Debasisha Mishra

Nonnegative splittings for rectangular matrices

The extension of the nonnegative splitting for rectangular matrices called proper nonnegative splitting is proposed first. Different convergence and comparison theorems for the proper nonnegative splittings are established. The notion of double nonnegative splitting is then generalized to rectangular matrices. Finally, different convergence and comparison results are presented for this decomposition. The case for singular square matrices is also studied.

Further results on generalized inverses of tensors via the Einstein product

Abstract The notion of the Moore–Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate on this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore–Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory.

Additional Results on Convergence of Alternating Iterations Involving Rectangular Matrices

ABSTRACT Different classes of matrix splittings of semi-monotone matrices lead to many comparison results, which are useful tools in examining the convergence rate of iterative methods for solving rectangular systems of linear equations in a faster way. In this context, the theory of alternating iterations for rectangular matrices was introduced recently, in order to arrive at the desired solution of accuracy or at the exact solution. In this article, we expand convergence theory of such alternating iterations and obtain comparison results for such iterations.

Further study of alternating iterations for rectangular matrices

Abstract Theory of matrix splittings is a useful tool in the analysis of iterative methods for solving systems of linear equations. When two splittings are given, it is of interest to compare the spectral radii of the corresponding iteration matrices. This helps to arrive at the conclusion that which splitting should one choose so that one can reach the desired solution of accuracy or the exact solution in a faster way. In the case of many splittings are provided, the comparison of the spectral radii is time-consuming. Such a situation can be overcome by introducing another iteration scheme which converges to the same solution of interest in a much faster way. In this direction, the theory of alternating iterations for real rectangular matrices is recently proposed. In this note, some more results to the theory of alternating iterations are added. A comparison result of two different alternating iteration schemes is then presented which will help us to choose the iteration scheme that will guarantee the faster convergence of the alternating iteration scheme. In addition to these results, a comparison result for proper weak regular splittings is also obtained.

Reverse-order law for the Moore–Penrose inverses of tensors

ABSTRACT Reverse-order law for the Moore–Penrose inverses of tensors is useful in the field of multilinear algebra. In this paper, we first prove some more identities involving the Moore–Penrose inverses of tensors. We then obtain a few necessary and sufficient conditions of the reverse-order law for the Moore–Penrose inverses of tensors via the Einstein product.

Two-stage iterations based on composite splittings for rectangular linear systems

Abstract In this paper, we introduce a two-stage method to solve rectangular linear systems that exhibits faster convergence than typical stationary iterative methods. Under suitable conditions, we prove convergence of the new method. The number of outer iterations can be reduced by using a few significant number of inner iterations for efficient computations. Further, we perform a comparison analysis, and establish that a higher number of inner iterations ensures a smaller spectral radius of the global iteration matrix. We also discuss the uniqueness of a proper splitting, and illustrate different comparison theorems for different subclasses of proper splittings.

Comparison results for proper multisplittings of rectangular matrices

The least square solution of minimum norm of a rectangular linear system of equations can be found out iteratively by using matrix splittings. However, the convergence of such an iteration scheme arising out of a matrix splitting is practically very slow in many cases. Thus, works on improving the speed of the iteration scheme have attracted great interest. In this direction, comparison of the rate of convergence of the iteration schemes produced by two matrix splittings is very useful. But, in the case of matrices having many matrix splittings, this process is time-consuming. The main goal of the current article is to provide a solution to the above issue by using proper multisplittings. To this end, we propose a few comparison theorems for proper weak regular splittings and proper nonnegative splittings first. We then derive convergence and comparison theorems for proper multisplittings with the help of the theory of proper weak regular splittings.