jacobi_rule

jacobi_rule, a Python code which generates a specific Gauss-Jacobi quadrature rule, based on user input.

The rule is written to three files for easy use as input to other programs.

The Gauss-Jacobi quadrature rule is used as follows:

        Integral ( A <= x <= B ) (B-x)^alpha (x-A)^beta f(x) dx
      

        Sum ( 1 <= i <= order ) w(i) * f(x(i))
      

Usage:

jacobi_rule ( order, alpha, beta, a, b, 'filename' )

• order is the number of points in the quadrature rule.
• alpha is the exponent of (B-x), which must be greater than -1.
• beta is the exponent of (x-A), which must be greater than -1.
• a is the left endpoint;
• b is the right endpoint.
• 'filename' specifies how the rule is to be reported: filename_w.txt, filename_x.txt, and filename_r.txt, containing the weights, abscissas, and interval limits.

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

Languages:

jacobi_rule is available in a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

alpert_rule, a Python code which sets up an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.

ccn_rule, a Python code which returns a nested Clenshaw Curtis quadrature rule.

chebyshev1_rule, a Python code which returns a Gauss-Chebyshev type 1 quadrature rule.

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clenshaw_curtis_rule, a Python code which returns a Clenshaw Curtis quadrature rule.

gegenbauer_rule, a Python code which returns a Gauss-Gegenbauer quadrature rule.

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gen_laguerre_rule, a Python code which returns a generalized Gauss-Laguerre quadrature rule.

hermite_rule, a Python code which returns a Gauss-Hermite quadrature rule.

jacobi_polynomial, a Python code which evaluates the Jacobi polynomial and associated functions.

laguerre_rule, a Python code which returns a Gauss-Laguerre quadrature rule.

legendre_rule, a Python code which returns a Gauss-Legendre quadrature rule.

line_felippa_rule, a Python code which returns a Felippa quadrature rule over the interior of a line segment in 1D.

patterson_rule, a Python code which returns a Gauss-Patterson quadrature rule.

quad_rule, a Python code which returns 1-dimensional quadrature rules.

truncated_normal_rule, a Python code which returns a quadrature rule for a normal probability density function (PDF), also called a Gaussian distribution, that has been truncated to [A,+oo), (-oo,B] or [A,B].

Reference:

1. Milton Abramowitz, Irene Stegun,
   Handbook of Mathematical Functions,
   National Bureau of Standards, 1964,
   ISBN: 0-486-61272-4,
   LC: QA47.A34.
2. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
3. Sylvan Elhay, Jaroslav Kautsky,
   Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of Interpolatory Quadrature,
   ACM Transactions on Mathematical Software,
   Volume 13, Number 4, December 1987, pages 399-415.
4. Jaroslav Kautsky, Sylvan Elhay,
   Calculation of the Weights of Interpolatory Quadratures,
   Numerische Mathematik,
   Volume 40, 1982, pages 407-422.
5. Roger Martin, James Wilkinson,
   The Implicit QL Algorithm,
   Numerische Mathematik,
   Volume 12, Number 5, December 1968, pages 377-383.
6. Arthur Stroud, Don Secrest,
   Gaussian Quadrature Formulas,
   Prentice Hall, 1966,
   LC: QA299.4G3S7.

Source Code:

• jacobi_rule.py, the source code.
• jacobi_rule.sh, runs all the tests.
• jacobi_rule.txt, the output file.

• jac_o4_a0.5_b1.5_r.txt, the region file;
• jac_o4_a0.5_b1.5_w.txt, the weight file;
• jac_o4_a0.5_b1.5_x.txt, the abscissa file;