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
is to be approximated by
        Sum ( 1 <= i <= order ) w(i) * f(x(i))
Generally, a Gauss-Hermite quadrature rule of n points will produce the exact integral when f(x) is a polynomial of degree 2n-1 or less.

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


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


The computer code and data files described and made available on this web page are distributed under the MIT license


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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  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:

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

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

Last revised on 26 May 2023.