chebyshev1_rule

chebyshev1_rule, a Fortran90 code which generates a specific Gauss-Chebyshev type 1 quadrature rule, based on user input.

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

The Gauss Chevbyshev type 1 quadrature rule is used as follows:

        Integral ( A <= x <= B ) f(x) / sqrt ( ( x - A ) * ( B - x ) ) dx
      

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

Usage:

chebyshev1_rule order a b filename

• order is the number of points in the quadrature rule.
• a is the left endpoint;
• b is the right endpoint.
• filename specifies the output files: filename_w.txt, filename_x.txt, and filename_r.txt, containing the weights, abscissas, and interval limits.

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

chebyshev1_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 codes:

chebyshev1_rule_test

f90_rule, a Fortran90 code which computes a quadrature rule which estimates the integral of a function f(x), which might be defined over a one dimensional region (a line) or more complex shapes such as a circle, a triangle, a quadrilateral, a polygon, or a higher dimensional region, and which might include an associated weight function w(x).

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:

• chebyshev1_rule.f90, the source code.
• chebyshev1_rule.sh, compiles the source code.