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Successive Over-Relaxation

Accelerates Gauss-Seidel iteration by over-relaxing each variable update.

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Quick Facts

Compact metadata for implementation and discovery.

Category: Numerical Methods
Difficulty: Intermediate
Family: Iterative Linear Solver
Tier: Core
Status: Verified
Review: Source Checked
Sources: 1
Formulas: 1
Relationships: 3
Code example: Ready
Visualization: Ready
Aliases: SOR

Overview

Successive Over-Relaxation blends the previous value with a scaled Gauss-Seidel update, often converging faster when the relaxation factor is chosen well.

Core Concepts

relaxation factorgauss-seidel updateiterative solverresidual reduction

Quick facts

Complexity

Runtime and memory behavior at a glance.

Best

O(n^2)

Average

O(iterations * n^2)

Worst

O(iterations * n^2)

Space

O(n)

When to Use

Use when solving equations needs relaxation factor.
Use when the input matches the assumptions behind iterative Linear Solver.

When Not to Use

Avoid when the problem is ill-conditioned and no scaling or stability checks are available.
Avoid when a simpler algorithm gives the required result with less setup.

What It Does

Accelerates Gauss-Seidel iteration by over-relaxing each variable update
Uses relaxation factor and gauss-seidel update to guide the work.
Produces approximate numeric result.

Use Cases

Solving equations
Approximating integrals
Working with matrices and differential equations

Inputs

Numeric function, matrix, or sampled data
Initial estimate or discretization
Tolerance

Outputs

Approximate numeric result
Error estimate or residual

Key Formulas (1)

SOR update

x_i = (1 − ω) · x_i + ω · x_{i gs}

x_i = (1 - omega) * x_i + omega * x_i_gs

Over-relaxes a Gauss-Seidel update by a chosen factor.

Pseudocode

1.Prepare the inputs and state needed for Successive Over-Relaxation.
2.Apply the relaxation factor step while the stopping condition has not been met.
3.Update intermediate state and discard invalid candidates.
4.Return approximate numeric result.

Canonical Example

Input: Successive Over-Relaxation input with a small representative case
Output: Approximate numeric result
Explanation: Successive Over-Relaxation applies relaxation factor to produce approximate numeric result.

Implementation Notes

Check convergence with residuals or estimated error, not iteration count alone.
Keep the relaxation factor step separate enough to test independently.

Common Pitfalls

Ignoring conditioning and residuals can produce plausible-looking but wrong numeric answers.
Forgetting to check edge cases can make Successive Over-Relaxation look correct on only ideal inputs.