Gauss–Christoffel quadrature

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1: 3.5 Quadrature

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§3.5(v) Gauss Quadrature

… ►In Gauss quadrature (also known as Gauss–Christoffel quadrature) we use (3.5.15) with nodes $x_k$ the zeros of $p_n$, and weights $w_k$ given by ►

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 remainder is given by …

2: 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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3: 18.2 General Orthogonal Polynomials

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§18.2(v) Christoffel–Darboux Formula

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

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Symbols:

$\equiv$: equals by definition, $(a,b)$: open interval, $y$: real variable, $p_n(x)$: polynomial of degree $n$ , $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $h_n$ and $k_n$

A&S Ref:

22.12.1

Referenced by:

§18.2(v), Erratum (V1.2.0) for Equations (18.2.12), (18.2.13)

Permalink:

http://dlmf.nist.gov/18.2.E12

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

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

See also:

Annotations for §18.2(v), §18.2 and Ch.18

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Confluent Form

… ►For usage of the zeros of an OP in Gauss quadrature see §3.5(v). … ►are the Christoffel numbers, see also (3.5.18). …