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I'm trying to learn Gauss Hermite Integration and was manually try to calculate the value of integral of $\frac{1}{1+x^2}$ from $-\infty$ to $+\infty$ The exact answer is simply $\pi$ ($\approx$ 3.14). But I keep getting answers that are a bit far off even with 5 nodes. Below are my calculations, could somebody please point anything obviously wrong or may be the method is more accurate for certain kind of functions.

Edit: even with 100 nodes, I only got as close as 2.93.

I got weights from - http://www.efunda.com/math/num_integration/findgausshermite.cfm

Gauss Hermite Quad of 1/(1+x^2)

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If you look at this description of Hermite-Gauss quadrature on MathWorld, its error term is $$ \frac{n!\sqrt\pi}{2^n(2n)!}f^{(2n)}(\xi), $$ where $\xi$ is some point in the region of integration.

When you integrate your function $$ \frac{e^{x^2}}{1+x^2}, \qquad I = \int \frac{e^{x^2}}{1+x^2}W(x)\,dx, \quad W(x) = e^{-x^2},$$ its $2n$-th derivate is extremely large, due to the $e^{x^2}$ term that you included to cancel out the weighting function $W(x)=e^{-x^2}$. Another way to look at this is that with these interpolating polynomials and this weight function, the function $(1+x^2)^{-1}$ is not easy to approximate: it doesn't decay nearly fast enough as $x\to\pm\infty$ for the weighting function to make sense. There is a mismatch between how quickly the weighting function goes to zero and your function.

Compare this, for example, with integrating $$ \int\frac{e^{-x^2}}{1+x^2}\,dx = \int \frac{1}{1+x^2}\,W(x)\,dx, $$ where the integrand $(1+x^2)^{-1}$ doesn't have such quickly growing derivatives, and for $n=5$ the approximate integral is $1.36343\ldots$, which is off by only $1.5\%$.

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