Legendre Root Finder (Golub-Welsch)
Computes roots of P_n(x) as eigenvalues of the symmetric tridiagonal Jacobi matrix.
Mathematics / Gauss-Legendre Quadrature
FREE TO EXPLORELABORATORY 28 · MATHEMATICS
Gauss-Legendre Quadrature: Optimal Orthogonal Nodes and Weights
Determine the optimal roots of Legendre orthogonal polynomials P_n(x), compute the Christoffel-Darboux quadrature weights, demonstrate algebraic exactness for all polynomials of degree ≤ 2n - 1, and verify the O(h⁶) super-convergence of the 3-point composite Gauss-Legendre rule. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
GAUSS QUADRATURE ERROR |I - G_3(H)|—
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
Standard Gauss-Legendre nodes are non-nested: doubling n requires discarding all previous function evaluations unless Gauss-Kronrod extensions are used. The simulation evaluates composite 3-point Gauss-Legendre quadrature for step sizes h = 0.20 / 2^{k-1}. The logarithmic regression ln(E) = α·ln(h) + ln(C) extracts the asymptotic order α = 6.000 with R² = 1.000000.
Computational tools & numerical modules.
Christoffel Weight Calculator
Evaluates weights w_j = 2 / [(1 - x_j²) (P'_n(x_j))²] ensuring all weights are strictly positive.
Affine Domain Mapper
Maps canonical Gauss nodes from [-1, 1] to physical sub-intervals [x_i, x_{i+1}] via x = (x_i + x_{i+1})/2 + (h/2) ξ_j.
Smooth Transcendental Benchmark
Evaluates the analytic test integral ∫₀¹ exp(x) cos(x) dx = 0.5 (e (sin(1) + cos(1)) - 1).
Newton-Cotes vs Gauss Comparator
Contrasts the error efficiency of 3-point Gauss-Legendre (degree 5 exact) against 3-point Simpson (degree 3 exact).
Asymptotic Order Verifier
Performs linear regression on (ln(h), ln(E_h)) to confirm the theoretical sixth-order slope α = 6.000.
FOLLOW THE METHOD
A reason behind every step.
Determine the optimal roots of Legendre orthogonal polynomials P_n(x), compute the Christoffel-Darboux quadrature weights, demonstrate algebraic exactness for all polynomials of degree ≤ 2n - 1, and verify the O(h⁶) super-convergence of the 3-point composite Gauss-Legendre rule. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Calculate Canonical 3-Point Nodes and Weights
Set reference nodes ξ = {-√(3/5), 0, √(3/5)} and weights w = {5/9, 8/9, 5/9} on [-1, 1].
02
Verify Degree 5 Algebraic Precision
Integrate test monomials x⁰, x¹, x², x³, x⁴, x⁵ and verify zero residual error to double machine precision.
03
Partition Domain into N Sub-Panels
Subdivide [0, 1] into N = 2^{k-1} sub-panels with panel width h = 1.0 / N across refinement levels k = 1 to 6.
04
Apply Affine Map & Accumulate Integral
Map nodes to each sub-panel [a_i, b_i] and accumulate G_3(h) = ∑_{i=1}^N (h/2) ∑_{j=1}^3 w_j f(a_i + (h/2)(1 + ξ_j)).
05
Evaluate Absolute Quadrature Error
Compare G_3(h) against the exact analytic integral and verify error reduction by 2⁶ = 64 per mesh halving.
06
Fit Logarithmic Convergence Slope
Plot ln(E_h) versus ln(h) and compute the linear regression slope to verify α = 6.000.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Gauss-Legendre Quadrature mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
ln(E_h) = α · ln(h) + ln(C)
| # | Subdivision Level (k) (step) | Gauss Quadrature Error |I - G_3(h)| (error) | Logarithmic Panel Width ln(h) | Logarithmic Quadrature Error ln(E_h) | Remove |
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