Title

A Uniform Error Bound for the Overrelaxation Methods

Document Type

Article

Publication Date

3-15-1997

Publication Title

Linear Algebra and its Applications

DOI

10.1016/S0024-3795(96)00313-8

ISSN

0024-3795

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., 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.

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