ppt on Numerical integration and Gaussian integration one point, two point and three point method.

1. Numerical Integration
Gaussian integration one point, two point and three point method.

2. What is Integration?
 The process of measuring the area under a curve.
Where: f(x) is the integrand
a= lower limit of integration
b= upper limit of integration

b
a
dxxfI )(

3. Gaussian quadrature
 In numerical analysis, a quadrature rule is an approximation of the definite
integral of a function, usually stated as a weighted sum of function values at
specified points within the domain of integration.
 An n-point Gaussian quadrature rule is a quadrature rule constructed to yield an
exact result for polynomials of degree 2n − 1 or less by a suitable choice of the
points xi and weights wi for i = 1, ..., n. The domain of integration for such a rule is
conventionally taken as [−1, 1], so the rule is stated as
 


n
i
ii xfwdxxf
0
1
1
)()(

4. One-Point Gaussian Quadrature Rule
 Consider a function f(x) over interval [-1,1] with sampling point The point
one formula is
 The formula of one point Gaussian quadrature rule,


1
1
11 )()( xfwdxxf


1
1
)0(2)( fdxxf
11, wx

5. Two-Point Gaussian Quadrature Rule
 Consider a function f(x) over interval [-1,1] with sampling point and
The two point formula is,
 The formula of one point Gaussian quadrature rule,
)()()( 22
1
1
11 xfwxfwdxxf 














1
1 3
1
3
1
)( ffdxxf
21, xx
21,ww

6. Three-Point Gaussian Quadrature Rule
 Consider a function f(x) over interval [-1,1] with sampling point and
The two point formula is,
 The formula of Three point Gaussian quadrature rule,
)()()()( 3322
1
1
11 xfwxfwxfwdxxf 

















5
3
9
5
)0(
9
8
5
3
-
9
5
)(
1
1
fffdxxf
321 ,, xxx
321 ,, www

7. Example 1
 Evaluate by one point , two point & Three point Gaussian
quadrature.
 Here,
 Using one point method
 Using Two point Method
dx
x

1
1
2
1
1
2
1
1
)(
x
xf


 
2
01
12
)0(2



 fdxxf
1
1
)(














1
1 3
1
3
1
)( ffdxxf
22
3
11
1
3
11
1














   3
1
3
1 1
1
1
1




5.14
6 

8.  Three point method

















5
3
9
5
)0(
9
8
5
3
-
9
5
)(
1
1
fffdxxf
   
5833.1
36
57
36
3225
9
8
8
5
9
2*5
9
8
1
1
1
1
9
5
2
5
3
2
5
3









































9. Example 2
 Evaluate by one point ,two point ,three point Gaussian formula.
Let
where a=0,b=1
now
 dt
t 
1
0
1
1





 





 

22
ab
x
ab
t
2
1
2
 xt dxdt 2
1
3
2
)(
1
)(
1
1
)(
1
1
)(
2
3
2
2
1
2








x
xf
xf
xf
t
tf
x
x

10.  t=0
 One point Method
 t=1
1
0
2
1
2
2
1
2




x
x
x
1
1
2
1
2
2
1
2



x
x
x
 




1
0
1
1
23
2
1
1 dx
x
dt
t


1
1
3
1
)( dxxf x


1
1
)0(2)( fdxxf
 
6666.0
3
2
3
12




11.  Two point method
 Three point method














1
1 3
1
3
1
)( ffdxxf
   
6923.0
331
3
331
3
3
1
3
1
3
1
3
1


























 
5
3
9
5
)0(
9
8
5
3
-
9
5
)(
1
1
fffdxxf
67167.0
3
5
3
1
9
5
27
8
3
5
3
1
9
5














