Superior convergence domains for the p-cyclic SSOR majorizer

Abstract

Thus 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 J A denote the block Jacobi method and let ν = ϱ(¦J A¦) . We establish regions in the ( ν, ω)-plane where ϱ( S ω A) ⩽ ϱ( Q ω A) < ¦ω − 1¦ [⩽ ϱ( L ω A)] . Here S ω A is the block SSOR iteration operator associated with A, L ω A is the block successive overrelaxation (SOR) iteration operator associated with A, and Q ω A is a convenient similarity transformation of the majorizing operator for S ω A . 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.