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Successive over-relaxation method - SOR

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m (Successive over-relaxation method moved to Successive over-relaxation method - SOR)

Revision as of 20:44, 14 December 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 SOR method can be expressed as :
x^{(k)}  = \left( {D - \omega L} \right)^{ - 1} \left( {\omega U + \left( {1 - \omega } \right)D} \right)x^{(k - 1)}  + \omega \left( {D - \omega L} \right)^{ - 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.
 \omega is extrapolation factor.

The pseudocode for the SOR 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)
  \sigma  = {{\left( {q_i  - \sigma } \right)} \over {a_{ii} }}
 x_i^{(k)}  = x_i^{(k - 1)}  + \omega \left( {\sigma  - x_i^{k - 1} } \right)
end (i-loop)
check if convergence is reached
end (k-loop)

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