Trapezoidal rule

The function f(x) (in blue) is approximated by a linear function (in red).

In mathematics, and more specifically in numerical analysis, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral

\int _{a}^{b}f(x)\,dx

The trapezoidal rule works by approximating the region under the graph of the function $f(x)$ as a trapezoid and calculating its area. It follows that

\int _{a}^{b}f(x)\,dx\approx (b-a)\left[{\frac {f(a)+f(b)}{2}}\right]

The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way.

Illustration of "chained trapezoidal rule" used on an irregularly-spaced partition of

[a,b]

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 $\{x_{k}\}$ be a partition of $[a,b]$ such that $a=x_{0}<x_{1}<\cdots <x_{N-1}<x_{N}=b$ and $\Delta x_{k}$ be the length of the $k$ -th subinterval (that is, $\Delta x_{k}=x_{k}-x_{k-1}$ ), then

\int _{a}^{b}f(x)\,dx\approx \sum _{k=1}^{N}{\frac {f(x_{k-1})+f(x_{k})}{2}}\Delta x_{k}

The approximation becomes more accurate as the resolution of the partition increases (that is, for larger $N$ ). When the partition has a regular spacing, as is often the case, the formula to be simplified for calculation efficiency.

As discussed below, it is also possible for place error bounds on the accuracy of the value of a definite interval estimated using a trapezoidal rule.

History

A 2016 paper reports that the trapezoid rule was in use in Babylon before 50 BC for integrating the velocity of Jupiter along the ecliptic.^[1]

Numerical implementation

Non-uniform grid

When the grid spacing is non-uniform, one can use the formula

\int _{a}^{b}f(x)\,dx\approx \sum _{k=1}^{N}{\frac {f(x_{k-1})+f(x_{k})}{2}}\Delta x_{k}

Uniform grid

For a domain discretized into $N$ equally spaced panels, considerable simplification may occur. Let

\Delta x_{k}=\Delta x={\frac {b-a}{N}}

the approximation to the integral becomes

\int _{a}^{b}f(x)\,dx\approx {\frac {\Delta x}{2}}\sum _{k=1}^{N}\left(f(x_{k-1})+f(x_{k})\right)

{}={\frac {\Delta x}{2}}(f(x_{0})+2\sum _{k=1}^{N-1}f(x_{k})+f(x_{N}))

{}={\frac {\Delta x}{2}}(f(x_{0})+2f(x_{1})+2f(x_{2})+2f(x_{3})+\dotsb +2f(x_{N-1})+f(x_{N}))

which requires fewer evaluations of the function to calculate.

Error analysis

An animation showing how the trapezoidal rule approximation improves with more strips for an interval with

a=2

and

b=8

. As the number of intervals

N

increases, so too does the accuracy of the result.

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

{\text{error}}=\int _{a}^{b}f(x)\,dx-{\frac {b-a}{N}}\left[{f(a)+f(b) \over 2}+\sum _{k=1}^{N-1}f\left(a+k{\frac {b-a}{N}}\right)\right]

There exists a number ξ between a and b, such that^[2]

{\text{error}}=-{\frac {(b-a)^{3}}{12N^{2}}}f''(\xi )

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:^[3]

An asymptotic error estimate for N → ∞ is given by

\text{error} = -\frac{(b-a)^2}{12N^2} \big[ f'(b)-f'(a) \big] + O(N^{-3}).

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.^[6]

Periodic functions

The trapezoidal rule often converges very quickly for periodic functions.^[7]^[6] This can be explained intuitively:

"When the function is periodic and one integrates over one full period, there are about as many sections of the graph that are concave up as concave down, so the errors cancel."^[3]

In the error formula above, f'(a) = f'(b), and only the O(N⁻³) term remains.

"Rough" functions

For various classes of functions that are not twice-differentiable, the trapezoidal rule has sharper bounds than Simpson's rule.^[8]

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.^[8]

Moreover, the trapezoidal rule tends to become extremely accurate when periodic functions are integrated over their periods, which can be analyzed in various ways.^[6]^[7]

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.

Notes

↑ Ossendrijver, Mathieu (Jan 29, 2016). "Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph". Science. 351: 482–484. PMID 26823423. doi:10.1126/science.aad8085.
↑ Atkinson (1989, equation (5.1.7))
1 2 (Weideman 2002, p. 23, section 2)
↑ Atkinson (1989, equation (5.1.9))
↑ Atkinson (1989, p. 285)
1 2 3 (Rahman & Schmeisser 1990)
1 2 (Weideman 2002)
1 2 (Cruz-Uribe & Neugebauer 2002)

References

Atkinson, Kendall E. (1989), An Introduction to Numerical Analysis (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50023-0
Rahman, Qazi I.; Schmeisser, Gerhard (December 1990), "Characterization of the speed of convergence of the trapezoidal rule", Numerische Mathematik, 57 (1): 123–138, ISSN 0945-3245, doi:10.1007/BF01386402
Burden, Richard L.; Faires, J. Douglas (2000), Numerical Analysis (7th ed.), Brooks/Cole, ISBN 0-534-38216-9
Weideman, J. A. C. (January 2002), "Numerical Integration of Periodic Functions: A Few Examples", The American Mathematical Monthly, 109 (1): 21–36, JSTOR 2695765, doi:10.2307/2695765
Cruz-Uribe, D.; Neugebauer, C. J. (2002), "Sharp Error Bounds for the Trapezoidal Rule and Simpson's Rule" (PDF), Journal of Inequalities in Pure and Applied Mathematics, 3 (4)

External links

The Wikibook A-level Mathematics has a page on the topic of: Trapezium Rule

Trapezium formula. I.P. Mysovskikh, Encyclopedia of Mathematics, ed. M. Hazewinkel
Notes on the convergence of trapezoidal-rule quadrature

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