A New Iterative Method to Find Polar Decomposition

doi:10.1007/s42967-024-00366-1

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (6): 2243-2256.doi: 10.1007/s42967-024-00366-1

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A New Iterative Method to Find Polar Decomposition

Salman Sheikhi, Hamid Esmaeili

Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan 6516738695, Iran

收稿日期:2023-09-04 修回日期:2023-12-18 发布日期:2025-12-24
通讯作者: Hamid Esmaeili, E-mail:esmaeili@basu.ac.ir E-mail:esmaeili@basu.ac.ir

A New Iterative Method to Find Polar Decomposition

Salman Sheikhi, Hamid Esmaeili

Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan 6516738695, Iran

Received:2023-09-04 Revised:2023-12-18 Published:2025-12-24
Contact: Hamid Esmaeili, E-mail:esmaeili@basu.ac.ir E-mail:esmaeili@basu.ac.ir

摘要/Abstract

摘要： 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.

关键词: Polar decomposition, Iterative method, Fourth-order convergent, Matrix multiplications

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.

Key words: Polar decomposition, Iterative method, Fourth-order convergent, Matrix multiplications

中图分类号:

Salman Sheikhi, Hamid Esmaeili. A New Iterative Method to Find Polar Decomposition[J]. Communications on Applied Mathematics and Computation, 2025, 7(6): 2243-2256.

参考文献

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[10] Kovarik, Z.: Some iterative methods for improving orthogonality. SIAM J. Numer. Anal. 7, 386–389 (1970)
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A New Iterative Method to Find Polar Decomposition

A New Iterative Method to Find Polar Decomposition

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