Gauss-Seidel method

(Difference between revisions)
 Revision as of 20:33, 15 December 2005 (view source)Tsaad (Talk | contribs) (fixed dot product notation)← Older edit Revision as of 20:46, 15 December 2005 (view source)Tsaad (Talk | contribs) (towards a uniform notation for linear systems : A*Phi = B)Newer edit → Line 1: Line 1: We seek the solution to set of linear equations:
We seek the solution to set of linear equations:
- :$A \cdot X = Q$
+ :$A \cdot \Phi = B$
For the given matrix '''A''' and vectors '''X''' and '''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 :
In matrix terms, the definition of the Gauss-Seidel method can be expressed as :
$[itex] - x^{(k)} = \left( {D - L} \right)^{ - 1} \left( {Ux^{(k - 1)} + q} \right) + \phi^{(k)} = \left( {D - L} \right)^{ - 1} \left( {U\phi^{(k - 1)} + b} \right)$
[/itex]
Where '''D''','''L''' and '''U''' represent the diagonal, lower triangular and upper triangular matrices of coefficient matrix '''A''' and k is iteration counter.
Where '''D''','''L''' and '''U''' represent the diagonal, lower triangular and upper triangular matrices of coefficient matrix '''A''' and k is iteration counter.
Line 19: Line 19: :::  $\sigma = 0$
:::  $\sigma = 0$
:::  for j := 1 step until i-1 do
:::  for j := 1 step until i-1 do
- ::::      $\sigma = \sigma + a_{ij} x_j^{(k)}$ + ::::      $\sigma = \sigma + a_{ij} \phi_j^{(k)}$ :::    end (j-loop)
:::    end (j-loop)
:::  for j := i+1 step until n do
:::  for j := i+1 step until n do
- ::::      $\sigma = \sigma + a_{ij} x_j^{(k-1)}$ + ::::      $\sigma = \sigma + a_{ij} \phi_j^{(k-1)}$ :::    end (j-loop)
:::    end (j-loop)
- :::    $x_i^{(k)} = {{\left( {q_i - \sigma } \right)} \over {a_{ii} }}$ + :::    $\phi_i^{(k)} = {{\left( {b_i - \sigma } \right)} \over {a_{ii} }}$ ::  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: $A \cdot \Phi = B$

For the given matrix A and vectors X and Q.
In matrix terms, the definition of the Gauss-Seidel method can be expressed as : $\phi^{(k)} = \left( {D - L} \right)^{ - 1} \left( {U\phi^{(k - 1)} + b} \right)$
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 $\sigma = \sigma + a_{ij} \phi_j^{(k)}$
end (j-loop)
for j := i+1 step until n do $\sigma = \sigma + a_{ij} \phi_j^{(k-1)}$
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)