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Convex data bracket the integral between composite midpoint and trapezoid values

Example

Let $f : [a, b] \to R$ be convex and Darboux integrable on $[a, b]$ , with $a < b$ , and let $n \geq 1$ be an integer. Then

$Q_{M} (f; a, b, n) \leq \int_{a}^{b} f (x) d x \leq T_{n} (f),$

where $Q_{M}$ is the composite midpoint value and $T_{n} (f)$ is the published composite trapezoidal value of Composite trapezoidal rule on the same interval and uniform grid. Darboux integrability is the explicit premise that makes the displayed Riemann integral defined. The proof uses convexity directly for both inequalities and assumes no differentiability.

Facts & Assumptions

Given: Reals $a < b$ , an integer $n \geq 1$ , and a convex Darboux-integrable $f : [a, b] \to R$ .

[F1]

Convex, strictly convex, concave, strictly concave, and midpoint-convex real functions on an interval: convexity means $f (λ x + (1 - λ) y) \leq λ f (x) + (1 - λ) f (y)$ for $0 \leq λ \leq 1$ .

[F2]

The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ , The Darboux and Riemann definitions agree: a bounded $f$ on $[a, b]$ is Darboux integrable with integral $I$ if and only if for every real $ε > 0$ there is a real $δ > 0$ such that $∣ S (f, P, ξ) - I ∣ < ε$ for every tagged partition of mesh below $δ$ , For bounded $f$ on $[a, b]$ and a partition $P$ : the infimum $m_{i}$ and supremum $M_{i}$ of $f$ on the $i$ -th subinterval, and the lower and upper Darboux sums $L (f, P) = \sum_{i} m_{i} Δ_{i}$ and $U (f, P) = \sum_{i} M_{i} Δ_{i}$ , and Partition of $[a, b]$ as a finite strictly increasing list $a = t_{0} < t_{1} < \dots < t_{n} = b$ , its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions: Darboux integrability means the lower and upper integrals agree, and this integral agrees with the tagged-Riemann definition.

[F3]

A finite sum is unchanged by a permutation of its index range: $\sum_{k < n} a_{π (k)} = \sum_{k < n} a_{k}$ for every bijection $π : n \to n$ : reordering a finite list of terms by a permutation does not change its sum.

[F4]

Composite midpoint rule on a uniform grid: the composite midpoint value is the sum of the panel midpoint values multiplied by the common width $h$ .

[F5]

Composite trapezoidal rule: the composite value is $h$ times the sum of endpoint values with half weights and interior node values with unit weights.

[F6]

If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ : pointwise order of integrable functions on an increasing interval implies the same order for their integrals.

[F7]

Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ : sums and scalar multiples of integrable functions are integrable, and their integrals are linear.

[F8]

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 $a, b, c$ : an integrable function remains integrable on subintervals, and its integral adds over adjacent subintervals.

[F9]

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: affine functions and polynomials are continuous.

[F10]

Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable: every continuous function on a closed interval (the case $q = 1$ ) is Riemann integrable.

[F11]

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: $(x^{k})^{'} = k x^{k - 1}$ for each integer $k \geq 1$ .

[F12]

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 $g (c) \neq 0$ : the sum, scalar-multiple, and product rules give the corresponding derivatives.

[F13]

The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ : an integrable derivative has integral equal to the endpoint increment of its antiderivative.

Verification

1.1F2F3F8construct

Fix a panel $[u, v] = [x_{j}, x_{j + 1}]$ , write $h = v - u$ and $m = (u + v) / 2$ , and let $g (x) : = f (u + v - x)$ . The restriction of $f$ to this panel is integrable by [F8]. Reflect any partition of $[u, v]$ about $m$ and reverse the order of its nodes; this gives an involution on the set of partitions. Each reflected subinterval has the same length and the same range values of $g$ as the corresponding original subinterval has for $f$ . Thus the lower and upper Darboux sums agree after reversing their finite lists of terms, by [F3]. Since the reflected partitions run through all partitions, [F2] shows that $g$ is integrable and $\int_{u}^{v} g = \int_{u}^{v} f$ .

1.2givenF1F5F6F8F9F10F11F12F13algebra

For each panel put $u = x_{j}$ , $v = x_{j + 1}$ and $h = v - u$ , and define the endpoint chord $L_{j} (x) : = \frac{v - x}{h} f (u) + \frac{x - u}{h} f (v)$ . For $x \in [u, v]$ , the coefficients are in $[0, 1]$ and sum to $1$ , so [F1] gives $f (x) \leq L_{j} (x)$ . The functions are integrable by the given premise and by continuity of the affine chord [F9] with [F10]. Monotonicity [F6] gives $\int_{u}^{v} f \leq \int_{u}^{v} L_{j}$ . Set $G_{j} (x) : = f (u) (x - u) + \frac{f (v) - f (u)}{2 h} (x - u)^{2} .$ The power, product, sum, and scalar rules [F11, F12] give $G_{j}^{'} = L_{j}$ , so [F13] gives $\int_{u}^{v} L_{j} = G_{j} (v) - G_{j} (u) = \frac{h}{2} (f (u) + f (v))$ . Summing over panels, additivity [F8] gives the integral over $[a, b]$ , and each interior node is counted twice with half weight while each endpoint is counted once with half weight. By [F5] this sum is $T_{n} (f)$ , so $\int_{a}^{b} f \leq T_{n} (f)$ .

2.1givenF1F6F7F10F11F12F13step 1.1

For every $x \in [u, v]$ , convexity [F1] at weight $1 / 2$ gives $2 f (m) \leq f (x) + g (x)$ . The constant function is integrable by [F10], and $f + g$ is integrable by [F7] and step 1.1. Monotonicity [F6] and linearity [F7] therefore give $\int_{u}^{v} 2 f (m) d x \leq \int_{u}^{v} (f + g) d x = 2 \int_{u}^{v} f d x$ . The function $G (x) = 2 f (m) x$ is an antiderivative of the constant integrand, by [F11] and [F12]. Hence [F13] gives $2 h f (m) \leq 2 \int_{u}^{v} f d x$ , so $h f (m) \leq \int_{u}^{v} f d x$ .

3.1step 2.1step 1.2F4F8∎

Summing the panel inequalities from step 2.1 by [F8] and using the composite midpoint formula [F4] gives $Q_{M} (f; a, b, n) \leq \int_{a}^{b} f$ . Combining this with step 1.2 proves the bracketing.

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Sources

Art B. Owen, Monte Carlo Theory, Methods and Examples, Chapter 7, §7.1 (standard reference, not scraped)
Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, §5.6 (standard reference, not scraped)