A uniform error bound for the overrelaxation methods

Abstract

Let Ax = b be a system of linear equations where A is symmetric and positive definite. Suppose that the associated block Jacobi matrix B is consistently ordered, weekly cyclic of index 2, and convergent [i.e., μ 1 ≔ ϱ( B) < 1]. Consider using the overrelaxation methods (SOR, AOR, MSOR, SSOR, or USSOR), x n + 1 = T ω x n + c ω for n ⩾ 0, to solve the system. We derive a uniform error bound for the overrelaxation methods, ∥x−x n∥ 2⩽ 1 [1+s(μ 1 2) + t(μ 1 2)] 2 x (t 0 + |t 1|μ 1 2) 2∥δ n∥ 2 − 2t 0〈δ n,δ n+1〉 +|t 1|μ 1 2∥δ n∥∥δ n+1∥+∥δ n+1∥ 2 where ∥ · ∥ = ∥ · ∥ 2, δ n = x n − x n − 1 , and s( μ 2) and t( μ 2) ≔ t 0 + t 1 μ 2 are two coefficients of the corresponding functional equation connecting the eigenvalues λ of T ω to the eigenvalues μ of B. As special cases of the uniform error bound, we will give two error bounds for the SSOR and USSOR methods.