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Jacobi-Gauss Quadrature


Jacobi-Gauss quadrature, also called Jacobi quadrature or Mehler quadrature, is a Gaussian quadrature over the interval [-1,1] with weighting function

 W(x)=(1-x)^alpha(1+x)^beta.
(1)

The abscissas for quadrature order n are given by the roots of the Jacobi polynomials P_n^((alpha,beta))(x). The weights are

w_i=-(A_(n+1)gamma_n)/(A_nP_n^((alpha,beta))^'(x_i)P_(n+1)^((alpha,beta))(x_i))
(2)
=(A_n)/(A_(n-1))(gamma_(n-1))/(P_(n-1)^((alpha,beta))(x_i)P_n^((alpha,beta))^'(x_i)),
(3)

where A_n is the coefficient of x^n in P_n^((alpha,beta))(x). For Jacobi polynomials,

 A_n=(Gamma(2n+alpha+beta+1))/(2^nn!Gamma(n+alpha+beta+1)),
(4)

where Gamma(z) is a gamma function. Additionally,

 gamma_n=1/(2^(2n)(n!)^2)(2^(2n+alpha+beta+1)n!)/(2n+alpha+beta+1)(Gamma(n+alpha+1)Gamma(n+beta+1))/(Gamma(n+alpha+beta+1)),
(5)

so

w_i=(2n+alpha+beta+2)/(n+alpha+beta+1)(Gamma(n+alpha+1)Gamma(n+beta+1))/(Gamma(n+alpha+beta+1))(2^(2n+alpha+beta+1)n!)/(V_n^'(x_i)V_(n+1)(x_i))
(6)
=(Gamma(n+alpha+1)Gamma(n+beta+1))/(Gamma(n+alpha+beta+1))(2^(2n+alpha+beta+1)n!)/((1-x_i^2)[V_n^'(x_i)]^2),
(7)

where

 V_m=P_n^((alpha,beta))(x)(2^nn!)/((-1)^n).
(8)

The error term is

 E_n=(Gamma(n+alpha+1)Gamma(n+beta+1)Gamma(n+alpha+beta+1))/((2n+alpha+beta+1)[Gamma(2n+alpha+beta+1)]^2)(2^(2n+alpha+beta+1)n!)/((2n)!)f^((2n))(xi)
(9)

(Hildebrand 1956).


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References

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 331-334, 1956.

Referenced on Wolfram|Alpha

Jacobi-Gauss Quadrature

Cite this as:

Weisstein, Eric W. "Jacobi-Gauss Quadrature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Jacobi-GaussQuadrature.html

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