Index-range monotonicity and index-proper splittings of matrices

Index-proper nonnegative splittings of matrices

“…For an index-proper splitting, the spectral radius of U D V is strictly less than 1 if and only if the above schemes converge to A D b and A D , respectively to the system Ax = b. More on index-proper splitting can be found in the recent articles [6,7]. The aim of this paper is to study the theory of nonnegative splitting 1 for square singular matrices using Drazin inverse.…”

“…When two splittings of A are given, it is of interest to compare the spectral radii of the corresponding iteration matrices. The comparison of asymptotic rates of convergence of the iteration matrices induced by two index-proper splittings of a given 104 CHINMAY KUMAR GIRI matrix, has been studied in the recent articles by Jena and Mishra, [6] and Jena and Pani, [7]. Jena and Mishra,[6] also obtained many results for nonnegativity of the Drazin inverse using different matrix splittings.…”

“…Here, only a few articles on the Drazin inverse are mentioned, but there is a vast amount of literature on it. (See the references [2,3,6,7,14] and the references cited therein. )…”

A note on index-proper multisplittings of matrices

“…For an index-proper splitting, the spectral radius of U D V is strictly less than 1 if and only if the above schemes converge to A D b and A D , respectively to the system Ax = b (see [9]). More on index-proper splittings can be found in the recent articles [9,11] and a simple method for constructing proper splittings can be found in [5]. Let us revisit some earlier results on index-proper splittings.…”

“…Here only a few articles on the Drazin inverse are mentioned, but there is a vast amount of literature on it. (See the references [9,11,17] and the references cited therein. )…”