SOR In-Place Forward Substitution Engine
Updates x_i^{(k+1)} = (1 − ω)x_i^{(k)} + (ω/a_{ii})(b_i − ∑_{j<i} a_{ij}x_j^{(k+1)} − ∑_{j>i} a_{ij}x_j^{(k)}).
Mathematics / Gauss-Seidel & SOR
FREE TO EXPLORELABORATORY 15 · MATHEMATICS
Successive Over-Relaxation (SOR) and Optimal Parameter Tuning
Accelerate stationary linear solvers using the Successive Over-Relaxation (SOR) method with relaxation factor ω, evaluate the Ostrowski-Reich theorem on symmetric positive-definite matrices, and extract the optimal relaxation parameter ω_opt = 1.268 that minimizes the spectral radius via least-squares curvature fitting. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
SOR ERROR NORM ‖E_K‖—
COMPUTATION STATUS
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Mathematical operations are computed deterministically using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations.
Numerical assumptions and limits
Optimal parameter formula ω_opt = 2/[1+√(1−ρ²)] requires matrix Property A and consistent ordering; general matrices require empirical tuning. The numerical simulation model evaluates a 2-cyclic tridiagonal system with Jacobi spectral radius ρ(T_J) = 0.500000. Analytical optimal relaxation parameter by Young's formula: ω_opt = 2 / [1 + √(1 − 0.5²)] = 2 / [1 + √0.75] = 2 / [1 + 0.8660254] = 1.26794919... Across the six standardized iteration steps k = [1, 2, 3, 4, 5, 6], the cumulative relaxation metric is y = ω_opt · k = [1.2679, 2.5359, 3.8038, 5.0718, 6.3397, 7.6077]. Ordinary least-squares linear regression on these coordinates produces an empirical slope m = 1.26795 and Y-intercept b = 0.000 with coefficient of determination R² = 1.000000. The empirical optimal parameter ω_opt = 1.268 matches the theoretical benchmark ω_opt = 1.26795 within 0.0000% deviation.
Computational tools & numerical modules.
Relaxation Parameter Governor (ω ∈ (0, 2))
Regulates the over-relaxation weighting factor ω to accelerate convergence.
Ostrowski-Reich Stability Monitor
Verifies that spectral radius ρ(T_ω) < 1 strictly holds across the interval 0 < ω < 2.
Young's Optimal Omega Analyzer
Computes David Young's analytical formula ω_opt = 2 / [1 + √(1 − ρ_J²)].
Interactive Spectral Radius Curve Canvas
Plots the V-shaped curve of ρ(T_ω) vs ω showing the sharp minimum at ω_opt.
High-Precision IEEE 754 Standard Unit
Provides analytical reference values to benchmark iteration performance.
FOLLOW THE METHOD
A reason behind every step.
Accelerate stationary linear solvers using the Successive Over-Relaxation (SOR) method with relaxation factor ω, evaluate the Ostrowski-Reich theorem on symmetric positive-definite matrices, and extract the optimal relaxation parameter ω_opt = 1.268 that minimizes the spectral radius via least-squares curvature fitting. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 steps.
01
Jacobi Spectral Radius Determination
Determine the base Jacobi spectral radius ρ(T_J) = 0.500 for the target tridiagonal 2-cyclic matrix A.
02
Theoretical Optimal Parameter Evaluation
Apply David M. Young's theorem (1950): compute ω_opt = 2 / [1 + √(1 − ρ(T_J)²)] = 2 / [1 + √(1 − 0.25)] = 2 / [1 + √0.75] = 2 / [1 + 0.866025] = 1.267949.
03
Execution with Gauss-Seidel Baseline (ω = 1.0)
Set ω = 1.000 (standard Gauss-Seidel). Observe spectral radius ρ_{GS} = ρ_J² = 0.2500, requiring 4 steps to achieve 10⁻⁴ accuracy.
04
Execution with Optimal SOR (ω = 1.268)
Set ω = 1.268. Evaluate SOR iteration: spectral radius drops to ρ_{SOR} = ω_opt − 1 = 1.267949 − 1 = 0.267949, speeding convergence dramatically.
05
Under-Relaxation and Over-Relaxation Sweep
Test parameter values ω ∈ {0.8, 1.0, 1.268, 1.5, 1.9}. Confirm that for ω ≥ 2, the algorithm diverges violently, verifying the Ostrowski-Reich boundary.
06
Linear Regression and Parameter Extraction
Plot cumulative relaxation metric y = 1.268 · k against iteration step k. Extract empirical slope m = 1.2680, confirming the optimal relaxation parameter.
YOUR NOTEBOOK · 0 OBSERVATIONS
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Points are your recorded observations. The solid regression line validates Gauss-Seidel & SOR mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
\omega_{\text{opt}} = \frac{2}{1 + \sqrt{1 - \rho(T_J)^2}} = \frac{2}{1 + \sqrt{1 - 0.5^2}} \approx 1.268
| # | Iteration Step k (steps) | SOR Error Norm ‖e_k‖ () | Iteration Step k | Cumulative Relaxation Weight (ω · k) | Remove |
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