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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.
Jacobi method is an iterative method for solving linear systems such as
where is an invertible matrix.
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
Theorem: If A is strictly diagonally dominant,
We decompose in the following way :
In the Jacobi’s method, we choose and (in the Gauss-Seidel Method, and ).
The -th line of is :
We obtain :
Let be the residual vector. We can write with calculated
For the stop criteria , we can use the residual vector, wich gives for a given precision :