ashfordeOU Numerical Integration (cross-cutting/numerics/numerical-integration)
Use when the task is integrating a function numerically: choosing the quadrature method, computing the integral estimate, and reporting the error estimate for the trapezoid rule.
Domain quick reference
- Composite trapezoid rule: I = (b-a)/(2n) * (f(a) + f(b) + 2 * sum_{i=1}^{n-1} f(a + i h)) with h = (b-a)/n. Error scales with h^2; cheap, robust, and the right default for an arbitrary or noisy integrand.
- Composite Simpson rule: I = (b-a)/(3n) * (f(a) + f(b) + 4 * sum_odd + 2 * sum_even), n even. Error scales with h^4; exact for polynomials of degree <= 3, so it beats trapezoid on smooth integrands with a mild endpoint cost.
- Gauss-Legendre quadrature: I = (b-a)/2 * sum w_i f((b-a)/2 x_i + (a+b)/2) with fixed nodes and weights on [-1, 1], n in {2, 3, 4, 5}. The n-point rule is exact for polynomials of degree <= 2n - 1; one short call resolves a smooth integrand that a composite rule would need many panels for.
- Error estimate: the trapezoid error scales with h^2, so combining the n and 2n estimates gives, to leading order, error(2n) = abs(I_2n - I_n) / 3 (Richardson extrapolation). Use it to pick n: refine until the estimate is below the tolerance.
- Method selection: trapezoid for arbitrary or noisy integrands and error estimation; Simpson for smooth integrands when n must stay moderate; Gauss-Legendre for smooth integrands when a single high degree of exactness matters and the integrand is cheap to evaluate.
- All functions are deterministic and stdlib-only; no network, no third-party numerical libraries.
- Textbook anchor: the composite rules and Gauss-Legendre tables are the classical numerical-analysis results (Abramowitz and Stegun, 25.4); integral of x^2 on [0, 2] = 8/3 = 2.6666666667 and integral of sin(x) on [0, pi] = 2.0 are the hand-computed checks.
Workflow
- Collect the integrand f, the interval [a, b], and the target tolerance.
- Select the method: trapezoid, Simpson, or Gauss-Legendre per the integrand shape (smooth vs noisy, degree of exactness needed).
- Compute the estimate with trapezoid, simpson, or gauss_legendre.
- Estimate the error with error_estimate_trapezoid and refine n until the estimate is below the tolerance.
- Report the estimate, the method, and the error estimate together.
Pitfalls
- Using Simpson with an odd n: the composite Simpson rule requires an even number of subintervals; an odd n raises, never silently drops a panel.
- Using a non-integer n: n must be an int; a float like 2.5 raises, never truncates.
- Refining without an error estimate: error_estimate_trapezoid gives the leading-order error for free; pick n against the tolerance instead of guessing.
- Expecting Gauss-Legendre to accept any n: only n in {2, 3, 4, 5} are in the fixed node table; other counts raise.
- Treating a single Gauss-Legendre call as a composite rule: it is one high-accuracy sample, exact for low-degree polynomials, not a convergence sequence.
Behavior contract (gate 3)
The composite trapezoid rule, composite Simpson rule, Gauss-Legendre quadrature, and the Richardson error estimate logic is exercised by the gate 3 contract test: scripts/test_numerical_integration.py against scripts/numerical_integration_logic.py (stdlib unittest, offline). Run: python3 scripts/test_numerical_integration.py
Compliance
- Standards referenced, not reproduced: NACA TR-824 anchors the compressible-flow tables that these quadrature rules support; the rules themselves are classical numerical-analysis methodology, summary-only per standards-map.yaml.
- compliance: STANDARDS-REF, gated: false.