From 21c5c13fb7ecbb9e029b765f840af0b9d291017f Mon Sep 17 00:00:00 2001 From: Andrew Cumming Date: Sun, 17 Sep 2023 12:55:55 -0400 Subject: [PATCH] Adds link for Gauss Laguerre to integration.md --- integration.md | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/integration.md b/integration.md index 4f881c7..ac83d1a 100644 --- a/integration.md +++ b/integration.md @@ -145,14 +145,20 @@ Modify your code to use the Gauss-Hermite coefficients and check that you can ge Hint: If you want to use `scipy.integrate.quad` again to get the value of the integral as a comparison, note that you can give it limits of $-\infty$ to $+\infty$ using `-np.inf` and `np.inf`. ``` +Another example is $W(x)=e^{-x}$ with integration limits $0$ to $\infty$. In this case, we need Gauss-Laguerre integration -- see [`numpy.polynomial.laguerre.laggauss`](https://numpy.org/doc/stable/reference/generated/numpy.polynomial.laguerre.laggauss.html). + + ## Integration challenge ```{admonition} Exercise: Average velocity of the Maxwell-Boltzmann distribution. Use Simpson's rule, Gaussian quadrature, and the general purpose integrator [`scipy.integrate.quad`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.quad.html) to evaluate the average velocity $\langle\left|v\right|\rangle$ for the [Maxwell-Boltzmann distribution](https://en.wikipedia.org/wiki/Maxwell–Boltzmann_distribution). + For each method, check the numerical error comparing to the analytic result. How many points do you need to get to $0.1$% accuracy? For Simpson's rule you can use your own implementation from above or you could try [`scipy.integrate.simpson`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.simpson.html)). + +For Gaussian quadrature, try both Gauss-Hermite and Gauss-Laguerre. Which one is best? ```