Trapezoidal rule

Trapezoidal rule
In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for numerical integration, i.e., approximating the definite integral:
The trapezoidal rule works by approximating the region under the graph of the function
as a trapezoid and calculating its area. It follows that
The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" (or "composite") trapezoidal rule is usually what is meant by "integrating with the trapezoidal rule". Let be a partition of such that and be the length of the -th subinterval (that is), then
When the partition has a regular spacing, as is often the case, that is, when all the have the same value the formula can be simplified for calculation efficiency by factoring out:.
The approximation becomes more accurate as the resolution of the partition increases (that is, for larger , all decrease).
As discussed below, it is also possible to place error bounds on the accuracy of the value of a definite integral estimated using a trapezoidal rule.
History
A 2016 Science paper reports that the trapezoid rule was in use in Babylon before 50 BCE for integrating the velocity of Jupiter along the ecliptic.
Numerical implementation
Non-uniform grid
When the grid spacing is non-uniform, one can use the formula
wherein
Uniform grid
For a domain discretized into equally spaced panels, considerable simplification may occur. Let
the approximation to the integral becomes
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.
An asymptotic error estimate for N → ∞ is given by
Further terms in this error estimate are given by the Euler–Maclaurin summation formula.
Several techniques can be used to analyze the error, including:
Fourier series
Residue calculus
Euler–Maclaurin summation formula
Polynomial interpolation
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.
Proof
First suppose that and . Let be the function such that is the error of the trapezoidal rule on one of the intervals, . Then
and
Now suppose that which holds if is sufficiently smooth. It then follows that
which is equivalent to
, or
Since and ,
and
Using these results, we find
and
Letting we find
Summing all of the local error terms we find
But we also have
and
so that
Therefore the total error is bounded by
Periodic and peak functions
The trapezoidal rule converges rapidly for periodic functions. This is an easy consequence of the Euler-Maclaurin summation formula, which says that
if is times continuously differentiable with period
where and is the periodic extension of the th Bernoulli polynomial. Due to the periodicity, the derivatives at the endpoint cancel and we see that the error is .
A similar effect is available for peak-like functions, such as Gaussian, Exponentially modified Gaussian and other functions with derivatives at integration limits that can be neglected. The evaluation of the full integral of a Gaussian function by trapezoidal rule with 1% accuracy can be made using just 4 points. Simpson's rule requires 1.8 times more points to achieve the same accuracy.
Although some effort has been made to extend the Euler-Maclaurin summation formula to higher dimensions, the most straightforward proof of the rapid convergence of the trapezoidal rule in higher dimensions is to reduce the problem to that of convergence of Fourier series. This line of reasoning shows that if is periodic on a -dimensional space with continuous derivatives, the speed of convergence is . For very large dimension, the shows that Monte-Carlo integration is most likely a better choice, but for 2 and 3 dimensions, equispaced sampling is efficient. This is exploited in computational solid state physics where equispaced sampling over primitive cells in the reciprocal lattice is known as Monkhorst-Pack integration.
"Rough" functions
For functions that are not in C2, the error bound given above is not applicable. Still, error bounds for such rough functions can be derived, which typically show a slower convergence with the number of function evaluations than the behaviour given above. Interestingly, in this case the trapezoidal rule often has sharper bounds than Simpson's rule for the same number of function evaluations.
Applicability and alternatives
The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas, of which the midpoint rule is similar to the trapezoid rule. Simpson's rule is another member of the same family, and in general has faster convergence than the trapezoidal rule for functions which are twice continuously differentiable, though not in all specific cases. However, for various classes of rougher functions (ones with weaker smoothness conditions), the trapezoidal rule has faster convergence in general than Simpson's rule.
Moreover, the trapezoidal rule tends to become extremely accurate when periodic functions are integrated over their periods, which can be analyzed in various ways. A similar effect is available for peak functions.
For non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally far more accurate; Clenshaw–Curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied accurately.
Example
The following integral is given:
Solution
See also
Gaussian quadrature
Newton–Cotes formulas
Rectangle method
Romberg's method
Simpson's rule
Notes
References
External links
Trapezium formula. I.P. Mysovskikh, Encyclopedia of Mathematics, ed. M. Hazewinkel
Notes on the convergence of trapezoidal-rule quadrature
An implementation of trapezoidal quadrature provided by Boost.Math
Numerical integration (quadrature)