Christoffel numbers
From Encyclopedia of Mathematics
Christoffel coefficients
The coefficients of a quadrature formula
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which is exact for algebraic polynomials of degrees . The interpolation nodes
of such a formula are the zeros of a polynomial
of degree
which is orthogonal on
relative to the distribution
to all polynomials of degree
; if
, the Christoffel numbers are uniquely determined. One has
,
and
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If the polynomials are orthonormal, the Christoffel numbers may be expressed as
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where is the leading coefficient of
. In the case
,
and
, the
are the Legendre polynomials, and
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These expressions are due to E.B. Christoffel [1]. For they were evaluated by C.F. Gauss. See also Gauss quadrature formula.
References
[1] | E.B. Christoffel, "Ueber die Gaussche Quadratur und eine Verallgemeinerung derselben" J. Reine Angew. Math. , 55 (1858) pp. 61–82 |
[2] | G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975) |
[3] | I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian) |
Comments
References
[a1] | F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974) |
How to Cite This Entry:
Christoffel numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Christoffel_numbers&oldid=46344
Christoffel numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Christoffel_numbers&oldid=46344
This article was adapted from an original article by N.P. KorneichukV.P. Motornyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article