Improved Qrginv Algorithm for Computing Moore-Penrose Inverse Matrices

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 real matrix A with rank . Numerical experiments show that the resulting pseudoinverse matrix is reasonably accurate and its computation time is significantly less than that obtained by Katsikis et al. 1. Introduction Let denote the set of all matrices over the field of real numbers, . The symbols , rank( ) will stand for the transpose and rank of , respectively. For a matrix , the Moore-Penrose inverse of , denoted by , is the unique matrix satisfying the following equations:(i) , (ii) , (iii) , (iv) . A lot of works concerning generalized inverses have been carried out, in finite and infinite dimensions (e.g., [1–3]). There are several methods for computing the Moore-Penrose inverse matrix [2, 4–8]. In a recent article [9], an improved method for the computation of the Moore-Penrose inverse matrix provided. 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 2 the improved version of this method is presented for computing the pseudoinverse of an real matrix with rank . In Section 3, the numerical results of some test matrices are given. Section 4 is devoted to the concluding remarks. 2. Improved Qrginv Method (IMqrginv) In [9], a method (qrginv) for computing the Moore-Penrose inverse of an arbitrary matrix was presented. They made use of the QR-factorization, as well as an algorithm based on a known reverse order law for generalized inverse matrices, and also they apply a method (ginv), presented in [4], based on a full rank Cholesky factorization of possibly singular symmetric positive matrices. In the current paper, we improved qrginv algorithm using the QR-factorization by Gram-Schmidt orthonormalization (GSO) and Theorem 1 for faster computing Moore-Penrose inverse of arbitrary matrices (including singular and rectangular). We should note that we invoke ginv algorithm. In order to support and state our achievement we need to remind Gram-Schmidt orthonormalization (GSO) and the QR-factorization. 2.1. The Gram-Schmidt Procedure Let us remind a generalization of the Gram-Schmidt