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Composite Simpson rule is exact through cubic degree

Statement

Let $a < b$ , let $n \geq 2$ be an even integer, and let $p (x) = c_{0} + c_{1} x + c_{2} x^{2} + c_{3} x^{3}$ with real coefficients. Then

$\int_{a}^{b} p (x) d x = Q_{S} (p; a, b, n),$

where $Q_{S}$ is the composite Simpson value of Composite Simpson rule on an even uniform grid.

Facts & Assumptions

Given: Reals $a < b$ , an even integer $n \geq 2$ , and $p (x) = c_{0} + c_{1} x + c_{2} x^{2} + c_{3} x^{3}$ .

[F1]

Composite Simpson rule on an even uniform grid: the composite value is $\frac{h}{3} \sum_{j < n / 2} (f (x_{2 j}) + 4 f (x_{2 j + 1}) + f (x_{2 j + 2}))$ .

[F2]

Peano kernel identity for one Simpson panel: on a panel, the error equals the integral of its Peano kernel times the fourth derivative.

[F3]

For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ : the integral over the partition is the sum of the integrals over its adjacent blocks.

[F4]

For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term: for $k \geq 1$ , $(x^{k})^{'} = k x^{k - 1}$ , and $x^{0}$ has derivative zero.

[F5]

Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : sums and scalar multiples of differentiable functions have the corresponding derivatives.

[F6]

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ : $p \in C^{4} ([a, b])$ requires that $p$ and its derivatives through order four exist and are continuous on $[a, b]$ .

[F7]

Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function: every polynomial function is continuous on its domain.

Proof

1.1givenF4F5F6F7algebra

Repeated use of [F4] and the sum/scalar rules in [F5] gives, for every real $x$ , $p^{'} (x) = c_{1} + 2 c_{2} x + 3 c_{3} x^{2}, p^{''} (x) = 2 c_{2} + 6 c_{3} x, p^{'''} (x) = 6 c_{3}, p^{(4)} (x) = 0.$ These formulas establish existence of all derivatives through order four, including their restrictions to the endpoints with the one-sided convention. Each of $p, p^{'}, p^{''}, p^{'''}$ and $p^{(4)}$ is a polynomial function, hence continuous by [F7]. Thus $p \in C^{4} ([a, b])$ by [F6], and $p^{(4)} \equiv 0$ .

2.1step 1.1F2

For every block $[x_{2 j}, x_{2 j + 2}]$ , the restriction of $p$ is $C^{4}$ and has fourth derivative zero. The Peano-kernel identity [F2] makes that block's Simpson error $\int_{x_{2 j}}^{x_{2 j + 2}} K_{x_{2 j}, x_{2 j + 2}} (x) 0 d x = 0$ .

3.1step 2.1F1F3algebra∎

By [F3], summing the exact block integrals gives $\int_{a}^{b} p$ . Summing the block Simpson values from [F1], each endpoint appears once, each odd interior node appears with coefficient $4$ , and each even interior node appears with coefficient $2$ . This is exactly the composite formula [F1], so the global rule is exact.

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Sources

Art B. Owen, Monte Carlo Theory, Methods and Examples, Chapter 7, §7.2 (standard reference, not scraped)
Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, §5.6 (standard reference, not scraped)