In numerical linear algebra, the method of successive over-relaxation (SOR) is a variant of the Gauss–Seidel method for solving a linear system of equations, resulting in faster convergence. A similar method can be used for any slowly converging iterative process.

2025年3月5日 · The successive overrelaxation method (SOR) is a method of solving a linear system of equations derived by extrapolating the Gauss-Seidel method. This extrapolation takes the form of a weighted average between the previous iterate and the computed Gauss-Seidel iterate successively for each component,

The Successive Over Relaxation (SOR) method improves on the convergence rate of the Gauss-Seidel method by applying a weighting factor to the updated estimates to adjust the extent of the change. Let \(\omega\) be a relaxation parameter in the range \([0,2]\) then the SOR method is

In numerical linear algebra, the method of successive over-relaxation (SOR) is a variant of the Gauss–Seidel method for solving a linear system of equations, resulting in faster convergence. It was devised simultaneously by David M. Young Jr. (1923--2008) and by Stanley P. Frankel (1919--1978) in 1950 for the purpose of automatically solving ...

2016年1月11日 · For the matrix Tj T j, he calculates the eigenvalues. We have the characteristic polynomial: det(Tj − λI) = −λ(λ2 − 0.625) λ1,2,3 = 0, ± 0.625− −−−√ (1) (1) det (T j − λ I) = − λ (λ 2 − 0.625) λ 1, 2, 3 = 0, ± 0.625. The spectral radius is given by the maximum magnitude eigenvalue: ρ(A) =max |λi| ρ (A) = max | λ i |. From (1) (1), we have:

The successive over-relaxation (SOR) method is another form of the Gauss-Seidel method in which the new estimate at iteration for the component is calculated as the weighted average of the previous estimate and the estimate using Gauss-Seidel :

The Successive Overrelaxation Method, or SOR, is devised by applying extrapolation to the Gauss-Seidel method. This extrapolation takes the form of a weighted average between the previous iterate and the computed Gauss-Seidel iterate successively for each component:

The symmetric successive over-relaxation (SSOR) iteration combines the "forward" SOR with a "reverse" SOR, much like the symmetric Gauss-Seidel does: \begin{equation*} \begin{array}{rcl} x^{(k+\frac{1}{2})} \amp = \amp M_F^{-1} ( N_F x^{(k)} + y ) \\ x^{(k+1)} \amp = \amp M_R^{-1} ( N_R x^{(k+\frac{1}{2})} + y ) . \end{array} \end{equation*}

In the SORU method, M = U + D/ω, where U is the strictly upper triangular part of A. The SOR and SORU methods use a more accurate approximation of the matrix, which leads to fewer iterations but slightly more work per iteration than in the Jacobi method.

2018年3月11日 · The successive overrelaxation method (SOR) is a method of solving a linear system of equations Ax=b derived by extrapolating the Gauss-Seidel method. More info at: http://mathworld.wolfram.com/SuccessiveOverrelaxationMethod.html