[SOLVED] Quadrature Degree of Precision on [-1,1] with n Even

**lvleph** · April 13th 2010, 09:38 AM

Prove that if the formula
$ \int_{-1}^1\! f(x)\, dx \approx \sum_{i=0}^n A_i f(x_i)\; (n\text{ is even}) $
is exact for all polynomials of degree $n,$ and if the nodes are symmetrically placed about the origin, then the formula is exact for all polynomials of degree $n+1.$

So, clearly if $f_n \in P_n$ then
$ p_n = \int_{-1}^1\! f_n(x)\, dx - \sum_{i=0}^n A_i f_n(x_i) = 0. $
But, obviously I am suppose to show that if $f_{n+1} \in P_{n+1}$ then
$ p_{n+1} = \int_{-1}^1\! f_{n+1}(x)\, dx - \sum_{i=0}^n A_i f_{n+1}(x_i) = 0. $
[strike]So, I was thinking that I had to show that $p_{n+1}$ has $n + 2$ roots and therefore $p_{n+1} \equiv 0$ . However, I cannot see how to do this.[/strike]

EDIT: I figured out a lot more. If $f_{n+1} \in P_{n+1}$ where $n$ is even then $\int_{-1}^1\! f_{n+1}(x)\, dx = 0$ Thus we see that $p_{n+1} = -\sum_{i=0}^n A_i f(x_i)$ . On the other hand, the $x_i$ are evenly distributed about zero and $f_{n+1}(x)$ is an odd function. Thus, $p_{n+1} = 0$ as long as the $A_i$ cancel out. Now I just have to ensure that these will cancel.

EDIT2: So I think there is a problem here. Since $n$ is even we have an odd number of points. Thus, one point is centered at zero, while the other points may result in a cancellation. However, there is no guarantee that the point at zero results in a zero function evaluation. hmmm.

EDIT3: By symmetry we know that the $A_i$ cancel. However, I am still stuck with the term at 0.

EDIT4: Wow! This is much simpler than I thought. We can think of a polynomial of degree $n+1$ , call it $f(x)$ , as the linear combination of lower degree monomials and the monomial $x^{n+1}$ . Thus $\int_{-1}^1\! f(x)\, dx = \sum_{i=0}^{n+1} c_i x^i\, dx$ where $c_i \in \mathbb{R}/\{0\}$ . Now if we look at the error $e = \int_{-1}^1\! f(x)\, dx - \sum_{i=0}^n A_i f(x_i)$ we see that this can be simplified to
$e = \int_{-1}^1\! c_{n+1} x^{n+1}\, dx + \int_{-1}^1\! p_n(x)\,dx - \sum_{i=0}^n A_i c_{n+1} x_i^{n+1} - \sum_{i=0}^n A_i p_n(x_i)$
where $p_n(x) \in P_n$ . But, we know that the quadrature for polynomials of degree n or less must be exact so
$e = c_{n+1}\left[\int_{-1}^1\! x^{n+1}\, dx - \sum_{i=0}^n A_i x_i^{n+1}\right]$ .
By symmetry and the fact that $x^{n+1}$ is an odd function the error further reduces to
$e = c_{n+1} A_{n/2 + 1} 0^{n+1} = 0$ .
Therefore, the quadrature is exact for polynomials of degree $n+1$ or less.

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