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:
- Fourier series
- Residue calculus
- 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.
Read more about this topic: Trapezoidal Rule
Famous quotes containing the words error and/or analysis:
“It is an error to believe that the Roman Pontiff can and ought to reconcile himself to, and agree with, progress, liberalism, and contemporary civilization.”
—Pope Pius IX (17921878)
“... the big courageous acts of life are those one never hears of and only suspects from having been through like experience. It takes real courage to do battle in the unspectacular task. We always listen for the applause of our co-workers. He is courageous who plods on, unlettered and unknown.... In the last analysis it is this courage, developing between man and his limitations, that brings success.”
—Alice Foote MacDougall (18671945)