Compute w_i and x_i (i=0,1,2) such that the formula

is exact is f is a polynomial of degree equal or less than 5. Give w_i and x_i to 12 significant digits.

In Mathematica, we can solve this problem by the following steps:

1) get the orthgonal polynomials

Orthogonalize[{1,x,x^2,x^3},Integrate[Cos[x*Pi/2]#1 #2,{x,-1,1}]&]

2) find roots for the 4th polynomial you got, which are x_i's

NumberForm[NRoots[-(((384-48 [Pi]^2+[Pi]^4) x)/([Pi]^2 (-8+[Pi]^2)))+x^3==0,x],12]

3) compute w_i with integration of cos(x*pi/2)*L_i(x) in the interval (-1,1), L_i(x) is the ith lagrange polynomial with respect to the nodes you found in step 2.

Results FYI,

x_0=-0.644641970907

x_1=0

x_2=0.644641970907

w_0=0.290197016802

w_1=0.692845511132

w_2=0.290197016802

 

Is it possible to solve the problem by introducing a polynomial system of equations? It is essential to recognize it as a gaussian quadrature problem and derive a gaussian quadrature formula. In general, the nonlinear system consists of those equations cannot be guarranteed to be solved in closed form in Mathematica.

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