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
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
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