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Gauss Chebyshev Scheme
The integrand in this method is f(x)/Ö(1-x2), that is
1
n
  
ò 1/Ö(1-x2)f(x) dx =
S
 lkfk  
-1
 k=0
  
is called the Gauss-Chebyshev quadrature scheme. These methods are exact for polynomials of degree upto 2n+1. The nodes xk's are found to be the roots of the Chebyshev polynomials
Tn+1(x) = cos((n+1)cos-1x) = 0
Thus, we get
xk = cos
(
(2k + 1)p
)
,
k = 0,1,. . .,n
2n + 2
Taking n = 2, we have
1    
ò 1/Ö(1-x2)f(x) dx = l0f(x0) + l1f(x1) + l2f(x2)  
-1    
Since the method is to be exact for f(x) = xi, i = 0(1)5, we get the system of equations


W0 + W1 + . . . + Wn = p
W0x0 + W1x1 + . . . + Wnxn = 0
W0x02 + W1x12 + . . . + Wnxn2 = p/2
.
.
.
W0x02n+1 + W1x12n+1 + . . . + Wnxn2n+1 = 0

We obtain 
xk = cos(2k +1)p
k = 0,1,. . .,n
or
x0 = Ö3/2 x1 = 0 xk = -Ö3/2
substituting the values of x0, x1 and x2 we get
l0 = l1 = l2 = p/3
Thus we get
1    
ò 1/Ö(1-x2)f(x) dx = p/3 [f(Ö3/2) + f(0) + f(-Ö3/2)]  
-1    
with the error term
R5 = C/6! fvi(x),
-1<x<1
where 
1    
C =  ò x6/Ö(1-x2) dx - ( l0x06+ l1x16 + l2x26 )    = p/32
-1    
It may be verified that in this method all the weights lk's are equal and are given by
lk = p/(n+1),
k = 0,1,. . .,n
Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting
Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
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