An efficient quadratically convergent iterative method to find the Moore–Penrose inverse

ABSTRACT

In this paper, we propose a new Schulz-type method to find the pseudo-inverse (also known as the Moore–Penrose inverse) of a singular or rectangular real (or complex) matrix. It is proved that the method converges quadratically. A wide set of numerical comparisons of our method with nine higher order methods shows that the average number of matrix–matrix multiplications and the average CPU time of proposed method are considerably less than those of other methods. So, our new method can be considered as a fast method. For each of sizes $n\times n$ and $n\times (n+10)$, n=100,200,300,400, ten random matrices were chosen to make these comparisons. So, overall 800 problems were solved.

KEYWORDS:

• Moore–Penrose inverse
• iterative method
• Schulz-type method
• second-orderconvergence
• matrix multiplication

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 15A09
• 65F30
• 65F50

Acknowledgments

The authors would like to thank the referees for their useful comments and constructive suggestions which substantially improved the quality of this paper.

Disclosure

No potential conflict of interest was reported by the authors.