Nonstationary multisplittings with general weighting matrices for non-Hermitian positive definite systems (Q1431840)

The author presents nonstationary multisplitting and two-stage multisplitting methods to solve a linear system \(Ax= b\) for matrices satisfying that there is a positive number \(r\) with \(0\leq r\leq 1\) such that \[ rx^H H(A)x- | x^HS(A)x|\geq 0, \quad\forall x\in{\mathcal C}^n, \] where \(H(A)= (A+ A^H)/2\) and \(S(A)= (A- A^H)/2\). Let \(A =M -N\) be a splitting for which \(M- (1+ r)H(A)/2\) is Hermitian positive, then the splitting is convergent. Hence many selections can be chosen for \(M\) and the splitting is convergent. Basing on this main result, the author constructs convergent multisplitting and two-stage multisplitting methods with weight matrices.