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## Jacobi method

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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.

{{{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->article36], 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 :