A uniform error bound for the overrelaxation methods Original Research Article

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 colon, equals varrho(B) < 1]. Consider using the overrelaxation methods (SOR, AOR, MSOR, SSOR, or USSOR), xn + 1 = Tωxn + cω for n greater-or-equal, slanted 0, to solve the system. We derive a uniform error bound for the overrelaxation methods, imageimageimage where short parallel · short parallel = short parallel · short parallel2, δn = xn − xn − 1, and s(μ2) and t(μ2) colon, equals 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.