Several numerical methods for computing unitary polar factor of a matrix
- Fazlollah Soleymani^{1},
- Farhad Khaksar Haghani^{2} and
- Stanford Shateyi^{3}Email author
https://doi.org/10.1186/s13662-015-0732-z
© Soleymani et al. 2015
Received: 26 April 2015
Accepted: 20 December 2015
Published: 4 January 2016
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.
Keywords
MSC
1 Preliminaries
The Hermitian factor H is always unique and can be written as \((A^{*}A)^{\frac{1}{2}}\), while the unitary factor U is unique if A is nonsingular; see for more [2].
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 \(A^{2}\) via \(A^{*}A\), and vice versa.
Practical interest in the polar decomposition stems mainly from the fact that the unitary polar factor of A is the nearest unitary matrix to A in any unitarily invariant norm. The polar decomposition is therefore of interest whenever it is required to orthogonalize a matrix [5]. To obtain more background in this topic, one may refer to [6–9].
Remark 1.1
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 (5), which requires the computation of one pseudo-inverse per computing cycle.
The other sections of this paper are organized as follows. In Section 2, we derive an iteration function for polar decomposition. Next, Section 3 discusses the convergence properties of this method. It is revealed that the rate of convergence is six since the proposed formulation transforms the singular values of the approximated matrices produced per cycle with a sixth rate to unity (one). This discloses that our method is quite rapid. Several other new iterative methods are constructed in Section 4. Many numerical experiments are provided to support the theoretical aspects of the paper in Section 5. Finally, conclusions are drawn in Section 6.
2 A numerical method
Theorem 2.1
Let \(\alpha\in D\) be a simple zero of a sufficiently differentiable function \(f:D\subseteq\mathbb{C}\rightarrow\mathbb{C}\) for an open interval D, which contains \(x_{0}\) as an initial approximation of α. Then the iterative expression (9) has sixth order of convergence.
Proof
The proof is based on Taylor expansions of the function f around the appropriate points and would be similar to those taken in [14]. As a consequence, it is skipped over. □
The iteration obtained after applying a nonlinear equation solver on the mapping (8) and its reciprocal, could be used for polar decomposition. But here, the experimental results show that the reciprocal form (10) is more stable in the presence of round-off errors.
3 Convergence properties
This section is dedicated to the convergence properties of (11) for finding the unitary polar factor of A.
Theorem 3.1
Assume that \(A\in\mathbb{C}^{m\times n}\) is an arbitrary matrix. Then the matrix iterates \(\{U_{k}\}_{k=0}^{k=\infty}\) of (11) converge to U.
Proof
The proof of this theorem follows the lines of the proofs given in [16]. As such, it is skipped over. □
Theorem 3.2
Let \(A\in\mathbb{C}^{m\times n}\) be an arbitrary matrix. Then the new method (11) is of sixth order to find the unitary polar factor of A.
Proof
Remark 3.1
The presented method is not a member of the Padé family of iterations given in [17] (and discussed deeply in [18]), with global convergence. As a result, it is interesting from both theoretical and computational point of views.
4 Some other iterative methods
As discussed in the preceding sections, the construction of the iterative methods for finding the unitary polar factor of a matrix mainly relies on the nonlinear equation solver which is going to be applied on the mapping (8).
Now, some may question that the construction (9) is straightforward, since it is the combination of two already known methods. It is here stated that the main goal is to attain a new scheme for a polar decomposition which has global convergence behavior and is new, i.e., it is not a member of the Padé family of iterations (or its reciprocal). So, the novelty and usefulness of (9) in terms of solving nonlinear equations is not of main interest here and the importance is focused on providing a novel and useful scheme for finding the unitary polar factor.
The error analysis of the new schemes (21) and (23) are similar to the case given in Section 3. As a result, they are not included here.
5 Numerical results
We now apply different numerical methods for finding the unitary polar factors of many randomly generated rectangular matrices with complex entries. In order to help the readers to re-run the experiments we used \(\mathtt{SeedRandom[12345]}\) for producing pseudo-random (complex) numbers.
