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m Paragraph 2 included an equation for the variance of the binomial distribution. The variance was improperly described a s n(1-p) I amended it to read np(1-p) as it should be.
 
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About|the quadrature rule for approximating integrals|the implicit trapezoidal rule for solving initial value problems|Trapezoidal rule (differential equations)|the explicit trapezoidal rule for solving initial value problems|Heun's method}}
[[Image:trapezoidal rule illustration.png|right|thumb|The function ''f''(''x'') (in blue) is approximated by a linear function (in red).]]
In [[numerical analysis]], the '''trapezoidal rule''' (also known as the '''trapezoid rule''' or '''trapezium rule''') is a technique for approximating the [[integral|definite integral]]
 
:<math> \int_{a}^{b} f(x)\,dx.</math>
 
The trapezoidal rule works by approximating the region under the graph of the function
<math>f(x)</math> as a [[trapezoid]] and calculating its area. It follows that
 
:<math> \int_{a}^{b} f(x)\, dx \approx (b-a) \left[\frac{f(a) + f(b)}{2} \right].</math>
 
== 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.<ref name="cun02">{{Harv|Cruz-Uribe|Neugebauer|2002}}</ref>
 
Moreover, the trapezoidal rule tends to become extremely accurate when [[periodic function]]s are integrated over their periods, which can be [[#Periodic functions|analyzed in various ways]].<ref name="rs90">{{Harv|Rahman|Schmeisser|1990}}</ref><ref name="w02">{{Harv|Weideman|2002}}</ref>
 
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.
 
== Numerical implementation ==
 
=== Uniform grid ===
 
For a domain discretized into ''N'' equally spaced panels, or ''N''+1 grid points ''a'' = ''x''<sub>1</sub> < ''x''<sub>2</sub> < ... < ''x''<sub>N+1</sub> = ''b'', where the grid spacing is ''h''=(''b''-''a'')/''N'', the approximation to the integral becomes
 
:<math> \int_{a}^{b} f(x)\, dx \approx \frac{h}{2} \sum_{k=1}^{N} \left( f(x_{k+1}) + f(x_{k}) \right)</math> <math>{}= \frac{b-a}{2N}(f(x_1) + 2f(x_2) + 2f(x_3) + \ldots + 2f(x_N) + f(x_{N+1})).</math>
 
=== Non-uniform grid ===
When the grid spacing is non-uniform, one can use the formula
 
:<math> \int_{a}^{b} f(x)\, dx \approx \frac{1}{2} \sum_{k=1}^{N} \left( x_{k+1} - x_{k} \right) \left( f(x_{k+1}) + f(x_{k}) \right).</math>
 
==Error analysis==
 
The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result:
 
:<math> \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]</math>
 
There exists a number ''ξ'' between ''a'' and ''b'', such that<ref>{{harvtxt|Atkinson|1989|loc=equation (5.1.7)}}</ref>
 
:<math> \text{error} = -\frac{(b-a)^3}{12N^2} f''(\xi)</math>
 
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:<ref name="w0223">{{Harv|Weideman|2002|loc=p. 23, section 2}}</ref>
# [[Fourier series]]
#[[Residue calculus]]
#[[Euler–Maclaurin summation formula]]:<ref>{{harvtxt|Atkinson|1989|loc=equation (5.1.9)}}</ref><ref>{{harvtxt|Atkinson|1989|loc=p. 285}}</ref>
An asymptotic error estimate for ''N'' → ∞ is given by
:<math> \text{error} = -\frac{(b-a)^2}{12N^2} \big[ f'(b)-f'(a) \big] + O(N^{-3}). </math>
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.<ref name="rs90" />
 
=== Periodic functions ===
The trapezoidal rule often converges very quickly for periodic functions.<ref name="w02" /> This can be explained intuitively as:
: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.<ref name="w0223" />
More detailed analysis can be found in.<ref name="rs90" /><ref name="w02" />
 
=== "Rough" functions ===
{{Expand section|date=January 2010}}
For various classes of functions that are not twice-differentiable, the trapezoidal rule has sharper bounds than [[Simpson's rule]].<ref name="cun02" />
 
==See also==
* [[Gaussian quadrature]]
* [[Newton–Cotes formulas]]
* [[Rectangle method]]
* [[Romberg's method]]
* [[Simpson's rule]]
 
==Notes==
<references/>
 
==References==
{{wikibooks|A-level Mathematics|C2/Integration#Trapezium Rule|Trapezium Rule}}
{{refbegin}}
* {{Citation | last1=Atkinson | first1=Kendall E. | title=An Introduction to Numerical Analysis | publisher=[[John Wiley & Sons]] | location=New York | edition=2nd | isbn=978-0-471-50023-0 | year=1989}}.
* {{ Citation | first1 = Qazi I. | last1 = Rahman | first2 = Gerhard | last2 = Schmeisser | title = Characterization of the speed of convergence of the trapezoidal rule | journal = Numerische Mathematik | issn = 0945-3245 | volume = 57 | issue = 1 |date=December 1990 | doi = 10.1007/BF01386402 | pages = 123–138 }}
* {{citation | first=Richard L. | last=Burden | coauthors=J. Douglas Faires | year=2000 | title=Numerical Analysis | edition=7th | publisher=Brooks/Cole | isbn=0-534-38216-9}}.
* {{ Citation | title = Numerical Integration of Periodic Functions: A Few Examples | first = J. A. C. | last = Weideman | journal = [[The American Mathematical Monthly]] | volume = 109 | issue = 1 |date=January 2002 | pages = 21–36 | doi = 10.2307/2695765 | jstor = 2695765 }}
* {{ Citation | journal = [http://jipam.vu.edu.au/ Journal of Inequalities in Pure and Applied Mathematics] | volume = 3 | issue = 4 | year = 2002 | title = Sharp Error Bounds for the Trapezoidal Rule and Simpson's Rule | first1 = D. | last1 = Cruz-Uribe | first2 = C.J. | last2 = Neugebauer | url = http://www.emis.de/journals/JIPAM/images/031_02_JIPAM/031_02.pdf }}
{{refend}}
 
==External links==
*[http://math.fullerton.edu/mathews/n2003/TrapezoidalRuleMod.html Trapezoidal Rule for Numerical Integration]
*[http://dedekind.mit.edu/~stevenj/trapezoidal.pdf Notes on the convergence of trapezoidal-rule quadrature]
 
{{DEFAULTSORT:Trapezoidal Rule}}
[[Category:Numerical integration (quadrature)]]

Latest revision as of 17:10, 22 April 2014

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