A Uniform Error Bound for the Overrelaxation Methods
Linear Algebra and its Applications
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., μ1 ≔ ϱ(B) < 1]. Consider using the overrelaxation methods (SOR, AOR, MSOR, SSOR, or USSOR), xn + 1 = Tωxn + cω for n ⩾ 0, to solve the system. We derive a uniform error bound for the overrelaxation methods, ∥x−xn∥2⩽1[1+s(μ12) + t(μ12)]2 x(t0+ |t1|μ12)2∥δn∥2− 2t0〈δn,δn+1〉 +|t1|μ12∥δn∥∥δn+1∥+∥δn+1∥2
where ∥ · ∥ = ∥ · ∥2, δn = xn − xn − 1, and s(μ2) and t(μ2) ≔ t0 + t1μ2 are two coefficients of the corresponding functional equation connecting the eigenvalues λ of Tω 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.
"A Uniform Error Bound for the Overrelaxation Methods."
Linear Algebra and its Applications, 254 (1-3): 315-333.
doi: 10.1016/S0024-3795(96)00313-8 source: https://www.sciencedirect.com/science/article/pii/S0024379596003138?via%3Dihub