A new iterative method to find the Moore–Penrose inverse

In this paper, we introduce a new efficient iterative method to find the Moore–Penrose inverse of a singular or rectangular real (or complex) matrix, which has five matrix-matrix multiplications per iteration. It is proved that the proposed method is fourth-order convergent to the Moore–Penrose inverse. Numerical results are obtained using random matrices of sizes n×n and n×(n+10), n=100,200,300,400. The average number of iterations, the average total number of matrix-matrix multiplications, the average precision, and the average elapsed execution time of two methods are computed using ten matrices in each dimension. Based on this comparison, our new method can be considered as an efficient fast method with low computational cost.