Successive Over-Relaxation

Updated 01 Oct 2026

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What it is
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  modifies a gauss-seidel component update using a relaxation factor $\omega$
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Formula
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  $x_i^{(k+1)}=(1-\omega)x_i^{(k)}+\omega x_{i,\mathrm{GS}}^{(k+1)}$ .
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Interpretation
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  $\omega=1$ : Gauss-Seidel Method
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  $0<\omega<1$ : under-relaxation.
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  $\omega>1$ : over-relaxation.
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Convergence
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  for symmetric positive-definite systems, $0<\omega<2$ guarantees convergence.
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  a poorly chosen factor can slow convergence or cause failure.
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  formulas for an “optimal” factor require additional matrix assumptions; they are not universal.

Linked references 2