In this paper we derive a determinantal formula of {1, 2} generalized inverses, for matrices over an integral domain and over a commutative ring. The corresponding results are derived for the set of matrices which have rank factorizations as well as for the matrices which do not have rank factorizations. The determinantal formula of {1, 2} inverses for matrices which do not have rank factorizations, is derived using the characterizations of the class of reflexive g-inverses from [10] and [19]. For the set of matrices which have rank factorizations, the determinantal formula of {1, 2} inverses is derived using a general representation of {1, 2} inverses and the general determinantal representation from [20]. Also, we examine the existence of this determinantal formula. Representations and conditions for the existence of {1, 2, 3} and {1, 2, 4} inverses are introduced for the set of matrices which allow a rank factorization. Determinantal representations of the Moore-Penrose inverse, the weighted Moore-Penrose inverse and the group inverse are derived for arbitrary matrices. Moreover, we investigate representations of the minors from A (1,2) , A † , A † M,N and A (1,2) by means of the expressions involving minors of A and the corresponding minors of randomly chosen matrices which satisfy specified conditions. If A allows a full-rank factorization, we obtain additional results for {1, 2, 3} and {1, 2, 4} inverses of A. Also, a determinantal representation of the corresponding solutions of a given linear system is investigated.
In this paper we extend the notion of the proper splitting of rectangular matrices introduced and investigated in (Berman, A. and Neumann, M., SIAM
Determining the step length in iterations of nonlinear minimization represents a problem that is not uniquely defined. Motivated by such uncertainty in defining step length, our intention was to use the capabilities of neutrosophy in this process. Our idea is to unify the usability and numerous applications of neutrosophic logic and the enormous importance of nonlinear optimization. An improvement of line search iteration for solving unconstrained optimization is proposed using appropriately defined Neutrosophic logic system in determining appropriate step size for the class of descent direction methods. The basic idea is to use an additional parameter that would monitor the behavior of the objective function and, based on that, correct the step length in known optimization methods. Mutual comparison and analysis of generated numerical results reveal better results generated by the suggested iterations compared to analogous available iterations considering the statistical ranking technique. Statistical measures show advantages of fuzzy improvements of considered line search optimization methods.
The present study is devoted to methods for the numerical solution to the system of equations AXB=D. In the case certain conditions are met, the classical gradient neural network (GNN) dynamics obtains fast convergence. However, if those conditions are not satisfied, solution to the equation does not exist and therefore the error function E(t):=AV(t)B-D cannot be equal to zero, which increases the CPU time required for the calculation. In this paper, the solution to the matrix equation AXB = D is studied using the novel Gradient Neural Network (GGNN) model, termed as GGNN(A,B,D). The GGNN model is developed using a gradient of the error matrix used in the development of the GNN model. The proposed method uses a novel objective function that is guaranteed to converge to zero, thus reducing the execution time of the Simulink implementation. The GGNN-based dynamical systems for computing generalized inverses are also discussed. The conducted computational experiments have shown the applicability and advantage of the developed method.
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