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  "body": "\n## Definition\n\nLet $a \u003c b$, let $f \\colon [a, b] \\to \\R$ be continuous, and let $m = (a + b)/2$.\n*Simpson's rule* is the [[quadrature rule|ADSVMCY77A]]\n$$S(f) = \\frac{b - a}{6} \\big( f(a) + 4 f(m) + f(b) \\big).$$\nFor an even integer $N \\ge 2$, let $h = (b - a)/N$ and $x_i = a + i h$.\nThe *composite Simpson's rule* is\n$$S_N(f) = \\frac{h}{3} \\Big( f(x_0) + 4 f(x_1) + 2 f(x_2) + 4 f(x_3) + \\cdots + 2 f(x_{N-2}) + 4 f(x_{N-1}) + f(x_N) \\Big),$$\nthe sum of $S$ over the subintervals $[x_{2j}, x_{2j+2}]$.\n\n## Remarks\n\n$S$ is the closed [[Newton–Cotes formula|BQPQAY1FDJ]] with $n = 2$, and its [[degree of exactness|ZYT1JNT1Y0]] is $3$, one more than the degree of the interpolating parabola, by symmetry.\nExample: for $f(x) = x^3$ on $[0, 1]$, $S(f) = \\frac{1}{6}\\big(0 + \\frac{1}{2} + 1\\big) = \\frac{1}{4} = \\int_0^1 x^3\\, dx$.\nNon-example of exactness: for $f(x) = x^4$ on $[0, 1]$, $S(f) = \\frac{1}{6}\\big(0 + \\frac{1}{4} + 1\\big) = \\frac{5}{24}$, while the integral is $\\frac{1}{5}$; so the degree is not $4$.\nIf $f$ has a continuous fourth derivative, then $\\int_a^b f - S_N(f) = -\\frac{(b - a) h^4}{180} f^{(4)}(\\xi)$ for some $\\xi \\in [a, b]$.\nFor $x^4$ on $[0, 1]$ with $N = 2$ this gives $-\\frac{1}{16 \\cdot 180} \\cdot 24 = -\\frac{1}{120}$, matching $\\frac{1}{5} - \\frac{5}{24}$.\n",
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