A note on double splittings of different monotone matrices

Comparison Theorems for Single and Double Splittings of Matrices

“…By investigating Theorem 6, it is easy to see that the conditioners, Theorem 6 are weaker than those of Theorem 4 [12]. That is, the result of Theorem 6 holds without −1 1 ≥ 0 and −1 2 ≥ 0.…”

“…Theorem 4 (see [12]). Let 1 and 2 be two nonsingular matrices with −1 1 ≥ 0 and −1 2 ≥ 0, 1 = 1 − 1 − 1 , and…”

“…Our basic purpose here is to derive some new comparison theorems for the spectral radius of double splittings of different matrices. Under suitable conditions, new comparison theorems are superior to the corresponding results in the recent paper [12]. Some comparison theorems between the spectral radius of single and double splittings of matrices are also established and are applied to the Jacobi and Gauss-Seidel double SOR method.…”

Some New Comparison Theorems for Double Splittings of Matrices

“…By investigating Corollary 2, it is easy to see that the conditioners of Corollary 2 are weaker than that of Theorem 3.1 [11]. That is, the result of Corollary 2 holds without A …”

“…In [12], some convergence results for double splittings of a non-Hermitian positive semidefinite matrix are established. In [11], a comparison theorem for double splittings of different monotone matrices is given. In [13], some convergence and comparison results for nonnegative double splittings of matrices are given.…”