The random matrices for different dimensions of \(m\times n\) are constructed by the following piece of Mathematica code (\(I=\sqrt{-1}\)):
SeedRandom[12345]; number = 15;
Table[A[l] = RandomComplex[{-10 - 10 I,
10 + 10 I}, {m, n}];, {l, number}];
Results of comparison for the dimension \(\pmb{m\times n=110\times 100}\) in terms of the number of iterations
Matrix No. | NM | ANM | KHM | PM1 | PM2 | PM3 |
---|---|---|---|---|---|---|
1 | 10 | 8 | 6 | 4 | 5 | 4 |
2 | 10 | 7 | 6 | 4 | 5 | 4 |
3 | 10 | 8 | 6 | 4 | 5 | 4 |
4 | 10 | 7 | 6 | 4 | 5 | 4 |
5 | 10 | 8 | 6 | 4 | 5 | 4 |
6 | 10 | 8 | 6 | 4 | 5 | 4 |
7 | 10 | 8 | 6 | 4 | 5 | 4 |
8 | 10 | 8 | 6 | 4 | 5 | 4 |
9 | 10 | 8 | 6 | 4 | 5 | 4 |
10 | 10 | 7 | 6 | 4 | 5 | 4 |
11 | 10 | 8 | 6 | 4 | 5 | 4 |
12 | 10 | 8 | 6 | 4 | 5 | 4 |
13 | 10 | 8 | 6 | 4 | 5 | 4 |
14 | 10 | 8 | 6 | 4 | 5 | 4 |
15 | 10 | 8 | 6 | 4 | 5 | 4 |
Results of comparison for the dimension \(\pmb{m\times n=110\times 100}\) in terms of the elapsed time
Matrix No. | NM | ANM | KHM | PM1 | PM2 | PM3 |
---|---|---|---|---|---|---|
1 | 0.040002 | 0.041002 | 0.020001 | 0.018001 | 0.019001 | 0.020001 |
2 | 0.039002 | 0.044002 | 0.029002 | 0.022001 | 0.018001 | 0.019001 |
3 | 0.042002 | 0.042002 | 0.020001 | 0.018001 | 0.018001 | 0.020001 |
4 | 0.039002 | 0.041002 | 0.020001 | 0.019001 | 0.018001 | 0.022001 |
5 | 0.040002 | 0.042002 | 0.020001 | 0.019001 | 0.018001 | 0.020001 |
6 | 0.041002 | 0.041002 | 0.020001 | 0.019001 | 0.018001 | 0.020001 |
7 | 0.039002 | 0.045003 | 0.023001 | 0.018001 | 0.027002 | 0.026002 |
8 | 0.050003 | 0.041002 | 0.020001 | 0.022001 | 0.019001 | 0.020001 |
9 | 0.039002 | 0.045003 | 0.020001 | 0.018001 | 0.021001 | 0.021001 |
10 | 0.043002 | 0.041002 | 0.020001 | 0.019001 | 0.018001 | 0.020001 |
11 | 0.040002 | 0.042002 | 0.020001 | 0.019001 | 0.019001 | 0.020001 |
12 | 0.040002 | 0.049003 | 0.022001 | 0.019001 | 0.018001 | 0.019001 |
13 | 0.040002 | 0.041002 | 0.020001 | 0.021001 | 0.018001 | 0.020001 |
14 | 0.041002 | 0.045003 | 0.020001 | 0.019001 | 0.018001 | 0.019001 |
15 | 0.040002 | 0.042002 | 0.020001 | 0.019001 | 0.019001 | 0.019001 |
Results of comparison for the dimension \(\pmb{m\times n=210\times 200}\) in terms of the elapsed time
Matrix No. | NM | ANM | KHM | PM1 | PM2 | PM3 |
---|---|---|---|---|---|---|
1 | 0.208012 | 0.240014 | 0.092005 | 0.088005 | 0.093005 | 0.091005 |
2 | 0.207012 | 0.240014 | 0.090005 | 0.086005 | 0.079005 | 0.090005 |
3 | 0.209012 | 0.221013 | 0.092005 | 0.088005 | 0.081005 | 0.094005 |
4 | 0.209012 | 0.243014 | 0.097005 | 0.106006 | 0.093005 | 0.090005 |
5 | 0.216012 | 0.240014 | 0.091005 | 0.107006 | 0.094005 | 0.089005 |
6 | 0.211012 | 0.216012 | 0.090005 | 0.088005 | 0.084005 | 0.090005 |
7 | 0.208012 | 0.243014 | 0.091005 | 0.105006 | 0.093005 | 0.092005 |
8 | 0.211012 | 0.225013 | 0.095005 | 0.091005 | 0.078005 | 0.088005 |
9 | 0.210012 | 0.238014 | 0.092005 | 0.089005 | 0.092005 | 0.088005 |
10 | 0.217012 | 0.239014 | 0.091005 | 0.094005 | 0.093005 | 0.089005 |
11 | 0.208012 | 0.218012 | 0.090005 | 0.086005 | 0.078004 | 0.089005 |
12 | 0.209012 | 0.240014 | 0.102006 | 0.087005 | 0.080005 | 0.089005 |
13 | 0.209012 | 0.244014 | 0.092005 | 0.086005 | 0.078005 | 0.089005 |
