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 $\int_{a}^{b}f(x)dx\approx\sum_{i=0}^{n}A_if(x_i)$ 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.
$\endgroup$ 71 Answer
$\begingroup$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.
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