On the exact p-cyclic SSOR convergence domains

Abstract

Suppose that A ∈ C n, n is a block p-cyclic consistently ordered matrix, and let B and S ω denote, respectively, the block Jacobi and the block symmetric successive overrelaxation (SSOR) iteration matrices associated with A. Neumaier and Varga found [in the ( ϱ(| B|), ω) plane] the exact convergence and divergence domains of the SSOR method for the class of H-matrices. Hadjidimos and Neumann applied Rouché's theorem to the functional equation connecting the eigenvalue spectra σ( B) and σ( S ω ) obtained by Varga, Niethammer, and Cai, and derived in the ( ϱ( B), ω) plane the convergence domains for the SSOR method associated with p-cyclic consistently ordered matrices, for any p ⩾ 3. In the present work it is further assumed that the eigenvalues of B p are real of the same sign. Under this assumption the exact convergence domains in the ( ϱ( B), ω) plane are derived in both the nonnegative and the nonpositive cases for any p ⩾ 3.