Английская Википедия:Gauss–Jacobi quadrature

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In numerical analysis, Gauss–Jacobi quadrature (named after Carl Friedrich Gauss and Carl Gustav Jacob Jacobi) is a method of numerical quadrature based on Gaussian quadrature. Gauss–Jacobi quadrature can be used to approximate integrals of the form

<math> \int_{-1}^1 f(x) (1 - x)^\alpha (1 + x)^\beta \,dx </math>

where ƒ is a smooth function on Шаблон:Math and Шаблон:Math. The interval Шаблон:Math can be replaced by any other interval by a linear transformation. Thus, Gauss–Jacobi quadrature can be used to approximate integrals with singularities at the end points. Gauss–Legendre quadrature is a special case of Gauss–Jacobi quadrature with Шаблон:Math. Similarly, the Chebyshev–Gauss quadrature of the first (second) kind arises when one takes Шаблон:Math. More generally, the special case Шаблон:Math turns Jacobi polynomials into Gegenbauer polynomials, in which case the technique is sometimes called Gauss–Gegenbauer quadrature.

Gauss–Jacobi quadrature uses Шаблон:Math as the weight function. The corresponding sequence of orthogonal polynomials consist of Jacobi polynomials. Thus, the Gauss–Jacobi quadrature rule on Шаблон:Math points has the form

<math> \int_{-1}^1 f(x) (1 - x)^\alpha (1 + x)^\beta \,dx \approx \lambda_1 f(x_1) + \lambda_2 f(x_2) + \ldots + \lambda_n f(x_n), </math>

where Шаблон:Math are the roots of the Jacobi polynomial of degree Шаблон:Math. The weights Шаблон:Math are given by the formula

<math>\lambda_i =

 -\frac{2n + \alpha + \beta + 2}
       {n + \alpha + \beta + 1}\,
  \frac{\Gamma(n + \alpha + 1)\Gamma(n + \beta + 1)}
       {\Gamma(n + \alpha + \beta + 1)(n + 1)!}\,
  \frac{2^{\alpha + \beta}}
       {P_{n}^{(\alpha,\beta)\,\prime}(x_i) P_{n+1}^{(\alpha,\beta)}(x_i)},

</math>

where Γ denotes the Gamma function and Шаблон:Math the Jacobi polynomial of degree n.

The error term (difference between approximate and accurate value) is:

<math>

E_n = \frac{\Gamma(n+\alpha+1) \Gamma(n+\beta+1) \Gamma(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)[\Gamma(2n+\alpha+\beta+1)]^2} \frac{2^{2+\alpha+\beta+1}}{(2n)!} f^{(2n)}(\xi), </math> where <math>-1 < \xi < 1</math>.

References

• Шаблон:Citation.

External links

• Jacobi rule - free software (Matlab, C++, and Fortran) to evaluate integrals by Gauss–Jacobi quadrature rules.
• Gegenbauer rule - free software (Matlab, C++, and Fortran) for Gauss–Gegenbauer quadrature