Jacobi-Gauss Quadrature

Also called Jacobi Quadrature or Mehler Quadrature. A Gaussian Quadrature over the interval $[-1,1]$ with Weighting Function $W(x)=(1-x)^\alpha(1+x)^\beta$. The Abscissas for quadrature order $n$ are given by the roots of the Jacobi Polynomials $P_n^{(\alpha,\beta)}(x)$. The weights are

$$w_i = -{A_{n+1}\gamma_n\over A_n{P_n^{(\alpha,\beta)}}'(x_i)P_{n+1}^{(\alpha,\beta)}(x_i)}$$

$$= {A_n\over A_{n-1}}{\gamma_{n-1}\over P_{n-1}^{(\alpha,\beta)}(x_i){P_n^{(\alpha,\beta)}}'(x_i)},$$ (1)

where $A_n$ is the Coefficient of $x^n$ in $P_n^{(\alpha,\beta)}(x)$. For Jacobi Polynomials,

$$A_n={\Gamma(2n+\alpha+\beta+1)\over 2^n n!\Gamma(n+\alpha+\beta+1)},$$ (2)

where $\Gamma(z)$ is a Gamma Function. Additionally,

$$\gamma_n={1\over 2^{2n}(n!)^2} {2^{2n+\alpha+\beta+1}n!\over... ...ma(n+\alpha+1)\Gamma(n+\beta+1)\over\Gamma(n+\alpha+\beta+1)},$$ (3)

so

$$w_i = {2n+\alpha+\beta+2\over n+\alpha+\beta+1}{\Gamma(n+\alpha+1)\Gamm... ...er\Gamma(n+\alpha+\beta+1)}{2^{2n+\alpha+\beta+1}n!\over V_n'(x_i)V_{n+1}(x_i)}$$ (4)

$$= {\Gamma(n+\alpha+1)\Gamma(n+\beta+1)\over\Gamma(n+\alpha+\beta+1)} {2^{2n+\alpha+\beta+1}n!\over(1-{x_i}^2)[V_n'(x_i)]^2},$$ (5)

where

$$V_m\equiv P_n^{(\alpha,\beta)}(x){2^nn!\over(-1)^n}.$$ (6)

The error term is

$$E_n={\Gamma(n+\alpha+1)\Gamma(n+\beta+1)\Gamma(n+\alpha+\bet... ...+\beta+1)]^2}{2^{2n+\alpha+\beta+1}n!\over (2n)!}f^{(2n)}(\xi)$$ (7)

(Hildebrand 1959).

References

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