Christoffel coefficients (or numbers)

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1: 3.5 Quadrature

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3.5.18 $$w_k={\int }_a^b\frac{p_n(x)}{(x-x_k)p_n^{\prime }(x_k)}w(x)dx.$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $w$: weight, $w_k$: weights and $p_n$: set of monic polynomials

Referenced by:

§18.2(x), §3.5(iv)

Permalink:

http://dlmf.nist.gov/3.5.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(v), §3.5 and Ch.3

►The $w_k$ are also known as Christoffel coefficients or Christoffel numbers and they are all positive. The remainder is given by …

2: 18.40 Methods of Computation

… ►In what follows we consider only the simple, illustrative, case that $\mu (x)$ is continuously differentiable so that $d\mu (x)=w(x)dx$, with $w(x)$ real, positive, and continuous on a real interval $[a,b].$ The strategy will be to: 1) use the moments to determine the recursion coefficients ${\alpha }_n,{\beta }_n$ of equations (18.2.11_5) and (18.2.11_8); then, 2) to construct the quadrature abscissas $x_i$ and weights (or Christoffel numbers) $w_i$ from the J-matrix of §3.5(vi), equations (3.5.31) and(3.5.32). …

3: 18.2 General Orthogonal Polynomials

… ►Then, with the coefficients (18.2.11_4) associated with the monic OP’s $p_n$, the orthonormal recurrence relation for $q_n$ takes the form … ►

§18.2(v) Christoffel–Darboux Formula

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Confluent Form

… ►are the Christoffel numbers, see also (3.5.18). … ►for certain coefficients $a_{n,j}$ with $s,t$ independent of $n$. …

4: Errata

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18.2.12 $$K_n(x,y)\equiv \sum _{\mathrm{\ell }=0}^n\frac{p_{\mathrm{\ell }}(x)p_{\mathrm{\ell }}(y)}{h_{\mathrm{\ell }}}=\frac{k_n}{h_n{k}_{n+1}}\frac{p_{n+1}(x)p_n(y)-p_n(x)p_{n+1}(y)}{x-y},$$ $x\ne y$

18.2.13 $$K_n(x,x)=\sum _{\mathrm{\ell }=0}^n\frac{{(p_{\mathrm{\ell }}(x))}^2}{h_{\mathrm{\ell }}}=\frac{k_n}{h_n{k}_{n+1}}\left(p_{n+1}^{\prime }(x)p_n(x)-p_n^{\prime }(x)p_{n+1}(x)\right)$$

The left-hand sides were updated to include the definition of the Christoffel–Darboux kernel $K_n(x,y)$.

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A sentence indicating that explicit representations for the coefficients $c_k$ are given in Volkmer (2023) was inserted just below (16.11.5).

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Equations: (3.3.3_1), (3.3.3_2), (5.15.9) (suggested by Calvin Khor on 2021-09-04), (8.15.2), Pochhammer symbol representation in (10.17.1) for $a_k(\nu )$ coefficient, Pochhammer symbol representation in (11.9.4) for $a_k(\mu ,\nu )$ coefficient, and (12.14.4_5).

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This subsection now identifies Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

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10.20.14 $$B_3(0)=-\frac{\mathrm{959\hspace{0.33em}71711\hspace{0.33em}84603}}{\mathrm{25\hspace{0.33em}47666\hspace{0.33em}37125\hspace{0.33em}00000}}{2}^{\frac{1}{3}}$$

Originally this coefficient was given incorrectly as $B_3(0)=-\frac{\mathrm{430\hspace{0.33em}99056\hspace{0.33em}39368\hspace{0.33em}59253}}{\mathrm{5\hspace{0.33em}68167\hspace{0.33em}34399\hspace{0.33em}42500\hspace{0.33em}00000}}{2}^{\frac{1}{3}}$. The other coefficients in this equation have not been changed.

Reported 2012-05-11 by Antony Lee.

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5: Bibliography G

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Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, pp. 251–270.

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External Links:

ISSN 0025-5718, Document, Link, MathReview (R. E. Barnhill)

Referenced by:

§18.39(iii).

Encodings:

BibTeX

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Variable-precision recurrence coefficients for nonstandard orthogonal polynomials. Numer. Algorithms 52 (3), pp. 409–418.

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External Links:

ISSN 1017-1398, Document, Link, MathReview Entry

Referenced by:

§18.39(iii), §18.40(ii).

Encodings:

BibTeX

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Tables of binomial coefficients and Stirling numbers. J. Res. Nat. Bur. Standards Sect. B 80B (1), pp. 99–171.

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External Links:

MathReview (M. S. Cheema), ZentralBlatt

Referenced by:

§26.21.

Encodings:

BibTeX

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