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Successive over-relaxation method - SOR

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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:

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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