Trapezoidal Rule - Error Analysis

Error Analysis

The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result:

There exists a number ξ between a and b, such that

It follows that if the integrand is concave up (and thus has a positive second derivative), then the error is negative and the trapezoidal rule overestimates the true value. This can also be seen from the geometric picture: the trapezoids include all of the area under the curve and extend over it. Similarly, a concave-down function yields an underestimate because area is unaccounted for under the curve, but none is counted above. If the interval of the integral being approximated includes an inflection point, the error is harder to identify.

In general, three techniques are used in the analysis of error:

1. Fourier series
2. Residue calculus
3. Euler–Maclaurin summation formula:

An asymptotic error estimate for N → ∞ is given by

Further terms in this error estimate are given by the Euler–Maclaurin summation formula.

It is argued that the speed of convergence of the trapezoidal rule reflects and can be used as a definition of classes of smoothness of the functions.