Comparison theorems for splittings of matrices
Let \(A\in \mathbb{C}^{n\times n}\) and \(x,b\in \mathbb{C}^n\) with \(x\) unknown. In order to solve a linear system of the form \(Ax=b\) with iterative methods, the author splits the matrix \(A\) into \(A=M-N\) (or \(A=M_{k}-N_{k}\), \(k=1,2,\dots ,m\) ``parallel multisplitting method'') with \(M\) nonsingular (\(M_{k}\) nonsingular) and then uses iterative formulas for solving the system. The transition matrices used in these iterative formula are functions of the matrices \(M,N\) (or \(M_{k},N_{k}\)). This paper suggests two different nonnegative splitting methods (nonnegative parallel multisplitting methods) and then, based on the nonnegative matrix theory, compares the asymptotic rates of convergence of its iteration matrices. The above comparisons are also applied for the special case where \(A\) is a Hermitian positive (semi)definite matrix.
scientific article; zbMATH DE number 1837367