JACOBI_RULE
Gauss-Jacobi Quadrature Rules

JACOBI_RULE is a MATLAB program 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 value of the exponent of (1-x), which must be greater than -1.
• beta is the value of the exponent of (1+x), 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 GNU LGPL license.

Languages:

JACOBI_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CCN_RULE, a MATLAB program which defines a nested Clenshaw Curtis quadrature rule.

CHEBYSHEV1_RULE, a MATLAB program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a MATLAB program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE, a MATLAB program which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, a MATLAB program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, a MATLAB program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, a MATLAB program which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, a MATLAB program which can compute and print a Gauss-Hermite quadrature rule.

INT_EXACTNESS, a MATLAB program which checks the polynomial exactness of a 1-dimensional quadrature rule for a finite interval.

INT_EXACTNESS_JACOBI, a MATLAB program which checks the polynomial exactness of a Gauss-Jacobi rule.

LAGUERRE_RULE, a MATLAB program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, a MATLAB program which computes a Gauss-Legendre quadrature rule.

PATTERSON_RULE, a MATLAB program which computes a Gauss-Patterson quadrature rule.

QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_RULES_JACOBI, a dataset directory which contains triples of files defining Gauss-Jacobi quadrature rules.

QUADRULE, a MATLAB library which contains 1-dimensional quadrature rules.

TANH_QUAD, a MATLAB library which sets up the tanh quadrature rule;

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.m the source code.

Examples and Tests:

• jac_o4_a0.5_b1.5_r.txt, the region file created by the command

  
              jacobi_rule ( 4, 0.5, 1.5, -1.0, +1.0, 'jac_o4_a0.5_b1.5' )
            

• jac_o4_a0.5_b1.5_w.txt, the weight file created by the command

  
              jacobi_rule ( 4, 0.5, 1.5, -1.0, +1.0, 'jac_o4_a0.5_b1.5' )
            

• jac_o4_a0.5_b1.5_x.txt, the abscissa file created by the command

  
              jacobi_rule ( 4, 0.5, 1.5, -1.0, +1.0, 'jac_o4_a0.5_b1.5' )
            

You can go up one level to the MATLAB source codes.