Vtyjkyuk

How many Gauss points are required if the Gauss quadrature rule should provide the exact value of the integral $I=int_-1^1f(x)dx$ for $f(x)=(x^2-1)^2$?

I am really not sure what theorem to use to solve this problem. What I can think of is a theorem about Gaussian quadrature with orthogonal polynomials as follows:

If a polynomial $p$ of degree $n+1$ is orthogonal to all polynomials of lower degree on the interval $[a,b]$ then it has $n+1$ distinct roots $x_i$ with $a<x_0<ldots<x_n<b$ and if one uses these roots to determine the weights $A_i$ in the approximate integration formula $int_a^bf(x)dxapproxsum_i=0^nA_if(x_i)$ so that it is exact for all polynomials of degree up to $n$, then it is in fact exact for all polynomials of degree up to $2n+1$

But somehow I still cannot relate this theorem to the problem.

Could anyone please lend some help?

Thanks.

As far is I know the correct formula for determining the number of Gauss points is given by:

$p + 1 = 2n$

or

$p = 2n-1$

where p is the degree of the polynomial and n are the number of Gauss points.

Since your problem involves a fourth degree polynomial, you need 5/2 gauss points. This problem would therefore require 3 integration points instead of 2:

$(4+1)/2 = 5/2$

I hope this might solve your problem. I tried it out on a simple fourth order polynomial which gave me the exact answer.