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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
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
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
We decompose in the following way :
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 :
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 :