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 
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] 

No comments:

Post a Comment

Our Latest Post

How to find log (alpha+ i beta), Where alpha and beta are real

Here is the video to show the details of solving this problem. It is an important problem for basic understanding about the logarithm of re...

Popular Post