Numerical Calculation
Part 3: Integration (Simpson’s method)
Dr.Entesar Ganash
Simpson’s Rule

Simpson’s rule is a method of numerical integration, which is a good deal more
accurate than the Trapezoidal rule.

It also divides the area under the function into vertical strips, but instead of
joining the points f ( xi ) with straight lines, every set of three such successive
points are fitted with a parabola.
Simpson’s Rule
f ( xi )
Simpson’s Rule

Dividing [a,b] into EVEN number n of sub intervals of equal length
h=(b-a)/n
a  x0
Simpson’s Rule

Simpson’s rule is
b
h
a f ( x) dx  3  f (a)  4 f ( x1 )  2 f ( x2 )  4 f ( x3 )  2 f ( x4 )  .....  4 f ( xn1 )  f (b)
Odd terms
Even terms
Example
Write a program to find the value of the following function using
Simpson's method, take n=60
1
2
x
 dx
0
Solving
program Simpson
Implicit none
Integer :: i ! counter
Real :: h, sum, x
Integer :: n ! the number of subintervals (EVEN No)
Real:: a,b
! the start & end integration term
a=0.0
b=1.0
n=60
h = (b-a)/real(n)
sum = f(a) + f(b)
Do i=1,n-1
x = a + i*h
if (i/2*2.NE.i)Then
sum = sum + 4*f(x) !ODD terms
Else
sum = sum + 2*f(x) !EVEN terms
End if
End Do
sum = h*sum/3.0
Print *,'The numerical Simpson value=',sum
CONTAINS
function f(t)
real ::f, t
f=t**2
end function f
End program Simpson
1
continue
Solving
The result:
Exercise
Write a program to find the value of the following function using Simpson's
method, take n=64
5
sin 2 ( x)
1 x dx
References
•Hahn, B.D., 1994, Fortran 90 For Scientists and Engineers, Elsevier.
•http://en.wikipedia.org/wiki/Simpson%27s_rule
•http://www.uobabylon.edu.iq/eprints/publication_4_13618_553.pdf
•Univ.,