Definite Integral Test Bed
Provides the smooth benchmark integrand f(x) = e^x with known analytic integral I = e - 1.
Mathematics / Composite Trapezoidal & Romberg Quadrature
FREE TO EXPLORELABORATORY 26 · MATHEMATICS
Composite Trapezoidal Quadrature & Romberg Extrapolation
Implement the composite trapezoidal quadrature rule, evaluate second-order asymptotic truncation error attenuation as panel width h decreases, and execute Richardson extrapolation to build the triangular Romberg integration tableau achieving high-order precision. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
QUADRATURE ERROR |I - T(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
The composite trapezoidal rule is inefficient for oscillatory or peaked integrands unless adaptive mesh refinement is employed. The simulation evaluates composite trapezoidal quadrature for step sizes h = 0.20 / 2^{k-1} on the exponential test function. The logarithmic fit ln(E) = α·ln(h) + ln(C) extracts the asymptotic order α = 2.000 with R² = 1.000000.
Computational tools & numerical modules.
Dyadic Mesh Subdivider
Generates sequence of step sizes h_k = (b - a)/2^k, reusing function evaluations from previous levels.
Composite Trapezoidal Integrator
Computes T(h) = h [0.5 f(a) + ∑ f(x_i) + 0.5 f(b)] with O(N) evaluations.
Euler-Maclaurin Analyzer
Calculates theoretical error coefficients c_2k = [B_{2k}/(2k)!] · [f^{(2k-1)}(b) - f^{(2k-1)}(a)] based on Bernoulli numbers.
Richardson Extrapolation Matrix
Cancels leading O(h²) error terms to systematically construct higher columns of the Romberg tableau.
Log-Log Asymptotic Slope Engine
Performs linear regression on (ln(h), ln(E_h)) to empirically confirm the second-order slope α = 2.000.
FOLLOW THE METHOD
A reason behind every step.
Implement the composite trapezoidal quadrature rule, evaluate second-order asymptotic truncation error attenuation as panel width h decreases, and execute Richardson extrapolation to build the triangular Romberg integration tableau achieving high-order precision. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Define Quadrature Domain & Integrand
Set up f(x) = exp(x) on [0, 1] with exact analytic reference value I = e - 1 = 1.718281828459045.
02
Initialize Dyadic Panel Spacing
For subdivision level k = 1 to 6, set panel count N = 2^{k-1} and spacing h = 1.0 / N.
03
Evaluate Composite Trapezoidal Sum
Compute T(h) = (h/2) [f(0) + 2 ∑_{i=1}^{N-1} f(i·h) + f(1)] in double precision registers.
04
Calculate Absolute Truncation Error
Compute E_h = |I - T(h)| and verify that halving h reduces the error by approximately a factor of 4 (2²).
05
Construct Romberg Extrapolation Columns
Apply the Richardson recurrence R(j, m) = [4^m R(j, m-1) - R(j-1, m-1)] / (4^m - 1) to eliminate O(h²), O(h⁴), O(h⁶) terms.
06
Extract Convergence Order α
Transform panel widths and errors into logarithmic coordinates (ln(h), ln(E_h)) and perform linear regression to confirm α = 2.000.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Composite Trapezoidal & Romberg Quadrature mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
ln(E_h) = α · ln(h) + ln(C)
| # | Subdivision Level (k) (step) | Quadrature Error |I - T(h)| (error) | Logarithmic Panel Width ln(h) | Logarithmic Quadrature Error ln(E_h) | Remove |
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