Predrag S. Stanimirović

On the generalized Drazin inverse and generalized resolvent

We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of a Banach algebra element is characterized in terms of the existence of a particularly chosen limit process. As an application, the computing of the Moore-Penrose inverse in >C*-algebras is considered. We investigate the generalized Drazin inverse as an outer inverse with prescribed range and kernel. Also, 2 × 2 operator matrices are considered. As corollaries, we get some well-known results.

Iterative method for computing the Moore-Penrose inverse based on Penrose equations

An iterative algorithm for estimating the Moore-Penrose generalized inverse is developed. The main motive for the construction of the algorithm is simultaneous usage of Penrose equations (2) and (4). Convergence properties of the introduced method as well as their first-order and second-order error terms are considered. Numerical experiment is also presented.

Two direct methods in linear programming

Abstract In this paper, we introduce two direct methods for solving some classes of linear programming problems. The first method produces the extreme vertex or a neighboring vertex with respect to the extreme point. The second method is based on the game theory. Both these methods can be used in the preparation of the starting point for the simplex method. The efficiency of the improved simplex method, whose starting point is constructed by these introduced methods, is compared with the original simplex method and the interior point methods, and illustrated by examples. Also, we investigate the elimination of excessive constraints.

Full-rank and determinantal representation of the Drazin inverse

Abstract In this article, we introduce a full-rank representation of the Drazin inverse A D of a given complex matrix A, which is based on an arbitrary full-rank decomposition of A l , l⩾k , where k is the index of A. Using this general representation, we introduce a determinantal representation of the Drazin inverse. More precisely, we represent elements of the Drazin inverse A D as a fraction of two expressions involving minors of the order rank (A k ) , k= ind (A) , taken from the matrices A and rank invariant powers A l , l⩾k . Also, we examine conditions for the existence of the Drazin inverse for matrices whose elements are taken from an integral domain. Finally, a few correlations between the minors of the Drazin inverse A D , powers of the Drazin inverse and the minors of the matrix A k , k= ind (A) , are explicitly derived.

Recurrent Neural Network for Computing the Drazin Inverse

This paper presents a recurrent neural network (RNN) for computing the Drazin inverse of a real matrix in real time. This recurrent neural network (RNN) is composed of n independent parts (subnetworks), where n is the order of the input matrix. These subnetworks can operate concurrently, so parallel and distributed processing can be achieved. In this way, the computational advantages over the existing sequential algorithms can be attained in real-time applications. The RNN defined in this paper is convenient for an implementation in an electronic circuit. The number of neurons in the neural network is the same as the number of elements in the output matrix, which represents the Drazin inverse. The difference between the proposed RNN and the existing ones for the Drazin inverse computation lies in their network architecture and dynamics. The conditions that ensure the stability of the defined RNN as well as its convergence toward the Drazin inverse are considered. In addition, illustrative examples and examples of application to the practical engineering problems are discussed to show the efficacy of the proposed neural network.

A higher order iterative method for computing the Drazin inverse.

A method with high convergence rate for finding approximate inverses of nonsingular matrices is suggested and established analytically. An extension of the introduced computational scheme to general square matrices is defined. The extended method could be used for finding the Drazin inverse. The application of the scheme on large sparse test matrices alongside the use in preconditioning of linear system of equations will be presented to clarify the contribution of the paper.

Symbolic computation of weighted Moore-Penrose inverse using partitioning method

Abstract We propose a method and algorithm for computing the weighted Moore–Penrose inverse of one-variable rational matrices. Continuing this idea, we develop an algorithm for computing the weighted Moore–Penrose inverse of one-variable polynomial matrix. These methods and algorithms are generalizations of the method for computing the weighted Moore–Penrose inverse for constant matrices, originated in Wang and Chen [G.R. Wang, Y.L. Chen, A recursive algorithm for computing the weighted Moore–Penrose inverse A MN † , J. Comput. Math. 4 (1986) 74–85], and the partitioning method for computing the Moore–Penrose inverse of rational and polynomial matrices introduced in Stanimirovic and Tasic [P.S. Stanimirovic, M.B. Tasic, Partitioning method for rational and polynomial matrices, Appl. Math. Comput. 155 (2004) 137–163]. Algorithms are implemented in the symbolic computational package MATHEMATICA .

Gauss-Jordan elimination method for computing outer inverses

This paper deals with the algorithm for computing outer inverse with prescribed range and null space, based on the choice of an appropriate matrix G and Gauss-Jordan elimination of the augmented matrix [G|I]. The advantage of such algorithms is the fact that one can compute various generalized inverses using the same procedure, for different input matrices. In particular, we derive representations of the Moore-Penrose inverse, the weighted Moore-Penrose inverse, the Drazin inverse as well as {2,4} and {2,3}-inverses. Numerical examples on different test matrices are presented, as well as the comparison with well-known methods for generalized inverses computation.

Partitioning method for rational and polynomial matrices

We propose an extension of the Greviles partitioning method for computing the Moore-Penrose inverse, which is applicable to the set of rational matrices. Also, we develop an algorithm for computing the Moore-Penrose inverse of given one-variable polynomial matrix, which is based on the Greviles method. Major problems arising in the implementation of this method are repetitive recomputations of the same values and simplification of rational and polynomial expressions which contain unknown variable. These algorithms are implemented in the symbolic computational package MATHEMATICA.

A class of numerical algorithms for computing outer inverses

The aim of this study is to present a class of numerical algorithms for finding outer inverses with prescribed range and null space. General convergence theorems are proven. Some particular cases of the general algorithm are introduced and demonstrated. Theoretical convergence speed and computational efficiency will further be supported by employing some numerical tests.