Trapezium formula

The special case of the Newton–Cotes quadrature formula in which two nodes are taken:

\begin{equation} \label{eq1} \int\limits _ { a } ^ { b } f ( x) dx \cong \ { \frac{b - a }{2} } [ f ( a) + f ( b)]. \end{equation}

If the integrand $ f $ differs strongly from a linear function, then formula \eqref{eq1} is not very exact. In this case the interval $ [ a, b] $ is divided into $ n $ subintervals $ [ x _ {i} , x _ {i + 1 } ] $, $ i = 0 \dots n - 1 $, of length $ h = ( b- a)/n $, and for the calculation of the integral over $ [ x _ {i} , x _ {i + 1 } ] $ one uses the trapezium formula

$$ \int\limits _ {x _ {i} } ^ { {x _ {i + 1} } } f ( x) dx \cong \ { \frac{h}{2} } [ f ( x _ {i} ) + f ( x _ {i + 1 } )]. $$

Summation of the left- and right-hand sides of this approximate equality with respect to $ i $ from 0 to $ n - 1 $ leads to the composite trapezium formula:

\begin{equation} \label{eq2} \int\limits _ { a } ^ { b } f ( x) dx \cong \ h \left [ { \frac{f ( a) }{2} } + f ( x _ {1} ) + \dots + f ( x _ {n - 1 } ) + { \frac{f ( b) }{2} } \right ] , \end{equation}

where $ x _ {j} = a + jh $, $ j = 0 \dots n $. The quadrature formula \eqref{eq2} is also called the trapezium formula (without adding the word composite). The algebraic degree of accuracy of the quadrature formula \eqref{eq2}, as well as of \eqref{eq1}, is equal to 1. The quadrature formula \eqref{eq2} is exact for the trigonometric functions

$$ \cos \frac{2 \pi }{b - a } kx,\ \ \sin \frac{2 \pi }{b - a } kx,\ \ k = 0 \dots n - 1. $$

In the case when $ b - a = 2 \pi $, formula \eqref{eq2} is exact for all trigonometric polynomials of order not exceeding $ n - 1 $; furthermore, its trigonometric degree of accuracy is equal to $ n - 1 $.

If the integrand $ f $ is twice-continuously differentiable on $ [ a, b] $, then the error $ R ( f ) $ of the quadrature formula \eqref{eq2}, that is, the difference between the integral and the quadrature sum, is given by

$$ R ( f ) = \ - { \frac{b - a }{12} } h ^ {2} f ^ { \prime\prime } ( \xi ), $$

where $ \xi $ is a point of $ [ a, b] $.

Comments

The complete formula

$$ \int\limits_{a}^{b} f(x) dx = h \left [ \frac{f(a)}{2} + f( x _ {1} ) + \dots + f( x _ {n-1} ) + \frac{f(b)}{2} \right ] - \frac{b-a}{12} h ^ {2} f ^ { \prime\prime } ( \xi ) $$

is often referred to as the trapezoidal rule.

References

[a1]	F.B. Hildebrand, "Introduction to numerical analysis" , Dover, reprint (1974) pp. 95ff
[a2]	W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, "Numerical recipes" , Cambridge Univ. Press (1986) pp. 105ff