Gauss-Seidel method
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(Difference between revisions)
(fixed dot product notation) |
(towards a uniform notation for linear systems : A*Phi = B) |
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We seek the solution to set of linear equations: <br> | We seek the solution to set of linear equations: <br> | ||
| - | :<math> A \cdot | + | :<math> A \cdot \Phi = B </math> <br> |
For the given matrix '''A''' and vectors '''X''' and '''Q'''. <br> | For the given matrix '''A''' and vectors '''X''' and '''Q'''. <br> | ||
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> | ||
| - | + | \phi^{(k)} = \left( {D - L} \right)^{ - 1} \left( {U\phi^{(k - 1)} + b} \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> | ||
| Line 19: | Line 19: | ||
::: <math> \sigma = 0 </math> <br> | ::: <math> \sigma = 0 </math> <br> | ||
::: for j := 1 step until i-1 do <br> | ::: for j := 1 step until i-1 do <br> | ||
| - | :::: <math> \sigma = \sigma + a_{ij} | + | :::: <math> \sigma = \sigma + a_{ij} \phi_j^{(k)} </math> |
::: end (j-loop) <br> | ::: end (j-loop) <br> | ||
::: for j := i+1 step until n do <br> | ::: for j := i+1 step until n do <br> | ||
| - | :::: <math> \sigma = \sigma + a_{ij} | + | :::: <math> \sigma = \sigma + a_{ij} \phi_j^{(k-1)} </math> |
::: end (j-loop) <br> | ::: end (j-loop) <br> | ||
| - | ::: <math> | + | ::: <math> \phi_i^{(k)} = {{\left( {b_i - \sigma } \right)} \over {a_{ii} }} </math> |
:: end (i-loop) | :: end (i-loop) | ||
:: check if convergence is reached | :: check if convergence is reached | ||
Revision as of 20:46, 15 December 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 Gauss-Seidel 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.
The pseudocode for the Gauss-Seidel algorithm:
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 i-1 do
-
- end (j-loop)
- for j := i+1 step until n do
-
- end (j-loop)
-
-
- end (i-loop)
- check if convergence is reached
- for i := 1 step until n do
- end (k-loop)
Return to Numerical Methods
