Iterative methods for Moore-Penrose inverse

Abstract

The Moore-Penrose inverse is one of the most important generalized inverses for arbitrary singular square or rectangular matrix. It finds many applications in engineering and applied sciences. The direct methods to find such inverse is expensive, especially for large matrices. Therefore, various numerical methods have been developed to compute the Moore-Penrose inverse. This thesis is mainly concerned with the development of iterative methods to compute the Moore-Penrose inverse. Besides our new results the thesis contains several recent known iterative methods. The convergence properties of these methods are presented. And, several numerical examples are given. Our own results involve new family of second-order iterative algorithms for computing the Moore-Penrose inverse. The construction of this algorithm is based on the usage of Penrose equations with approximations for p-th root for a product of the matrix with its inverse approximations. Convergence properties are considered. Numerical results are also presented and a comparison with Newton’s method is made. It is observed that the new methods require less number of iterations than that of Newton’s method. In addition, numerical experiments show that these methods are more effective than Newton’s method when the number of columns increases than the number of rows. In addition, we establish a new iterative scheme by using a square of the product of the matrix with its inverse approximations. By convergence analysis, we show that this scheme is also a second order. Several numerical tests are made. It is observed that the above family is more effective than this method.