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The convergence properties of the Gauss–Seidel method are dependent on the matrix . Namely, the procedure is known to converge if either: Namely, the procedure is known to converge if either: A {\displaystyle \mathbf {A} } is symmetric positive-definite , [ 6 ] or
If Scarborough criterion is not satisfied then Gauss–Seidel method iterative procedure is not guaranteed to converge a solution. This criterion is a sufficient condition, [3] not a necessary one. If this criterion is satisfied then it means equation will be converged by at least one iterative method. The Scarborough criterion is used as a ...
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 .
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 Jacobi method is a simple relaxation method. 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
An early iterative method for solving a linear system appeared in a letter of Gauss to a student of his. He proposed solving a 4-by-4 system of equations by repeatedly solving the component in which the residual was the largest [ citation needed ] .
In most cases, preconditioning is necessary to ensure fast convergence of the conjugate gradient method. If M − 1 {\displaystyle \mathbf {M} ^{-1}} is symmetric positive-definite and M − 1 A {\displaystyle \mathbf {M} ^{-1}\mathbf {A} } has a better condition number than A {\displaystyle \mathbf {A} } , a preconditioned conjugate gradient ...
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.