JACOBI_RULE_SS
Gauss-Jacobi Quadrature Rules

JACOBI_RULE_SS is a C++ program which generates a specific Gauss-Jacobi quadrature rule, based on user input.

The rule can be output as text in a standard programming language, or the data can be written to three files for easy use as input to other programs.

The Gauss-Jacobi quadrature rule is designed to approximate integrals on the interval [-1,1], with a weight function of the form (1-x)^ALPHA * (1+x)^BETA. ALPHA and BETA are real parameters that must be greater than -1.

Gauss-Jacobi quadrature assumes that the integrand we are considering has a form like:

        Integral ( -1 <= x <= +1 ) (1-x)^alpha (1+x)^beta f(x) dx

The standard Gauss-Jacobi quadrature rule is used as follows:

        Integral ( -1 <= x <= +1 ) (1-x)^alpha (1+x)^beta f(x) dx

is to be approximated by

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

Usage:

jacobi_rule_ss order alpha beta output

where

order is the number of points in the quadrature rule. A typical value might be 4, 8, or 16.
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.
output specifies how the rule is to be reported:
- C++, print as C++ text;
- F77, print as FORTRAN77 text;
- F90, print as FORTRAN90 text;
- MAT, print as MATLAB text;
- file, written to three files, file_w.txt, file_x.txt, and file_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.

Related Data and Programs:

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

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

CLENSHAW_CURTIS_RULE is a C++ program which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, is a C++ program which can compute and print a Gauss-Gegenbauer quadrature rule.

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

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

HERMITE_RULE, is a C++ program which can compute and print a Gauss-Hermite quadrature rule.

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

INT_EXACTNESS_JACOBI, is a C++ program which checks the polynomial exactness of a Gauss-Jacobi rule.

INTLIB is a FORTRAN90 library which contains a variety of routines for numerical estimation of integrals in 1D.

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

LAGUERRE_RULE, is a C++ program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, is a C++ program which computes a Gauss-Legendre quadrature rule.

PATTERSON_RULE, is a C++ program which computes a Gauss-Patterson quadrature rule.

PRODUCT_FACTOR is a C++ program which constructs a product rule from distinct 1D factor rules.

PRODUCT_RULE is a C++ program which constructs a product rule from identical 1D factor rules.

QUADPACK is a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.

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

QUADRATURE_RULES_JACOBI is a dataset directory of triples of files defining Gauss-Jacobi quadrature rules.

QUADRULE is a C++ library which contains 1-dimensional quadrature rules.

TANH_SINH_RULE, a C++ program which computes and writes out a tanh-sinh quadrature rule of given order.

TEST_INT is a FORTRAN90 library which defines functions that may be used as test integrands for quadrature rules in 1D.

Reference:

Milton Abramowitz, Irene Stegun,
Handbook of Mathematical Functions,
National Bureau of Standards, 1964,
ISBN: 0-486-61272-4,
LC: QA47.A34.
Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
Arthur Stroud, Don Secrest,
Gaussian Quadrature Formulas,
Prentice Hall, 1966,
LC: QA299.4G3S7.

Source Code:

jacobi_rule_ss.cpp, the source code.

Examples and Tests:

output_cpp.txt, printed output from the command


            jacobi_rule_ss 4 0.0 0.0 C++

output_f77.txt, printed output from the command


            jacobi_rule_ss 4 1.0 0.0 F77

output_f90.txt, printed output from the command


            jacobi_rule_ss 4 0.0 1.0 F90

output_mat.txt, printed output from the command


            jacobi_rule_ss 8 1.0 1.0 MAT

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


            jacobi_rule_ss 4 0.5 1.5 jac_o4_a0.5_b1.5

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


            jacobi_rule_ss 4 0.5 1.5 jac_o4_a0.5_b1.5

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


            jacobi_rule_ss 4 0.5 1.5 jac_o4_a0.5_b1.5

List of Routines:

MAIN is the main program for JACOBI_RULE_SS.
JACOBI_COMPUTE computes a Gauss-Jacobi quadrature rule.
JACOBI_HANDLE computes the requested Gauss-Jacobi rule and outputs it.
JACOBI_RECUR finds the value and derivative of a Jacobi polynomial.
JACOBI_ROOT improves an approximate root of a Jacobi polynomial.
R8_ABS returns the absolute value of an R8.
R8_EPSILON returns the R8 roundoff unit.
R8_GAMMA evaluates Gamma(X) for a real argument.
R8MAT_WRITE writes an R8MAT file with no header.
TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.

Last revised on 02 February 2010.

JACOBI_RULE_SS Gauss-Jacobi Quadrature Rules