👁 Image

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)
👁 Image	👁 =	👁 (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)

👁 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)
👁 Image	👁 =	👁 (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)