Superior convergence domains for the p-cyclic SSOR majorizer

AbstractThus far for an n × n complex nonsingular matrix A, the symmetric successive overrelaxation (SSOR) majorizing operator has been used to establish convergence properties of the SSOR method mostly in the case where A is an H-matrix. In this paper we use (actually a similarity transformation of) the SSOR majorizer to investigate convergence properties of the block SSOR method when A is a block p-cyclic matrix. Let JA denote the block Jacobi method and let ν = ϱ(¦JA¦). We establish regions in the (ν, ω)-plane where ϱ(SωA) ⩽ ϱ(QωA) < ¦ω − 1¦ [⩽ ϱ(LωA)]. Here SωA is the block SSOR iteration operator associated with A, LωA is the block successive overrelaxation (SOR) iteration operator associated with A, and QωA is a convenient similarity transformation of the majorizing operator for SωA. Of special interest to us are the values of ν for which the above inequality holds for the corresponding values of the relaxation parameter ω(A) = 2(1 + ν), the latter being an important quantity in the SOR-SSOR theory for H-matrices