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## Jacobi method

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

### Iterative method

Jacobi 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 Jacobi algorithm will converge to the solution of the system

### Jacobi Method

We decompose in the following way :

with
the diagonal
the strictly lower triangular part of
the strictly upper triangular part of .

In the Jacobi’s method, we choose and (in the Gauss-Seidel Method, and ).

The -th line of is :

We obtain :

### Residual vector

Let be the residual vector. We can write with calculated
like follows

### Stop criteria

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