Recipe

Apply the composite Simpson's rule

Use when: you need \int_a^b f(x)\,dx to a specific accuracy and f is smooth enough that a piecewise-parabola fit tracks it well

Goal: a single numerical approximation S_n to the integral, built from n+1 evaluations of f

Taught in Simpson's rule and degree of precision

1. 1

   Decide how many panels to split [a,b] into. n must be even. The 1,4,2,4,\dots,4,1 weight pattern pairs panels two at a time. An odd n leaves one panel with no defined pair.

   Example

   For \displaystyle\int_0^2 e^x\,dx (true value e^2-1\approx6.389056), take n=4.

   • n is already even — proceed directly to Step 2.
   • n is given odd (or computed from a target step size and lands odd) — round up to the next even integer before continuing. The resulting h shrinks slightly, which only helps accuracy.

   Why: the weight pattern's own derivation fits one parabola through each PAIR of panels. An odd number of panels leaves a leftover one the pattern was never built to weight.

2. 2

   h=\dfrac{b-a}{n}, and x_i = a+ih for i=0,\dots,n.

   Example

   h=\dfrac{2-0}{4}=0.5, giving grid points x_0,\dots,x_4=0,\,0.5,\,1,\,1.5,\,2.

   ⚡ Shortcut

   Doubling n later never wastes the first pass's work — say, going from 4 to 8 panels, to check whether the answer has converged. Every OLD grid point is still a grid point of the new, finer grid. Only the new midpoints between them are genuinely new evaluations. Section 4's Romberg integration is built entirely around this same reuse, applied to the trapezoidal rule instead of Simpson's rule.

   Why: an evenly spaced grid is what makes the weight pattern in Step 3 apply the same way everywhere. Every inner point's weight depends only on whether its index is odd or even, never on where it sits.

3. 3

   Compute f(x_0), f(x_1), \dots, f(x_n) once each.

   Example

   f(x)=e^x: f(0)=1, f(0.5)\approx1.648721, f(1)\approx2.718282, f(1.5)\approx4.481689, f(2)\approx7.389056.

   Why: every later step is pure arithmetic on these n+1 numbers. No further evaluations of f are needed once this list exists.

4. 4

   Odd-indexed points (x_1,x_3,\dots) get weight 4. Even inner points (x_2,x_4,\dots,x_{n-2}) get weight 2. The two endpoints get weight 1.

   Example

   1(1)+4(1.648721)+2(2.718282)+4(4.481689)+1(7.389056)=38.347261.

   Why: this exact pattern is what makes each pair of panels match its own fitted parabola's area exactly (Section 2's own degree-of-precision point). It isn't an arbitrary smoothing weight — it falls directly out of the parabola's own closed-form area.

5. 5

   S_n = \frac{h}{3}\Big[\text{the weighted sum from Step 4}\Big].

   Example

   S_4=\frac{0.5}{3}(38.347261)\approx6.391210,

   against the true value e^2-1\approx6.389056 — an error of about 0.00215.

   Why: this is the constant every pair of panels' own exact parabola-area formula carries. It's derived once and reused for every pair, rather than re-derived per pair.