This document discusses Gaussian quadrature formulas, which approximate definite integrals of functions by using weighted sums of function values at specified points. It presents the one-point, two-point, and three-point Gaussian quadrature formulas. The one-point formula is exact for polynomials up to degree 1, the two-point formula is exact for polynomials up to degree 3, and the three-point formula is exact for polynomials up to degree 5. Examples are provided to demonstrate applying the formulas.

1. Vadodara Institute of Engineering
Active Learning Assignment
Sub :- Numerical And Statistical Methods (2140706)
Presented by:-
Maitree Patel 15CE048
Meet Patel 15CE049
Nikita Patel 15CE051
Computer Engineering - 1

2. Outline:
Gaussian Quadrature Formula
- One Point Gaussian Quadrature Formula
- Two Point Gaussian Quadrature Formula
- Three Point Gaussian Quadrature Formula

3. Gaussian Quadrature Formula
• An n point Gaussian quadrature formula is a quadrature formula
constructed to given an exact result of polynomials of degree 2n-1 or
less by a suitable of the points xᵢ and weights wᵢ for i = 1, 2, …..,n.
• Gauss quadrature formula can be expressed as
 
 

1
1 1
)()(
n
i
ii xfwdxxf

4. One Point Gaussian Quadrature Formula
• Consider a function f(x) over the interval [-1, 1] with sampling point x₁
and weight w₁.
• The one-point Gauss quadrature formula is
…………………..eq.(1)
This formula can be exact for polynomials of degree up to 2n-1 = 2(1)-
1=1, i.e., it is exact for f(x)=1 and x.
 11
1
1
)( xfwdxxf 

5. • Substituting f(x) in Eq.(1) successively,
…………..eq.(2)
…………………eq.(3)
1
1
1
)( wdxxf 
1
1
1
wx 
12 w
11
1
1
xwxdx 
11
1
1
2
2
xw
x


110 xw

6. This equation is known as one-point Gauss quadrature formula.
This Formula is exact for polynomial up to degree one.
12 w
10 x
 02)(........
1
1
fdxxfhence 

7. Two Point Gaussian Quadrature Formula
• Consider a function f(x) over the interval [-1, 1] with sampling points
x₁, x₂ and weights w₁, w₂ respectively.
• Two Point Gaussian Quadrature Formula is
• This formula can be exact for polynomials of degree up to 2n-1 =
2(2)-1=3, i.e., it is exact for f(x)=1,x, x²,

1
1
)( dxxf )()( 2211 xfwxfw 
3
x

1
1
1dx 21 ww 
)1.(........................... eq212 ww 
21
1
1
wwx 

8. ……………………eq.2
……………………eq.3
2211
1
1
2
2
xwxw
x


22110 xwxw 
2
2
21
2
1
1
1
2
xwxwdxx 
2
2
21
2
1
3
2
xwxw 
2
2
21
2
1
1
1
3
3
xwxw
x


2
3
21
3
1
1
3
xwxwdxx 
2211
1
1
xwxwxdx 

9. ……………………….eq.3
Solving eq. 1,2,3 and 4,
w₁ = w₂ = 1
This equation is known as Two Point Gaussian Quadrature Formula.
This Formula is exact for polynomial up to degree three.
2
3
21
3
1
1
1
4
4
xwxw
x


2
3
21
3
10 xwxw 
1
3
1
x

2
3
1
x

1
1
)( dxxf )
3
1
()
3
1
( ff 


10. Three Point Gaussian Quadrature Formula
• Consider a function f(x) over the interval [-1, 1] with sampling points x₁, x₂, x₃
and weights w₁, w₂, w₃ respectively.
• Three Point Gaussian Quadrature Formula is
• This formula can be exact for polynomials of degree up to 2n-1 = 2(2)-1=3, i.e., it
is exact for f(x)=1,x, x², , x⁴ and x⁵.

1
1
)( dxxf )()()( 332211 xfwxfwxfw 
3
x

1
1
1dx 321 www 
321
1
1
wwwx 
3210 www  )1.(................. eq

11. …………………….eq.2
……………………eq.3
332211
1
1
2
2
xwxwxw
x


3322110 xwxwxw 
3
3
32
2
21
2
1
1
1
2
xwxwxwdxx 
3
2
32
2
21
2
1
3
2
xwxwxw 
3
3
32
2
21
2
1
1
1
3
3
xwxwxw
x


3
3
32
3
21
3
1
1
1
3
xwxwxwdxx 
332211
1
1
xwxwxwxdx 

12. ……………………….eq.4
......……………………eq.5
………………………….eq.6
3
3
32
3
21
3
1
1
1
4
4
xwxwxw
x


3
3
32
3
21
3
10 xwxwxw 
3
4
32
4
21
4
1
1
1
5
5
xwxwxw
x


3
4
32
4
21
4
1
1
1
4
xwxwxwdxx 
3
4
32
4
21
4
1
5
2
xwxwxw 
3
5
32
5
21
5
1
1
1
5
xwxwxwdxx 
3
5
32
5
21
5
10 xwxwxw 

13. Solving eq. 1,2,3 and 4,5 and 6,
• This equation is known as Two Point Gaussian Quadrature Formula.
• This Formula is exact for polynomial up to degree 5.

1
1
)( dxxf )
5
3
(
9
5
)0(
9
8
)
5
3
(
9
5
fff 
9
5
1 w
9
8
2 w
9
5
3 w
5
3
1 x 02 x
5
3
3 x

14. Example :
• Evaluate
Solution: f(x)=
By the one-point Gaussian formula


1
1
2
1
1
dx
x
2
1
1
x


1
1
2
1
1
dx
x
 02 f








01
1
2
2

15. By the two-point Gaussian formula
= 1.5
By the three-point Gaussian formula


1
1
2
1
1
dx
x
)
3
1
()
3
1
( ff 
















3
1
1
1













3
1
1
1


1
1
2
1
1
dx
x
)
5
3
(
9
5
)0(
9
8
)
5
3
(
9
5
fff 
5833.1
5
3
1
1
9
5
01
1
9
8
5
3
1
1
9
5





































16. References
• http://numericalmethods.eng.usf.edu/topics/gauss_quadrature.html
• Book: Numerical Methods for engineers by Author : S C Chapra and R
P Canale : Publisher : McGrow Hill International Edition

17. THANK YOU