History of Nodes and weights of Gauss–Lobatto quadrature

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when	who	what
2026-09-10 09:20	bmatschke interactive	the differential operator is upright; d is not a variable	current reviewed
2026-09-05 11:27	bmatschke interactive	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
2026-09-03 03:00	zeta3 table-critique@1.37+04b9737f	act on the critique: the interior nodes interlace with the (n-1)-point Gauss rule, not the n-point one (checked for every n); n=9 central weight in lowest terms; drop the comment restating the symmetry and the issue-tracker citation; link the Kronrod table; a hundred digits, every ball supports them
2026-09-03 02:39	zeta3 with claude (agent run 20260903T0221 table-build@1.42+4513583d	Gauss-Lobatto nodes and weights for 2 <= n <= 30: roots of P_{n-1}' isolated over Q[x], endpoints and rational weights exact, weights in ball arithmetic, every rule checked for its degree of exactness with a control before being sent
2026-09-03 02:39	zeta3 table-build@1.42+4513583d	checking that this table can be written to
2026-09-03 02:38	zeta3 with claude (agent run 20260903T022137Z) table-build@1.42+4513583d	draft: nodes and weights of Gauss-Lobatto quadrature, proposal 5 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–Lobatto quadrature Definition: For $n\geq 2$ the Gauss–Lobatto quadrature rule with $n$ points is the-  unique rule $\int_{-1}^{1} f(x)\,dx \approx \sum_{k=1}^{n} w_k f(x_k)$ with $x_1=-1$-  and $x_n=1$ that is exact for every polynomial $f$ of degree at most $2n-3$ CITE{Wiki}.-  Listed are its nodes $x_1<\cdots<x_n$ and its weights $w_k$.+  unique rule $\int_{-1}^{1} f(x)\,\mathrm{d}x \approx \sum_{k=1}^{n} w_k f(x_k)$+  with $x_1=-1$ and $x_n=1$ that is exact for every polynomial $f$ of degree at most+  $2n-3$ CITE{Wiki}. Listed are its nodes $x_1<\cdots<x_n$ and its weights $w_k$. Parameters:   n:

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     at the endpoints $P_{n-1}(\pm1)^2=1$, so $w_1=w_n=\dfrac{2}{n(n-1)}$.   formula-secondary: $w_k=\dfrac{q(x_k)}{\omega_n'(x_k)}$, where $\omega_n=(x^2-1)P_{n-1}'=\dfrac{n(n-1)}{2n-1}\,(P_n-P_{n-2})$-    is the polynomial whose roots are the nodes and $q(x)=\int_{-1}^{1}\dfrac{\omega_n(t)-\omega_n(x)}{t-x}\,dt+    is the polynomial whose roots are the nodes and $q(x)=\int_{-1}^{1}\dfrac{\omega_n(t)-\omega_n(x)}{t-x}\,\mathrm{d}t     =\dfrac{n(n-1)}{2n-1}\,(q_n-q_{n-2})$ with $q_n$ the HREF{Secondary_polynomials_of_the_Legendre_polynomials}[secondary     polynomial of $P_n$].-  formula-exactness: $\sum_{k=1}^{n} w_k x_k^m=\int_{-1}^{1}x^m\,dx=\dfrac{1+(-1)^m}{m+1}$+  formula-exactness: $\sum_{k=1}^{n} w_k x_k^m=\int_{-1}^{1}x^m\,\mathrm{d}x=\dfrac{1+(-1)^m}{m+1}$     for $0\leq m\leq 2n-3$, and not for $m=2n-2$.-  formula-remainder: $\int_{-1}^{1}f(x)\,dx-\sum_{k=1}^{n}w_kf(x_k) =-\dfrac{n(n-1)^3\,2^{2n-1}\,((n-2)!)^4}{(2n-1)\,((2n-2)!)^3}\,f^{(2n-2)}(\xi)$+  formula-remainder: $\int_{-1}^{1}f(x)\,\mathrm{d}x-\sum_{k=1}^{n}w_kf(x_k) =-\dfrac{n(n-1)^3\,2^{2n-1}\,((n-2)!)^4}{(2n-1)\,((2n-2)!)^3}\,f^{(2n-2)}(\xi)$     for some $\xi\in(-1,1)$ CITE{AS}.   formula-symmetry: $x_{n+1-k}=-x_k$, $w_{n+1-k}=w_k$, $w_k>0$ and $\sum_{k=1}^{n}w_k=2$.

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     nodes are the roots of the Jacobi polynomial $P^{(1,1)}_{n-2}$, orthogonal on     $[-1,1]$ for the weight $1-x^2$.-  formula-lagrange: $w_k=\int_{-1}^{1}\ell_k(x)\,dx$ with $\ell_k(x)=\prod_{j\neq+  formula-lagrange: $w_k=\int_{-1}^{1}\ell_k(x)\,\mathrm{d}x$ with $\ell_k(x)=\prod_{j\neq     k}\frac{x-x_j}{x_k-x_j}$ the Lagrange basis polynomial of the nodes, as for every     interpolatory rule.