Neighborhoods of Dominant Convergence for the SSOR Method
Michael Neumann · SIAM Journal on Algebraic and Discrete Methods · 1986
Let A be an $n \times n$ nonsingular irreducible 3-cyclic H-matrix and let $J^A$, $L_\omega ^A$ and $S_\omega ^A $ denote, respectively, the Jacobi, the SOR, and the SSOR iteration matrices associated with A. In this paper we show that if the spectral radius $\rho ( | J^A | ) \in ( 0,r_0 )$, where $r_0$ is the unique root of the cubic $17r^3 + r^2 - r - 1$ in the interval (0, 1), then there exists a neighborhood $\Omega _{\omega ( A )} $ of $\omega ( A ) : = 2 /( 1 + \rho ( | J^A | )$ such that \[ \rho ( S_\omega ^A ) < | \omega - 1 | \leqq \rho ( L_\omega ^A )\quad \forall \omega \in \Omega _{\omega ( A )} . \]