Tuesday, 10 December 2013

Simpson's Rule

Approximate evaluation of a definite integral :
Simpson's Rule .

In many cases , a definite integral can not be obtained either because the quantity to be integrated can not be expressed as a mathematical function , or because the indefinite integral of the unction itself can not be determined directly . In such cases formula of approximation are used . One such important formula is Simpson's Rule . By this rule the definite integral of any function is expressed in terms of the individual values of any number of ordinates within the interval , by assuming that the function within each of the small ranges into which the whole interval may be divided can be represented to a sufficient degree of approximation by a parabolic function .
 


In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:
 \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{6}\left[f(a) + 4f\left(\frac{a+b}{2}\right)+f(b)\right].
Simpson's rule also corresponds to the three-point Newton-Cotes quadrature rule.



Suppose that the interval [a, b] is split up in n subintervals, with n an even number. Then, the composite Simpson's rule is given by
\int_a^b f(x) \, dx\approx 
\frac{h}{3}\bigg[f(x_0)+2\sum_{j=1}^{n/2-1}f(x_{2j})+
4\sum_{j=1}^{n/2}f(x_{2j-1})+f(x_n)
\bigg],
where x_j=a+jh for j=0, 1, ..., n-1, n with h=(b-a)/n; in particular, x_0=a and x_n=b. The above formula can also be written as
\int_a^b f(x) \, dx\approx\frac{h}{3}\bigg[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+\cdots+4f(x_{n-1})].

The error committed by the composite Simpson's rule is bounded (in absolute value) by
\frac{h^4}{180}(b-a) \max_{\xi\in[a,b]} |f^{(4)}(\xi)|,
where h is the "step length", given by h=(b-a)/n.     

In other words this Simpson's Rule can be written as  :

h/3 [sum of the extreme ordinates  +  2.sum of the remaining odd ordinates   +  4.sum of the even ordinates] 





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