On the spectrum of the SOR operator for symmetric positive definite matrices

AbstractOne attractive feature of the classical successive overrelaxation (SOR) iterative method is its algorithmic simplicity. Whereas for linear systems with symmetric positive definite matrices A, the theorem of Ostrowski guarantees convergence as long as the relaxation parameter ω∈(0,2), results on the optimal choice of ω can only be obtained under certain additional assumtions on A. Here, for positive definite systems with coefficient matrices A=I−L−LT in which L is not necessarily strictly lower triangular, we study the behavior of the spectrum of the SOR operator Lω as ω→0 and ω→2 and describe enclosure sets which may be used to estimate the spectral radius of Lω for 0<ω