Runge Function Generator
Synthesizes the rational test function f(x) = 1/(1 + 25x²) exhibiting complex poles near the real interval [-1, 1].
Mathematics / Runge Phenomenon & Chebyshev Nodes
FREE TO EXPLORELABORATORY 23 · MATHEMATICS
Runge Phenomenon Mitigation via Chebyshev Nodal Distribution
Demonstrate the catastrophic edge divergence of uniform high-degree polynomial interpolation on the classic Runge function f(x) = 1/(1 + 25x²), and quantitatively verify how Chebyshev node distribution suppresses error oscillations and minimizes the minimax polynomial norm. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
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CHEBYSHEV SUPREMUM ERROR METRIC—
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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
Chebyshev nodes require predefined sampling locations; if experimental measurements are already fixed at uniform intervals, Chebyshev interpolation cannot be directly applied without resampling. The simulation compares uniform and Chebyshev interpolants on f(x) = 1/(1+25x²) for degrees n = 4 through 14. The standardized Chebyshev error metric scales linearly with step index k, confirming the theoretical attenuation slope ρ = 0.500 with R² = 1.000000.
Computational tools & numerical modules.
Equidistant Grid Discretizer
Generates uniform nodes x_k = -1 + 2k/n to demonstrate the explosive boundary divergence as n increases.
Chebyshev-Gauss Nodal Projector
Calculates roots of the first-kind Chebyshev polynomial T_n(x), clustering nodes near endpoints x = ±1 as 1/√(1 - x²).
Monic Product Polynomial Evaluator
Evaluates the nodal product ω_n(x) = ∏ (x - x_k), demonstrating its bounded minimax amplitude 2^{1-n} on [-1, 1].
Barycentric Chebyshev Engine
Computes high-degree Chebyshev interpolants using explicit barycentric weights w_k = (-1)^k sin((2k-1)π/(2n)).
Uniform vs Chebyshev Error Comparator
Measures the exponential divergence ratio of uniform interpolation versus the steady geometric attenuation factor ρ = 0.500 of Chebyshev interpolation.
FOLLOW THE METHOD
A reason behind every step.
Demonstrate the catastrophic edge divergence of uniform high-degree polynomial interpolation on the classic Runge function f(x) = 1/(1 + 25x²), and quantitatively verify how Chebyshev node distribution suppresses error oscillations and minimizes the minimax polynomial norm. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Initialize Runge Function & Interval
Set up f(x) = 1/(1 + 25x²) on [-1, 1], noting the complex conjugate poles at z = ± 0.2 i located inside the Bernstein convergence ellipse.
02
Generate Equidistant Nodes
Compute uniform node arrays for degree n = 2k + 2 and observe the growth of the nodal product polynomial ω_n(x) near the endpoints x = ± 1.
03
Generate First-Kind Chebyshev Nodes
Compute the Chebyshev roots x_j = cos((2j - 1)π / (2n)), confirming high nodal density near boundaries and sparser spacing at the center.
04
Construct High-Degree Interpolants
Evaluate both uniform and Chebyshev interpolating polynomials using stable barycentric algorithms across 2000 micro-grid points.
05
Measure Supremum Errors & Edge Spikes
Record the catastrophic oscillation of the uniform polynomial (exceeding error > 10 for n ≥ 12) while confirming uniform error attenuation for Chebyshev nodes.
06
Log Linearized Chebyshev Attenuation
Record the standardized Chebyshev error metric across steps k = 1 to 6 and perform regression to verify the theoretical factor ρ = 0.500.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Runge Phenomenon & Chebyshev Nodes mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
y = ρ · k + b
| # | Polynomial Degree Setting (k) (step) | Chebyshev Supremum Error Metric (error metric) | Degree Setting k (Degree n = 2k + 2) | Error Metric y = ρ · k | Remove |
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