Jacobi method

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(Difference between revisions)

Revision as of 05:15, 15 September 2005

We seek the solution to set of linear equations:

$A \bullet X = Q$

For the given matrix A and vectors X and Q.
In matrix terms, the definition of the Jacobi method can be expressed as :
$x^{(k)} = D^{ - 1} \left( {L + U} \right)x^{(k - 1)} + D^{ - 1} q$
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 $X^{0}$ 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

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

end if

end (j-loop)

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

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 X are used.

@@ Line 26: / Line 26: @@
 :    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. <br>

Jacobi method

From CFD-Wiki

Revision as of 05:15, 15 September 2005

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