Comparison Theorems of Spectral Radius for Splittings of Matrices

Abstract

A class of the iteration method from the double splitting of coefficient matrix for solving the linear system is further investigated. By structuring a new matrix, the iteration matrix of the corresponding double splitting iteration method is presented. On the basis of convergence and comparison theorems for single splittings, we present some new convergence and comparison theorems on spectral radius for splittings of matrices.

Reuben Louis Rosenberg (1909–1986) and the Stein-Rosenberg theorem

“…They generalize the original results of Varga [123] of 1960 and those of Woźnicki's thesis [127]. Let us mention that, in 1993, Woźnicki introduced a double splitting of the form A = P − R − S and the corresponding iterative method [128], which was later studied by Cui-Xia Li and Su-Hua Li [71]. As noted by Woźnicki in [131]: from this time a renewed interest in comparison theorems, proven under progressively weaker hypotheses for different splittings, has been permanently observed in the literature.…”

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].…”

“…We call a splitting convergent if the associated iterative scheme convergent. Several types of splittings and numerous comparative studies can be found in the literature [26,29,30,32,45,51] and the reference therein.…”

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.…”