Successive Over-Relaxation

Numerical Analysis · Axiom Academy

LESSON Successive Over-Relaxation (SOR) Accelerating convergence with a relaxation parameter What if we could predict where the iteration is heading and take a bigger step? That's the idea behind SOR : blend the old value with a weighted Gauss-Seidel update. If Gauss-Seidel is moving toward the solution too slowly, we can "over-correct" by taking a bigger step in the same direction. The magic is choosing the right ω. Too small → slow convergence. Too large → divergence! For some special matrices (like those from certain PDEs), we can compute the optimal ω analytically: where ρ(G J ) is the spectral radius of the Jacobi iteration matrix. ω = 1: Reduces to Gauss-Seidel 0 Under-relaxation (useful for diverging GS) 1 Over-relaxation (faster convergence) ω ≥ 2 or ω ≤ 0: Always diverges You've learned SOR—the turbocharged version of Gauss-Seidel!

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