A class of iterative methods for computing polar decomposition

Abstract

In this article, there is offered a parametric class of iterative methods for computing the polar decomposition of a matrix. Each iteration of this class needs only one scalar-by-matrix and three matrix-by-matrix multiplications. It is no use computing inversion, so no numerical problems can be created because of ill-conditioning. Some available methods can be included in this class by choosing a suitable value for the parameter. There are obtained conditions under which this class is always quadratically convergent. The numerical comparison performed among six quadratically convergent methods for computing polar decomposition, and a special method of this class, chosen based on a specific value for the parameter, shows that the number of iterations of the special method is considerably near that of a cubically convergent Halley's method. Ten n×n matrices with n=5, 10, 20, 50, 100 were chosen to make this comparison.

Keywords:

• polar decomposition
• iterative method
• quadratic convergence

2000 AMS Subject Classifications :

• 65F30
• 65F25

Acknowledgement

The author thanks the referee for his important suggestions, which essentially improved the first version of the paper.