In calculus, the trapezoidal rule (informally trapezoid rule; or in British English 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. This is easily calculated by noting that the area of the region is made up of a rectangle with width and height , and a triangle of width and height .
Therefore,
The rule can also be derived by replacing the integrand with the equation of the line joining points and , which using the two point form of the equation of a line, is
Therefore,
as before.
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
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 approximation becomes more accurate as the resolution of the partition increases (that is, for larger , all decrease).
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:
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
Evidence discovered in Babylonian cuneiform tablets circa 200 BCE predicting the motion of Jupiter along the ecliptic suggests that the trapezoid rule was in use for numerical approximation long before calculus itself was invented.
Numerical implementation
Non-uniform grid When the grid spacing is non-uniform, one can use the formula
where or more a computationally efficient formula
where are the corresponding forward, backward, and central differences.
Uniform grid For a domain partitioned by equally spaced points, considerable simplification may occur.
Let and for The approximation to the integral becomes
Sometimes this expression is written as
where the symbol indicates that the first and last terms are halved.
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 sign of 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 then
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.
"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. Convergence usually is exponential or faster. A similar effect is available for peak functions.
For infinitely smooth non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and Clenshaw–Curtis quadrature generally give more precise answers for the same number of function evaluations; 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.
Algorithm The following pseudocode describes the Trapezoidal rule.
input: Real valued function, f Lower limit of integration, a Upper limit of integration, b Number of intervals, n output: An approximation of h ← (b - a) / n sum ← f(a) + f(b) for i ← 1 to n - 1 do x ← a + (i × h) sum ← sum + (2 × f(x)) end for return (h / 2) × sum
Numerical examples Approximating the natural logarithm of 3 Since
we can use the trapezoidal rule to approximate the integral, thereby generating an approximation of .
Applying the rule with segments gives
which has absolute error of and a relative error of .
Applying the rule with segments gives
which has absolute error of and a relative error of .
Suppose we wish to determine the number of intervals required to approximate to within an accuracy of . The error term is
where , , and is some number between and . Since is a decreasing function, on the internal its maximum occurs at , which is . We therefore have
We seek such that
which gives
Solving for gives
so intervals are required to guarantee an accuracy of .
Approximating the integral of a product The following integral is given:
Solution
See also Gaussian quadrature Newton–Cotes formulas Rectangle method Romberg's method Simpson's rule Clenshaw–Curtis quadrature Tai's model
Notes
References
External links
I. P. Mysovskikh, Trapezium formula, Encyclopedia of Mathematics, ed. M. Hazewinkel Notes on the convergence of trapezoidal-rule quadrature An implementation of trapezoidal quadrature provided by Boost.Math