Trigonometric Benchmark Manifold
Evaluates f(x) = sin(x) on [0, π] with exact analytic reference integral I = 2.000000.
Mathematics / Simpson Quadrature Rules
FREE TO EXPLORELABORATORY 27 · MATHEMATICS
Simpson's 1/3 and 3/8 Rules: Parabolic Quadrature Convergence
Implement composite Simpson's 1/3 and 3/8 quadrature rules, demonstrate why quadratic interpolation gains an extra order of algebraic accuracy for cubic polynomials, and verify the fourth-order O(h⁴) truncation error convergence rate under systematic mesh halving. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
SIMPSON QUADRATURE ERROR |I - S(H)|—
COMPUTATION STATUS
Ready to execute Step = 1.
Adjust Step, then execute computation.
Mathematical operations are computed deterministically using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations.
Numerical assumptions and limits
Simpson's rule requires equidistant node spacing; if data points are non-uniformly distributed, spline quadrature or Gaussian quadrature must be used. The simulation evaluates composite Simpson quadrature for step sizes h = 0.20 / 2^{k-1}. The maximum error is measured against the analytic integral of sin(x). The logarithmic regression ln(E) = α·ln(h) + ln(C) extracts the asymptotic order α = 4.000 with R² = 1.000000.
Computational tools & numerical modules.
Even-Panel Mesh Discretizer
Partitions [a, b] into N = 2·2^{k-1} panels with uniform step size h = (b - a)/N.
Simpson 1/3 Accumulator
Evaluates S_{1/3} = (h/3) [f(a) + 4 ∑_{odd} f(x_i) + 2 ∑_{even} f(x_i) + f(b)].
Simpson 3/8 Cubic Accumulator
Executes the 3/8 rule on groups of three panels, accommodating odd panel remainders.
Peano Error Kernel Evaluator
Visualizes the sign-definite Peano error kernel K(t) = ∫ (x - t)_+² dx showing fourth-order behavior.
Log-Log Regression Slope Processor
Calculates ordinary least squares slope on (ln(h), ln(E_h)) to confirm α = 4.000.
FOLLOW THE METHOD
A reason behind every step.
Implement composite Simpson's 1/3 and 3/8 quadrature rules, demonstrate why quadratic interpolation gains an extra order of algebraic accuracy for cubic polynomials, and verify the fourth-order O(h⁴) truncation error convergence rate under systematic mesh halving. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Initialize Trigonometric Integrand
Set f(x) = sin(x) on [0, π] with exact analytic reference value I = 2.000000000000000.
02
Set Even Discretization Panels
For subdivision level k = 1 to 6, set panel count N = 4 · 2^{k-1} and step size h = π / N.
03
Execute Simpson 1/3 Quadrature
Accumulate endpoint, odd, and even nodes using weights 1, 4, 2, 4, ..., 1 and scale by h/3.
04
Execute Simpson 3/8 Quadrature
Evaluate the cubic composite rule using weights 1, 3, 3, 2, 3, 3, ..., 1 and scale by 3h/8 for comparison.
05
Measure Absolute Quadrature Residual
Compute E_h = |2.0 - S(h)| and verify that halving h attenuates the error by a factor of 16 (2⁴ = 16).
06
Fit Asymptotic Order of Convergence
Plot ln(E_h) versus ln(h) and compute the linear regression gradient to verify α = 4.000.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Simpson Quadrature Rules mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
ln(E_h) = α · ln(h) + ln(C)
| # | Subdivision Level (k) (step) | Simpson Quadrature Error |I - S(h)| (error) | Logarithmic Panel Width ln(h) | Logarithmic Quadrature Error ln(E_h) | Remove |
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