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## Gauss-Seidel method

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.

### Iterative method

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.

### Algorithm

If is solution of then

### Error

Let be the error vector

We put , which gives

### Convergence

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

### Gauss-Seidel Method

We decompose in the following way :

with
the 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:

### Stop criteria

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