14 | 0.210012 | 0.239014 | 0.091005 | 0.086005 | 0.079005 | 0.094005 |
15 | 0.216012 | 0.239014 | 0.097006 | 0.105006 | 0.093005 | 0.088005 |
Results of comparison for the dimension \(\pmb{m\times n=410\times 400}\) in terms of the elapsed time
Matrix No. | NM | ANM | KHM | PM1 | PM2 | PM3 |
---|---|---|---|---|---|---|
1 | 1.131065 | 1.151066 | 0.581033 | 0.619035 | 0.542031 | 0.607035 |
2 | 1.144065 | 1.150066 | 0.587034 | 0.597034 | 0.522030 | 0.610035 |
3 | 1.190068 | 1.144065 | 0.578033 | 0.587034 | 0.532030 | 0.632036 |
4 | 1.153066 | 1.144065 | 0.589034 | 0.591034 | 0.524030 | 0.598034 |
5 | 1.135065 | 1.147066 | 0.581033 | 0.586033 | 0.538031 | 0.607035 |
6 | 1.145066 | 1.148066 | 0.588034 | 0.599034 | 0.527030 | 0.602034 |
7 | 1.134065 | 1.152066 | 0.587034 | 0.593034 | 0.532030 | 0.599034 |
8 | 1.123064 | 1.157066 | 0.577033 | 0.594034 | 0.518030 | 0.617035 |
9 | 1.137065 | 1.149066 | 0.589034 | 0.593034 | 0.520030 | 0.604035 |
10 | 1.127064 | 1.140065 | 0.503029 | 0.593034 | 0.521030 | 0.614035 |
11 | 1.129065 | 1.144065 | 0.577033 | 0.591034 | 0.524030 | 0.596034 |
12 | 1.119064 | 1.147066 | 0.593034 | 0.592034 | 0.522030 | 0.600034 |
13 | 1.139065 | 1.152066 | 0.591034 | 0.590034 | 0.527030 | 0.600034 |
14 | 1.124064 | 1.142065 | 0.587033 | 0.593034 | 0.522030 | 0.595034 |
15 | 1.126064 | 1.155066 | 0.597034 | 0.605035 | 0.522030 | 0.619035 |
Results of comparison for the dimension \(\pmb{m\times n=510\times 500}\) in terms of the number of iterations
Matrix No. | NM | ANM | KHM | PM1 | PM2 | PM3 |
---|---|---|---|---|---|---|
1 | 12 | 9 | 7 | 5 | 6 | 5 |
2 | 12 | 9 | 7 | 5 | 6 | 5 |
3 | 12 | 9 | 7 | 5 | 6 | 5 |
4 | 12 | 9 | 7 | 5 | 6 | 5 |
5 | 12 | 9 | 7 | 5 | 6 | 5 |
6 | 12 | 9 | 7 | 5 | 6 | 5 |
7 | 12 | 9 | 7 | 5 | 6 | 5 |
8 | 12 | 9 | 7 | 5 | 6 | 5 |
9 | 12 | 9 | 7 | 5 | 6 | 5 |
10 | 12 | 9 | 7 | 5 | 6 | 5 |
Results of comparison for the dimension \(\pmb{m\times n=510\times 500}\) in terms of the elapsed time
Matrix No. | NM | ANM | KHM | PM1 | PM2 | PM3 |
---|---|---|---|---|---|---|
1 | 2.160124 | 2.222127 | 1.067061 | 1.116064 | 0.926053 | 1.074061 |
2 | 2.185125 | 2.181125 | 1.041060 | 1.129065 | 0.924053 | 1.102063 |
3 | 2.142123 | 2.199126 | 1.029059 | 1.174067 | 0.957055 | 1.121064 |
4 | 2.154123 | 2.111121 | 1.113064 | 1.121064 | 0.955055 | 1.096063 |
5 | 2.139122 | 2.131122 | 1.086062 | 1.049060 | 0.961055 | 1.077062 |
6 | 2.130122 | 2.195126 | 1.084062 | 1.050060 | 0.939054 | 1.075061 |
7 | 2.126122 | 3.235185 | 1.077062 | 1.134065 | 0.938054 | 1.076062 |
8 | 2.126122 | 2.098120 | 1.108063 | 1.083062 | 0.957055 | 1.076062 |
9 | 2.100120 | 2.147123 | 1.054060 | 1.084062 | 0.994057 | 1.069061 |
10 | 2.157123 | 2.210126 | 1.052060 | 1.076062 | 0.966055 | 1.073061 |
Results of comparison for the dimension \(\pmb{m\times n=510\times 500}\) in terms of the elapsed time
Matrix No. | NM | ANM | KHM | PM1 | PM2 | PM3 |
---|---|---|---|---|---|---|
1 | 2.006115 | 1.940111 | 0.972056 | 0.984056 | 0.878050 | 1.003057 |
2 | 1.967113 | 1.937111 | 0.969055 | 0.983056 | 0.876050 | 1.013058 |
3 | 1.967113 | 1.918110 | 0.972056 | 0.994057 | 0.878050 | 1.003057 |
4 | 1.982113 | 1.912109 | 0.980056 | 0.996057 | 0.876050 | 1.012058 |
5 | 2.099120 | 1.932111 | 0.968055 | 0.992057 | 0.886051 | 1.011058 |
