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 Incoming links: Symmetric trapezoidal exactness for odd functions

Past exam of the numerical analysis course of the University of Oxford / 2015 / 1 / c / ii / Solution 2026-10-31

Use the nonpolynomial function

f (x) = 1 + sin x

on

[- 1, 1]

. Its exact integral is

\int_{- 1}^{1} (1 + sin x) d x = 2,

because the sine is an odd function.

For every uniform partition of an interval with

N \geq 1

panels, the quadrature nodes satisfy

x_{N - j} = - x_{j}

. Their composite trapezoidal rule quadrature weights agree, so the two sampled sine values cancel. If there is a node at zero its sine value is already zero. The constant term is integrated exactly, giving

T_{N} (1 + sin x) = 2 for every N \geq 1.

This is symmetric trapezoidal exactness for odd functions, applied after adding a constant function so that the resulting definite integral is nonzero. It is exactness for this particular integrand, rather than a claim that every nonpolynomial function is integrated exactly.

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