jacobi_rule (original) (raw)

jacobi_rule, a MATLAB 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

is to be approximated by

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

a b

Usage:

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

where

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 ina C++ version anda FORTRAN90 version anda MATLAB version.

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

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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.
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.
Jaroslav Kautsky, Sylvan Elhay,
Calculation of the Weights of Interpolatory Quadratures,
Numerische Mathematik,
Volume 40, 1982, pages 407-422.
Roger Martin, James Wilkinson,
The Implicit QL Algorithm,
Numerische Mathematik,
Volume 12, Number 5, December 1968, pages 377-383.
Arthur Stroud, Don Secrest,
Gaussian Quadrature Formulas,
Prentice Hall, 1966,
LC: QA299.4G3S7.

Source Code:

<jacobi%5Frule.m> the source code.