Some New Comparison Theorems for Double Splittings of Matrices

Abstract

Abstract:In this paper, we further investigate the double splitting iterative methods for solving linear systems. Building on the previous work by Song and Song [Convergence for nonnegative double splittings of matrices, Calcolo, (2011)

Comparison results for proper double splittings of rectangular matrices

“…So, 0 < ρ(W 2 ( )) < 1. Then by using equations (11) and (12), as in the proof of the Theorem 3.1, we can show that ρ(W 2 ( ))A x 1 ( ) ≥ 0.…”

“…Song and Song [17] have studied convergence and comparison theorems for nonnegative double splittings of a real square nonsingular matrices. 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.…”

Regularized iterative method for ill-posed linear systems based on matrix splitting

“…In addition to these, Song and Song [44] introduced the double nonnegative splitting to discuss the iterative solution of the nonsingular system Ax = b. Further, the comparison results of [44] have been extended by the authors of [26], [27], and [30]. For convenience, we have renamed the double nonnegative splitting as the double weak splitting of type I.…”

“…Motivated by the work of the authors [26], [27], [30] and [44], we have introduced the double weak splitting of type II for symmetric matrices. In connection to double weak splitting of type II, we have extended a few results of [44].…”

Comparison Theorems of Spectral Radius for Splittings of Matrices

“…In [12], some convergence results for the double splitting of a non-Hermitian positive semidefinite matrix are established. Further, some comparison theorems for double splittings of different monotone matrices are given in [13,14] and some convergence and comparison results for nonnegative double splittings of matrices are given in [4,15]. In this paper, by structuring a new matrix, the iteration matrix of the corresponding iteration method from double splitting of coefficient matrix is presented.…”