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Christoffel Number

A Christoffel number is one of the weights lambda_i in Gaussian quadrature, including Gauss-Jacobi mechanical quadrature. They satisfy

lambda_1+lambda_2+...+lambda_n=int_a^bdalpha(x)
(1)
=alpha(b)-alpha(a)
(2)

and are given by

lambda_nu=int_a^b[(p_n(x))/(p_n^'(x_nu)(x-x_nu))]^2dalpha(x)
(3)
lambda_nu=-(k_(n+1))/(k_n)1/(p_(n+1)(x_nu)p_n^'(x_nu))
(4)
=(k_n)/(k_(n-1))1/(p_(n-1)(x_nu)p_n^'(x_nu))
(5)
(lambda_nu)^(-1)=[p_0(x_nu)]^2+...+[p_n(x_nu)]^2,
(6)

where p_n are orthonormal polynomials for dalpha with positive leading coefficients k_n, and x_nu are the zeros of p_n. The inverse sum is the diagonal value of the Christoffel-Darboux kernel.

See also

Christoffel-Darboux Kernel, Christoffel Function, Cotes Number, Gauss-Jacobi Mechanical Quadrature, Gaussian Quadrature, Hermite's Interpolating Polynomial

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References

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 47-48, 1975.Yakimiw, E. "Accurate Computation of Weights in Classical Gauss-Christoffel Quadrature." J. Comput. Phys. 129, 406-430, 1996.

Referenced on Wolfram|Alpha

Christoffel Number

Cite this as:

Weisstein, Eric W. "Christoffel Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChristoffelNumber.html

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