Jacobi method

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(Difference between revisions)

Revision as of 20:48, 15 December 2005

We seek the solution to set of linear equations:

$A \cdot \Phi = B$

In matrix terms, the definition of the Jacobi method can be expressed as :
$\phi^{(k)} = D^{ - 1} \left( {L + U} \right)\phi^{(k - 1)} + D^{ - 1} B$
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 $\Phi^{0}$ to the solution

for k := 1 step 1 untill convergence do

for i := 1 step until n do

$\sigma = 0$

for j := 1 step until n do

if j != i then

$\sigma = \sigma + a_{ij} \phi_j^{(k-1)}$

end if

end (j-loop)

$\phi_i^{(k)} = {{\left( {b_i - \sigma } \right)} \over {a_{ii} }}$

end (i-loop)

check if convergence is reached

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 $\Phi$ are used.

Return to Numerical Methods

@@ Line 11: / Line 11: @@
 === Algorithm ===
 ----
-:    Chose an intital guess <math>X^{0}</math> to the solution <br>
+:    Chose an intital guess <math>\Phi^{0}</math> to the solution <br>
 :    for k := 1 step 1 untill convergence do <br>
 ::   for i := 1 step until n do <br>

Jacobi method

From CFD-Wiki

Revision as of 20:48, 15 December 2005

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