Knowledge base dedicated to Linux and applied mathematics.

Home > Mathematics > Linear Systems > **Gauss-Seidel method**

All the versions of this article: [English] [français]

We will study an iterative method for solving linear systems: the Gauss-Seidel method. The aim is to build a sequence of approximations that converges to the true solution.

The Gauss-Seidel method is an iterative method for solving linear systems such as

For this, we use a sequence which converges to the fixed point(solution) .

For given, we build a sequence such with .

where is an invertible matrix.

where is an affine function.

If is solution of then

Let be the error vector

We put , which gives

The algorithm converges if (null matrix).

**Theorem**: if and only if the spectral radius of the matrix

checks:

we remind that where represent the eigenvalues of .

**Theorem**: If A is strictly diagonally dominant,

then for all the Gauss-Seidel algorithm will converge to the solution of the system

We decompose in the following way :

withthe diagonal

the strictly lower triangular part of

the strictly upper triangular part of .

In the Gauss-Seidel method we choose and (in the Jacobi method, et ).

We obtain:

For the stop criteria , we can use the residual vector, wich gives for a given precision :