Comparison Theorems for Single and Double Splittings of Matrices

Abstract

Some comparison theorems for the spectral radius of double splittings of different matrices under suitable conditions are presented, which are superior to the corresponding results in the recent paper by Miao and Zheng (2009). Some comparison theorems between the spectral radius of single and double splittings of matrices are established and are applied to the Jacobi and Gauss-Seidel double SOR method.

Some New Comparison Theorems for Double Splittings of Matrices

“…In fact, Corollaries 2 and 3 are mainly results in [16], which implies that Theorems 5 and 6 extend the results of Corollaries 2 and 3 in [16] …”

Comparison Theorems of Spectral Radius for Splittings of Matrices

“…In [4], some comparison theorems for the double splitting (4) through investigating the matrix splitting defined by (14) were obtained, which were described as follows.…”

“…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.…”

Alternating Stationary Iterative Methods Based on Double Splittings

“…Such type of splitting leads to the iterative scheme x k+1 = P −1 Rx k − P −1 Sx k−1 + P −1 b, k > 0 for solving the non-singular linear system (1.1), when n = m. Shen and Huang [36] and Miao et al [26] studied the convergence and comparison of the above iterative scheme for monotone matrices (A ∈ R n×n is monotone [13] if and only if A −1 exists and A −1 ≥ 0). Moreover, several convergence and its comparison results exist in the literature for different types of double splittings (see [19], [20], [21], [25], [36], [37], [39], [41], [45]). In 2019, Li et al [23] proposed an alternating scheme using double splittings of a matrix to find an approximate solution of a real non-singular linear system of equations.…”