6 | 1.969113 | 1.919110 | 0.977056 | 0.984056 | 0.889051 | 1.003057 |
7 | 1.974113 | 1.919110 | 0.975056 | 0.983056 | 0.881050 | 1.015058 |
8 | 1.967113 | 1.920110 | 0.969055 | 0.999057 | 0.877050 | 1.011058 |
9 | 1.976113 | 1.920110 | 0.992057 | 1.003057 | 0.875050 | 1.004057 |
10 | 1.970113 | 1.932111 | 0.975056 | 0.990057 | 0.876050 | 1.012058 |
To give an answer to the key question: whether the increasing order convergence is worth in view of increasing the matrix multiplications in each iteration, it is requisite to incorporate the notion of efficiency index, \(p^{1/\theta}\), whereas p and θ stand for the rate of convergence and the computational cots per cycle, respectively. This is achieved by assuming that each matrix-matrix multiplication cost 1-unit while the cost for one regular matrix inverse is 1.5-unit and one matrix Moore-Penrose inverse is 3-unit. Consequently, the efficiency indices for the discussed methods are: \(E(\mbox{4})\simeq1.2599\), \(E(\mbox{6})\simeq1.2210\), \(E(\mbox{11})\simeq1.2698\), \(E(\mbox{21})\simeq1.2866\), and \(E(\mbox{23})\simeq1.2962\).
Results of comparison for the dimension \(\pmb{m\times n=600\times 600}\) in terms of the elapsed time
Matrix No. | NM | ANM | KHM | PM1 | PM2 | PM3 |
---|---|---|---|---|---|---|
1 | 1.281073 | 1.320075 | 1.687096 | 1.866107 | 1.568090 | 1.870107 |
2 | 1.301074 | 1.322076 | 1.695097 | 1.853106 | 1.554089 | 1.878107 |
3 | 1.216070 | 1.333076 | 1.706098 | 1.867107 | 1.562089 | 1.578090 |
4 | 1.286074 | 1.324076 | 1.703097 | 1.847106 | 1.561089 | 1.895108 |
5 | 1.296074 | 1.325076 | 1.716098 | 1.858106 | 1.557089 | 1.858106 |
6 | 1.370078 | 1.319076 | 1.905109 | 1.830105 | 1.768101 | 1.873107 |
7 | 1.286074 | 1.320075 | 1.708098 | 1.857106 | 1.566090 | 1.583091 |
8 | 1.531088 | 1.317075 | 1.906109 | 2.123121 | 1.766101 | 1.868107 |
9 | 1.375079 | 1.314075 | 1.917110 | 1.851106 | 1.762101 | 1.888108 |
10 | 1.288074 | 1.321076 | 1.711098 | 1.841105 | 1.562089 | 1.855106 |
The acquired numerical results agree with the theoretical discussions given in Sections 2 and 3, overwhelmingly. As a result, we can state that PM1-PM3 reduce the number of iterations and time in finding the polar decomposition.
6 Concluding remarks
In this paper, we developed high-order methods for matrix polar decomposition. It has been shown that the convergence is global. Many numerical tests (of various dimensions) have been provided to show the performance of the new method.
In 1991, Kenney and Laub [17] proposed a family of rational iterative methods for sign (subsequently for polar decomposition), based on Padé approximation. Their principal Padé iterations are convergent globally. Thus, we have convergent methods of arbitrary orders for sign (subsequently for polar decomposition). However, here we tried to propose new methods, which are interesting from theoretical point of view and are not members of Padé family. Numerical results have demonstrated the behavior of the new algorithms.
Declarations
Acknowledgements
The authors thank the anonymous referees for their suggestions which helped to improve the quality of the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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