Knowledge base dedicated to Linux and applied mathematics.

Home > Mathematics > Linear Systems > **Jacobi method**

All the versions of this article: [English] [français]

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 :

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 :