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Gauss quadrature Scheme


1
n
Gauss Quadrature formulae approximates the integral 
 òf(x) dx in the form 
S
 Wi f(xi)
-1
i = 0

where n is an integer has to be taken depending on the accuracy of the solution needed, Wi are the weights to be determined and xi are the unknown points where the function values are to be used. There are many ways to find weights Wi and nodal points xi. Few of the important methods among them are Gauss - Legendre, Gauss - Chebyshev, Gauss - Laguere and Gauss - Hermite Quadrature formulae.
1. Gauss - Legendre formula:

In this method f(x) is taken as xk for k = 0, 1, . . ., 2(n+1)-1 and is integrated between -1 and 1 to obtain a system of non-linear equations of order 2(n+1). By solving this system we can obtain the unknowns Wi and Xi. Since f(x) is taken as xk where k takes values upto 2(n+1)-1 the resultant quadrature rule can integrate exactly any polynomial of the degree lessthan 2(n+1).

For n = 1, the quadrature formula is 

1
n
1
  ò f(x) dx =
S
 Wi f(xi) = 
S
 Wi f(xi)
-1
i = 0
i = 0

 
   
 
= 
 W0 f(x0) + W1 f(x1)   for n = 1
   
1    
if f(x) = 1 ò f(x) dx =
2
  =    W0 + W1
-1    

 
1    
if f(x) = x ò x dx =
0
  =    W0 x0+ W1x1
-1    

 
1    
if f(x) = x2 ò x2 dx =
2/3
  =    W0x02 + W1x12
-1    

 
1    
if f(x) = x3 ò x3 dx =
0
  =    W0x03 + W1x13
-1    

 
Þ
W0
 = -
x1
W0
 = -
x13
W1
x0
W1
x03
Þ
x1
 = 
x13
Þ
  
x12 = x02
  
x0
x03
    

(x1 - x0)(x1 + x0) = 0

 Þx1 = -x0ÞW0 = W1= 1 & x0 = -x1 = 1/Ö3


1    
so for n = 1 ò f(x) dx =
f(-1/Ö3) + f(1/Ö3)
    
-1    
This formula can integrate exactly any polynomial of degree upto 2(1 +1)-1 =3.

For any arbitrary 'n' the above method will have W0, W1, . . . , Wn & x0, x1, . . ., xn that is 2(n+1) unknowns and equal number of non-linear equations of the form

W0 + W1 + . . . + Wn = 2
W0x0 + W1x1 + . . . + Wnxn = 0
W0x02 + W1x12 + . . . + Wnxn2 = 2/3
.
.
.
W0x02n+1 + W1x12n+1 + . . . + Wnxn2n+1 = 0
In principle these 2(n+1) non-linear equations have to be solved to get the 2(n+1) unknowns. However, due to the non-linearity of the system the conventional methods to solve the linear systems does not work here and hence we need an alternate means to find Wi & xi for large n. 
It is observed  that the nodes xk's are the roots of the Legendre polynomial Pn+1(x) = 0 where
Pn+1(x) = 
1
dn+1
 [(x2 - 1)n+1],
n=0,1. . .
2n+1(n+1)!
  
dxn+1
The nodes and the corresponding weights for the Gauss-Legendre integration method are given as
n
nodes xk
weights lk
 
1
± 0.5773502692
1.0000000000
 
2
0.0000000000
0.888888889
± 0.7745966692
± 0.555555556
 
3
± 0.3399810436
0.6521451549
± 0.8611363116
0.3478548451
 
4
0.0000000000
0.5688888889
± 0.5384693101
0.4786286705
± 0.9061798459
0.2369268851
 
 5
± 0.2386191861
0.4679139346
± 0.6612093865
0.3607615730
± 0.9324695142
0.1713244924
     

Work out Gauss-Legendre here :

Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting
Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
Numerical Solution of Partial Differential Equations