Gauss-Seidel method

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 linear system of 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 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 Gauss–Seidel method is an iterative technique for solving a square system of n linear equations with unknown x:

It is defined by the iteration

where $\mathbf {x} ^{(k)}$ $\mathbf {x} ^{(k)}$ is the kth approximation or iteration of $\mathbf {x} ,\,\mathbf {x} ^{(k+1)}$ $\mathbf {x} ,\,\mathbf {x} ^{(k+1)}$ is the next or k + 1 iteration of $\mathbf {x}$ $\mathbf {x}$ , and the matrix A is decomposed into a lower triangular component $L_{*}$ $L_{*}$ , and a strictly upper triangular component U: $A=L_{*}+U$ $A=L_{*}+U$ .

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