hermite_rule

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

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

The Gauss-Hermite quadrature rule is used as follows:

        c * Integral ( -oo < x < +oo ) f(x) exp ( - b * ( x - a )^2 ) dx
      

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

The value of C in front of the integral depends on the user's choice of the SCALE parameter:

• scale=0, then C = 1; this is the standard choice for Gauss-Hermite quadrature.
• scale=1, then C is a normalization factor so that f(x)=1 will integrate to 1. This implies in turn that the weights will sum to 1. This choice is appropriate when using the rule to compute probabilities.

Usage:

hermite_rule ( order, a, b, scale 'filename' )

• order is the number of points in the quadrature rule.
• a is the center point (default 0);
• b is the scale factor (default 1);
• scale is the normalization option (0/1). If 1, then the weights are normalized to have unit sum;
• 'filename' specifies the output filenames: 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:

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

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hermite_exactness, a Python code which tests the exactness of Gauss-Hermite quadrature rules.

hermite_polynomial, a Python code which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

jacobi_rule, a Python code which returns a Gauss-Jacobi quadrature rule.

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

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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:

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

HERM_O4 is a Hermite rule of order 4, created by the command

        hermite_rule ( 4, 0.0, 1.0, 0, 'herm_o4' )
      

• herm_o4_r.txt, the region file;
• herm_o4_w.txt, the weight file;
• herm_o4_x.txt, the abscissa file;