Jacobi-Gauss Quadrature
المؤلف:
Hildebrand, F. B
المصدر:
Introduction to Numerical Analysis. New York: McGraw-Hill
الجزء والصفحة:
pp. 331-334
5-12-2021
769
Jacobi-Gauss Quadrature
Jacobi-Gauss quadrature, also called Jacobi quadrature or Mehler quadrature, is a Gaussian quadrature over the interval ![[-1,1]](https://mathworld.wolfram.com/images/equations/Jacobi-GaussQuadrature/Inline1.gif)
with weighting function
 |
(1)
|
The abscissas for quadrature order 
are given by the roots of the Jacobi polynomials 
. The weights are
where 
is the coefficient of 
in 
. For Jacobi polynomials,
 |
(4)
|
where 
is a gamma function. Additionally,
 |
(5)
|
so
where
 |
(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)](https://mathworld.wolfram.com/images/equations/Jacobi-GaussQuadrature/NumberedEquation5.gif) |
(9)
|
(Hildebrand 1956).
REFERENCES:
Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 331-334, 1956.
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