Mathematical Sciences: Faculty Presentations (1991-2022)

A Uniform Error Bound for Overrelaxation Methods

Document Type

Presentation

Presentation Date

8-16-1995

Copyright

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Abstract or Description

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.

Sponsorship/Conference/Institution

International Linear Algebra Society Annual Conference (ILAS)

Location

Atlanta, GA

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