# Gauss-Seidel method

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## Gauss-Seidel method

[¦gau̇s ′zīd·əl ‚meth·əd]
(mathematics)
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The thermal boundary conditions in Equations 70 to 72 are incorporated to solve Equation 64 for the bush temperature by using Gauss-Seidel method.
It is noteworthy that the SOR method is equivalent to the well known Gauss-Seidel method [25] when [omega] = 1, which implies that the Gauss-Seidel method is a special case of the SOR method.
The RALS algorithm can be viewed as a proximal regularization of a three-block Gauss-Seidel method for minimizing f(A, B, C).
For the solution of the system of linear equations obtained from the fourth order compact scheme, if we use classical iterative methods like Jacobi and Gauss-Seidel method this will slow the convergence due to large linear system.
The Gauss-Seidel iteration method will have better convergent speed than Jacobi iteration method, but it is hard to parallelize the Gauss-Seidel method.
To improve the convergence characteristic of Gauss-Seidel method, the Newton-Raphson method was introduced.
Our considered approach is called Gauss-Seidel method which is an iterative procedure that is based on a modification of the Jacobi method.
We are solving this equation by Multigrid method with sixth-order compact finite-difference scheme using Gauss-Seidel method as a smoother.
Inspired by the advantage of Gauss-Seidel method against Jacobi method for linear equations, the dynamic differential evolution (DDE) [14] was developed.
In accordance to the present algorithms (Anderson, 1995; Glover & Sarma, 1994; Press et al., 1992; Rajcic, 1988) programs have been created in C++ programming language for Gauss-Seidel method, Fast-Decoupled method and Newton-Rapson method of Solution of Nonlinear Algebraic Equations.
*See Section 4 for more details on the origin of the graphs and on the experiments setting; let us just mention here that the conjugate gradient acceleration was used in all cases and that all software were used with default parameter, with the exception of Boomer AMG, considered with the same smoother as AGMG (that is, with a single forward Gauss-Seidel sweep as pre-smoother, and a single backward Gauss-Seidel sweep as post-smoother), and a coarsest grid solver being 5 iterations of the symmetric Gauss-Seidel method with a coarsest grid of size at most 100 (instead of 1 by default; this way we circumvent issues stemming from the singularity of the coarsest grid matrix, which boils down to the zero 1 x 1 matrix when the coarsest grid contains a single unknown).

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