Katsikis et al. presented a computational method in order to calculate the Moore-Penrose inverse of an arbitrary matrix (including singular and rectangular) (2011). In this paper, an improved version of this method is presented for computing the pseudo inverse of an
Let
A lot of works concerning generalized inverses have been carried out, in finite and infinite dimensions (e.g., [
In this paper, we aim to improve their method so that it can be used for any kind of matrices, square or rectangular, full rank or not. The numerical examples show that our method is competitive in terms of accuracy and is much faster than the commonly used methods and can also be used for large sparse matrices.
This paper is organized as follows. In Section
In [
In the current paper, we improved
Let us remind a generalization of the Gram-Schmidt orthonormalization process (shortly GSO) which is applied for singular matrices. Let
The integer
Let us remind the QR-factorization for arbitrary matrices (including singular and rectangular).
Let the orthonormal set
One obtains from (
The following theorem is due to MacDuffe [
If
As a direct consequence of Theorem
Let
With
The function
In this section, we compare the performance of the proposed method (
We are computing the performance of the proposed method
Error and computational time results for random singular matrices.
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Method | Time |
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qrginv | 0.0317 |
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IMqrginv | 0.0137 |
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qrginv | 0.1176 |
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IMqrginv | 0.0786 |
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qrginv | 1.0584 |
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IMqrginv | 0.8236 |
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qrginv | 10.8336 |
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IMqrginv | 8.2976 |
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qrginv | 81.1429 |
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IMqrginv | 55.0746 |
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In this section we use a set of singular test matrices that includes singular matrices of size
Error and computational time results for
Method | Time |
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qrginv | 0.0376 |
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IMqrginv | 0.0187 |
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Error and computational time results for
Method | Time |
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qrginv | 0.2816 |
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IMqrginv | 0.2334 |
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Error and computational time results for
Method | Time |
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qrginv | 0.0377 |
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IMqrginv | 0.0165 |
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Error and computational time results for
Method | Time |
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qrginv | 0.0279 |
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IMqrginv | 0.0099 |
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Error and computational time results for
Method | Time |
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qrginv | 0.0244 |
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IMqrginv | 0.0102 |
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Error and computational time results for
Method | Time |
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qrginv | 0.0276 |
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IMqrginv | 0.0138 |
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Error and computational time results for
Method | Time |
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qrginv | 0.0247 |
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IMqrginv | 0.0117 |
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Error and computational time results for
Method | Time |
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qrginv | 0.0154 |
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IMqrginv | 0.0035 |
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For sparse matrices, we have chosen some matrices from Matrix-Market collection [
Test problem information.
Matrix |
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Cond |
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WELL1033 | 1033 | 320 | 4732 | Not available |
WELL1850 | 1850 | 712 | 8758 | Not available |
ILCC1033 | 1033 | 320 | 4732 | Not available |
ILCC1850 | 10850 | 712 | 8758 | Not available |
WATT1 | 1856 | 1856 | 11360 |
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GR-30-30 | 900 | 900 | 4322 |
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ADD20 | 2395 | 2395 | 17319 |
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NOSE3 | 960 | 960 | 8402 |
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SHERMAN1 | 1000 | 1000 | 3750 |
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Error and computational time results for
Method | Method | Time |
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qrginv | 0.6277 |
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IMqrginv | 0.3300 |
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qrginv | 3.8271 |
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IMqrginv | 1.7559 |
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qrginv | 0.6119 |
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IMqrginv | 0.3786 |
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qrginv | 3.9197 |
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IMqrginv | 1.6791 |
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qrginv | 10.5110 |
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6.1068 |
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IMqrginv | 3.1018 |
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6.1068 |
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qrginv | 3.1095 |
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IMqrginv | 1.6834 |
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qrginv | 60.9176 |
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IMqrginv | 59.0933 |
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qrginv | 4.3836 |
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IMqrginv | 3.0214 |
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qrginv | 4.3150 |
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IMqrginv | 1.3069 |
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In this paper we have presented a new method, called
Function IMqrginv = IMqrginv(
The author declares that there is no conflict of interests regarding the publication of this paper.
The author deeply indebted to referees whose comments helped in improving the paper. Also, the author would like to thank Professor Davod Khojasteh Salkuyeh from Guilan University for his worth comments on this paper.