# Successive over-relaxation method - SOR

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We seek the solution to set of linear equations: $A \cdot \Phi = B$

In matrix terms, the definition of the SOR method can be expressed as : $\phi^{(k)} = \left( {D - \omega L} \right)^{ - 1} \left( {\omega U + \left( {1 - \omega } \right)D} \right)\phi^{(k - 1)} + \omega \left( {D - \omega L} \right)^{ - 1} B$
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 a relaxation factor.

The pseudocode for the SOR algorithm:

### Algorithm

Chose an intital guess $\Phi^{0}$ 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) $\sigma = {{\left( {b_i - \sigma } \right)} \over {a_{ii} }}$ $\phi_i^{(k)} = \phi_i^{(k - 1)} + \omega \left( {\sigma - \phi_i^{k - 1} } \right)$
end (i-loop)
check if convergence is reached
end (k-loop)

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