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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.

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  • $\begingroup$ You have it: Gauss Ian quadrature uses exactly those nodes and weights. Then your polynomial is degree 4 so... $\endgroup$ – Ian Apr 27 '16 at 10:54
  • $\begingroup$ @Ian so it has 5 distinct roots. So 5 Gauss points? $\endgroup$ – user71346 Apr 27 '16 at 10:59
  • $\begingroup$ No, think about that $2n+1$ thing... $\endgroup$ – Ian Apr 27 '16 at 13:07
  • $\begingroup$ @Ian. If $n$ is 3, then $2(3)+1=7$ points? But $2n+1$ is the degree of the polynomials, not the number of Gauss points? $\endgroup$ – user71346 Apr 27 '16 at 13:32
  • $\begingroup$ With $n+1$ points you exactly integrate polynomials of degree $2n+1$, that is the important message of your paragraph above. $\endgroup$ – Ian Apr 27 '16 at 13:39
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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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