CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Wiki > Gauss-Seidel method

Gauss-Seidel method

From CFD-Wiki

(Difference between revisions)
Jump to: navigation, search
Line 6: Line 6:
In matrix terms, the definition of the Gauss-Seidel method can be expressed as : <br>
In matrix terms, the definition of the Gauss-Seidel method can be expressed as : <br>
<math>  
<math>  
-
x^k  = \left( {D - L} \right)^{ - 1} \left( {Ux^{k - 1}  + Q} \right)
+
x^{(k)} = \left( {D - L} \right)^{ - 1} \left( {Ux^{(k - 1)}  + Q} \right)
</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 05:07, 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 Gauss-Seidel method can be expressed as :
x^{(k)} = \left( {D - L} \right)^{ - 1} \left( {Ux^{(k - 1)} + Q} \right)
Where D,L and U represent the diagonal, lower triangular and upper triangular matrices of coefficient matrix A and k is iteration counter.

The pseudocode for the Gauss-Seidel algorithm:

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 i-1 do
\sigma = \sigma + a_{ij} x_j^{(k)}
end (j-loop)
for j := i+1 step until n do
\sigma = \sigma + a_{ij} x_j^{(k-1)}
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)
Retrieved from "https://ns.cfd-online.com/Wiki/Gauss-Seidel_method"
My wiki