Convergence and comparison theorems for single and double decompositions of rectangular matrices

Properweak regular splitting and its application to convergence of alternating iterations

“…below characterize semi-monotone matrices in terms of these class of splittings. Noted next result is proved in [9] which contains equivalent convergence condition for iteration scheme (1.2).…”

“…The authors of [3] also obtained several convergence criteria for different subclasses of proper splitting. Recently, Jena et al [9] revisited the same theory.…”

Comparison results for proper double splittings of rectangular matrices

“…Recently, Jena et al [10] extended the notion of regular and weak regular splittings to rectangular matrices and the respective definitions are given next. A decomposition A = U − V of A ∈ R m×n is called a proper regular splitting if it is proper splitting such that U † ≥ 0 and V ≥ 0.…”

“…A matrix A ∈ R m×n is called semi-monotone if A † ≥ 0. The authors of [10] have considered proper regular splitting of semi monotone matrix A and obtained some comparison results. Now, we turn our focus on to the comparison results for double splittings that are available in the literature.…”

“…Li and Wu [11] have obtained some comparison theorems for double splittings of a matrix. Jena et al [10] and Mishra [13] have introduced the notions of double proper regular splittings and double proper weak regular splittings and derived some comparison theorems. Recently, Alekha kumar and Mishra [1] have considered proper nonnegative double splittings of nonnegative matrix and derived certain comparison theorems.…”