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In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a system of linear equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel .
The Gauss–Seidel method is an improvement upon the Jacobi method. Successive over-relaxation can be applied to either of the Jacobi and Gauss–Seidel methods to speed convergence. Multigrid methods
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 .
function phi = V_Cycle (phi,f,h) % Recursive V-Cycle Multigrid for solving the Poisson equation (\nabla^2 phi = f) on a uniform grid of spacing h % Pre-Smoothing phi = smoothing (phi, f, h); % Compute Residual Errors r = residual (phi, f, h); % Restriction rhs = restriction (r); eps = zeros (size (rhs)); % stop recursion at smallest grid size, otherwise continue recursion if smallest_grid_size ...
Gauss–Seidel method. Successive over-relaxation (SOR) — a technique to accelerate the Gauss–Seidel method Symmetric successive over-relaxation (SSOR) — variant of SOR for symmetric matrices; Backfitting algorithm — iterative procedure used to fit a generalized additive model, often equivalent to Gauss–Seidel; Modified Richardson ...
The Jacobi and Gauss–Seidel methods for solving a linear system converge if the matrix is strictly (or irreducibly) diagonally dominant. Many matrices that arise in finite element methods are diagonally dominant.
The Stein-Rosenberg theorem, proved in 1948, states that under certain premises, the Jacobi method and the Gauss-Seidel method are either both convergent, or both divergent. If they are convergent, then the Gauss-Seidel is asymptotically faster than the Jacobi method.
The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems.