Several numerical methods for computing unitary polar factor of a matrix

Fazlollah Soleymani,Farhad Khaksar Haghani,Stanford Shateyi

doi:10.1186/s13662-015-0732-z

Fazlollah Soleymani, Farhad Khaksar Haghani + Show 1 more

Open Access

https://doi.org/10.1186/s13662-015-0732-z

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Abstract

We present several numerical schemes for computing the unitary polar factor of rectangular complex matrices. Error analysis shows high orders of convergence. Many experiments in terms of number of iterations and elapsed times are reported to show the efficiency of the new methods in contrast to the existing ones.

Highlights

Where H is a Hermitian positive semi-definite matrix of order n and U ∈ Cm×n is a subunitary matrix [ ]
These integral formulas reveal that any property or iterative method involving the matrix sign function can be transformed into one for the polar decomposition by replacing A via A∗A, and vice versa
We point out that here we focus mainly on computing the unitary polar factor of rectangular matrices, since the high-order methods discussed in this work will not require the computation of pseudo-inverse and is better than the corresponding Newton’s version ( ), which requires the computation of one pseudo-inverse per computing cycle

Summary

Introduction

Where H is a Hermitian positive semi-definite matrix of order n and U ∈ Cm×n is a subunitary matrix [ ]. These integral formulas reveal that any property or iterative method involving the matrix sign function can be transformed into one for the polar decomposition by replacing A via A∗A, and vice versa. This is known as one of the good ways in the literature for constructing an initial value to ensure the convergence of iterative Newton-type methods for finding the unitary polar factor of A.

Results

Conclusion