Гаусс-Зейдель әдісі: Сызықтық теңдеулер жүйесін итеративті шешу

Iterative method used to solve a linear system of equations
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, and is similar to the Jacobi method. Though it can be applied to any matrix with non zero elements on the diagonals, convergence is only guaranteed if the matrix is either strictly diagonally dominant, or symmetric and positive definite. It was only mentioned in a private letter from Gauss to his student Gerling in 1823. A publication was not delivered before 1874 by Seidel.

The element wise formula for the Gauss–Seidel method is similar to that of the Jacobi method. The computation of uses the elements of that have already been computed, and only the elements of that have not been computed in the (k+1) th iteration. This means that, unlike the Jacobi method, only one storage vector is required as elements can be overwritten as they are computed, which can be advantageous for very large problems. However, unlike the Jacobi method, the computations for each element are generally much harder to implement in parallel, since they can have a very long critical path, and are thus most feasible for sparse matrices. Furthermore, the values at each iteration are dependent on the order of the original equations. Gauss Seidel is the same as successive over relaxation with .