Numerical Integration Playground

1. Pick a function and an integration rule

Function:

a₀ 1.00 a₁ 0.60 a₂ -0.40 a₃ 0.50 a₄ 0.00 a₅ 0.00

Rule: N / n: 2

Drag the red dots left/right to place samples by hand; weights are auto-chosen by matching monomial moments.

f(ξ) sample point ξ_i weight stem (height ∝ W_i · f) exact integral region

Exact ∫ f dξ : — Estimate Σ W f : — Error : — Rel. error : — Polynomial deg : — Rule exact thru deg : —

Gauss–Legendre with n points is exact for every polynomial of degree ≤ 2n−1. Try the “polynomial of degree 5” preset with Gauss n=3: the error should be zero to machine precision.

2. Moment-matching gauges — the heart of the derivation

Each gauge compares Σ_i W_i ξ_i^k (computed from your points) to ∫₋₁¹ ξ^k dξ (the true moment). A rule with n points has exactly 2n free parameters (the positions ξ_i and weights W_i), so it can match 2n moments — i.e. it is exact for every polynomial of degree ≤ 2n−1.

3. Monte-Carlo “rain” (physics-flavored sampling)

raindrop caught (below curve) missed (above curve) bounding box

Drops so far : 0 Caught below f : 0 MC estimate : — Exact : — MC error : — Gauss n=2 error : —

Drops fall from the top of the bounding box at random ξ. Those below the curve are “caught” (green). The integral is approximated by (hits / total) × (box area). This is the slowest method on the page — error shrinks like 1/√N. Notice how many drops it takes to beat a 2-point Gauss rule.

4. Convergence — how fast does each rule get it right?

For the currently-selected function, the plot shows the absolute error of each rule as the number of evaluations N grows. Both axes are log-scaled. On a smooth function, Gauss quadrature converges exponentially while the midpoint rule only manages O(1/N²). On the non-smooth functions (|ξ| and Runge), the story is messier — try them and see.

— Gauss–Legendre — Simpson (composite) — Midpoint (composite)