Jacobi method
From CFD-Wiki
(Difference between revisions)
| Line 6: | Line 6: | ||
In matrix terms, the definition of the Jacobi method can be expressed as : <br> | In matrix terms, the definition of the Jacobi method can be expressed as : <br> | ||
<math> | <math> | ||
| - | x^{(k)} = D^{ - 1} \left( {L + U} \right)x^{(k - 1)} + D^{ - 1} | + | x^{(k)} = D^{ - 1} \left( {L + U} \right)x^{(k - 1)} + D^{ - 1} Q |
</math><br> | </math><br> | ||
Where '''D''','''L''' and '''U''' represent the diagonal, lower triangular and upper triangular matrices of coefficient matrix '''A''' and k is iteration counter.<br> | Where '''D''','''L''' and '''U''' represent the diagonal, lower triangular and upper triangular matrices of coefficient matrix '''A''' and k is iteration counter.<br> | ||
Revision as of 18:04, 28 September 2005
We seek the solution to set of linear equations:
For the given matrix A and vectors X and Q.
In matrix terms, the definition of the Jacobi method can be expressed as :
Where D,L and U represent the diagonal, lower triangular and upper triangular matrices of coefficient matrix A and k is iteration counter.
Algorithm
- Chose an intital guess
to the solution
- for k := 1 step 1 untill convergence do
- for i := 1 step until n do
-
- for j := 1 step until n do
- if j != i then
-
- end if
- if j != i then
- end (j-loop)
-
-
- end (i-loop)
- check if convergence is reached
- for i := 1 step until n do
- end (k-loop)
Note: The major difference between the Gauss-Seidel method and Jacobi method lies in the fact that for Jacobi method the values of solution of previous iteration (here k) are used, where as in Gauss-Seidel method the latest available values of solution vector X are used.
