History of Nodes and weights of Gauss–Hermite quadrature

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when	who	what
2026-09-10 09:20	bmatschke	the differential operator is upright; d is not a variable	current reviewed
2026-09-05 11:27	bmatschke	the completeness note is read as part of a sentence and was a noun phrase with a telegraphic appendix; say which rules are here, what n indexes, and that each is listed in full
2026-09-04 14:24	bmatschke	who asked for a table is not a fact about the mathematics; the issue is answered in the issue
2026-09-02 20:41	bmatschke	the Sage program's sqrt(pi) was symbolic, so every weight printed as a multiple of sqrt(pi) rather than as the number written beside it; take the square root in the interval field. Also fix at most n-1, which is the floor for n distinct points
2026-09-02 13:41	zeta3 with claude (agent run 20260902T131953Z)	audit: the Definition says what is listed; that the nodes are the roots of H_n moves to the first comment
2026-09-02 13:37	zeta3 with claude (agent run 20260902T1319	Gauss-Hermite nodes and weights for n <= 30: roots of H_n isolated over Z[x], weights in ball arithmetic with sqrt(pi), every rule checked for its degree of exactness with a control before being sent
2026-09-02 13:37	zeta3	checking that this table can be written to
2026-09-02 13:37	zeta3 with claude (agent run 20260902T131953Z)	draft: nodes and weights of Gauss-Hermite quadrature, proposal 2 of BATCH-2026-09-02

What changed between 2026-09-05 11:27 and 2026-09-10 09:20

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 Title: Nodes and weights of Gauss–Hermite quadrature Definition: For $n\geq 1$ the Gauss–Hermite quadrature rule with $n$ points is the-  unique rule $\int_{-\infty}^{\infty} f(x)\,e^{-x^2}\,dx \approx \sum_{k=1}^{n} w_k-  f(x_k)$ that is exact for every polynomial $f$ of degree at most $2n-1$ CITE{Wiki}.+  unique rule $\int_{-\infty}^{\infty} f(x)\,e^{-x^2}\,\mathrm{d}x \approx \sum_{k=1}^{n}+  w_k f(x_k)$ that is exact for every polynomial $f$ of degree at most $2n-1$ CITE{Wiki}.   Listed are its nodes $x_1<\cdots<x_n$ and its weights $w_k$. Parameters:

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     CITE{AS}, the two equal because $H_n'=2nH_{n-1}$. Also $\dfrac{1}{w_k}=\sum_{j=0}^{n-1}\dfrac{H_j(x_k)^2}{2^j\,j!\,\sqrt{\pi}}$,     the Christoffel function at the node.-  formula-exactness: $\sum_{k=1}^{n} w_k x_k^m=\int_{-\infty}^{\infty}x^m e^{-x^2}\,dx$,+  formula-exactness: $\sum_{k=1}^{n} w_k x_k^m=\int_{-\infty}^{\infty}x^m e^{-x^2}\,\mathrm{d}x$,     which is $\Gamma\bigl(\frac{m+1}{2}\bigr)=\sqrt{\pi}\,\dfrac{(m-1)!!}{2^{m/2}}$     for even $m$ and $0$ for odd $m$, for $0\leq m\leq 2n-1$, and not for $m=2n$.

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     summing to $\sqrt{2\pi}$.   formula-normal: For $Y$ normally distributed with mean $\mu$ and variance $\sigma^2$,-    $\mathbb{E}\,f(Y)=\dfrac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{\infty} f(y)\,e^{-(y-\mu)^2/(2\sigma^2)}\,dy\approx\dfrac{1}{\sqrt{\pi}}+    $\mathbb{E}\,f(Y)=\dfrac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{\infty} f(y)\,e^{-(y-\mu)^2/(2\sigma^2)}\,\mathrm{d}y\approx\dfrac{1}{\sqrt{\pi}}     \sum_{k=1}^{n}w_k\,f\bigl(\sqrt{2}\,\sigma x_k+\mu\bigr)$ CITE{Wiki}.   formula-interlacing: 'The nodes of the rules with $n$ and $n-1$ points interlace: