584X Journal Article 2026 02 06 This paper presents a novel approach for determining the polar decomposition of a complex (real) matrix, accompanied by a thorough examination of its convergence. The proposed class of methods' structure is inspired by Petkovic's idea for computing the Moore-Penrose inverse of a matrix A using polynomials. This class of methods, rooted in matrix multiplications, circumvents the need for matrix inversion. Several categories of methods with varying convergence orders are introduced and scrutinized. The efficacy of these proposed methods is demonstrated through numerical experiments, comparing them with alternative approaches. The study focuses on random matrices of dimensions $n\times n$, where $n$ takes values of $80$, $100$, $120$, $150$, $180$, and $200$ and several ill-conditioned matrices. The assessment includes key metrics such as the average number of iterations, matrix multiplications, and execution time for each method under consideration. The results affirm the efficiency of certain proposed methods in comparison to others. https://jmm.guilan.ac.ir/article_9736_177c99afbfd4c6d466d2c36f73dca7bd.pdf