Nonnegative splittings for rectangular matrices

Three-step alternating iterations for index 1 and non-singular matrices

“…A method of construction of proper splitting can be found in [27] while its uniqueness is shown very recently in [28]. The result produced below is a combination of Theorem 5.2, [25] and Theorem 4.1, [27], and is also a special case of Theorem 3.2 & 3.3, [14] and Theorem 3.1, [33] when index 1 matrices are considered.…”

Index-proper nonnegative splittings of matrices

“…We now introduce another splitting which covers a larger class of matrices in comparison to the above two splittings. When k = 1, the above definition coincides with the definition of proper Gnonnegative splitting (Definition 5.1, [10]). So index-proper nonnegative splitting is an extension of proper G-nonnegative splitting.…”

Reverse proper splittings of rectangular matrices

“…The authors of [14] and [17] studied the convergence and comparison results for proper regular splittings, proper weak regular splittings and proper nonnegative splittings. Unlike the above splittings, we now introduce a few new splittings which are subclasses of reverse proper splittings.…”

“…A splitting A = U − V of A ∈ R m×n is called a proper weak regular splitting if it is a proper splitting such that U † ≥ 0 and U † V ≥ 0. Thereafter, Mishra [17] introduced another splitting and is recalled next. A splitting A = U − V of A ∈ R m×n is called a proper nonnegative splitting if it is a proper splitting such that U † V ≥ 0.…”