I want to find a quadrature approximation of $\int f(x)e^{-\frac{x^2}{2}}$dx. Using the Hermite Polynomials 1, x, $x^2-1$, $x^3-3x$, $x^4 - 6x^2 + 3$. that is exact for polynomials up to order 7.
Now I know by definition it will be exact for polynomials up to order 7 as they are exact up to order 2n - 1, for degree n polynomials. Which holds as our largest degree is 4, and 2*4 - 1 = 7.
I know I want to find some weight function $\alpha_j$ such that $\int f(x)w(x)$ = $\sum^n_{j=1}\alpha_jf(x_j)$.
So I take my set of Hermite polynomials to be my f(x), but how do I find $\alpha_j$. I imagine it has something to do with the Hermite polynomials being orthonormal but I'm not sure of the next step. Any help appreciated.
I assume your integration range is the entire real line $(-\infty,\infty)$. You can integrate all $4$ Hermite polynomials exactly using $\int_{-\infty}^\infty H_n(x)e^{-\frac{1}{2}x^2}dx=\sqrt{2\pi}\delta_{n,0}$. By expanding the integral into the sum $$ \sum_{i=0}^n \alpha_i H_n(x_i)=\sqrt{2\pi}\delta_{n,0}. $$ Choosing the $x_i$ as the roots of $H_4(x)$ ie, $x\in\{x_i|H_4(x)=0\}$ gives a matrix equation you can solve for $\{\alpha_i\}$ $$ \begin{bmatrix} 1 & 1 & 1 & 1 \\ H_1(x_0) & H_1(x_1) & H_1(x_2) & H_1(x_3) \\ H_2(x_0) & H_2(x_1) & H_2(x_2) & H_2(x_3) \\ H_3(x_0) & H_3(x_1) & H_3(x_2) & H_3(x_3) \\ \end{bmatrix} \begin{bmatrix} \alpha_0\\ \alpha_1\\ \alpha_2\\ \alpha_3 \end{bmatrix}= \begin{bmatrix} \sqrt{2\pi} \\ 0 \\0 \\0 \end{bmatrix} $$