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:
“No consensus of men can make an error erroneous. We can only find or commit an error, not create it. When we commit an error, we say what was an error already.”
—Josiah Royce (1855–1916)
“Whatever else American thinkers do, they psychologize, often brilliantly. The trouble is that psychology only takes us so far. The new interest in families has its merits, but it will have done us all a disservice if it turns us away from public issues to private matters. A vision of things that has no room for the inner life is bankrupt, but a psychology without social analysis or politics is both powerless and very lonely.”
—Joseph Featherstone (20th century)