R16_HERMITE_RULE
Gauss-Hermite Quadrature Rules

R16_HERMITE_RULE is a FORTRAN90 program which generates a specific Gauss-Hermite quadrature rule, based on user input.

The rule is computed using "quadruple real precision" arithmetic. This means that an attempt is made to compute the results to about 30 decimal digits.

The related program HERMITE_RULE uses the more common double precision real arithmetic, which has about 15 digits of accuracy.

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

The Gauss Hermite quadrature rule is used as follows:

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

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

Usage:

r16_hermite_rule order a b 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);
• 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 GNU LGPL license.

Languages:

R16_HERMITE_RULE is available in a FORTRAN90 version.

Related Data and Programs:

CCN_RULE, a FORTRAN90 program which defines a nested Clenshaw Curtis quadrature rule.

CHEBYSHEV1_RULE, a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE, a FORTRAN90 program which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, a FORTRAN90 program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, a FORTRAN90 program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, a FORTRAN90 program which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, a FORTRAN90 program which can compute and print a Gauss-Hermite quadrature rule.

INT_EXACTNESS, a FORTRAN90 program which checks the polynomial exactness of a 1-dimensional quadrature rule for a finite interval.

INT_EXACTNESS_HERMITE, a FORTRAN90 program which checks the polynomial exactness of a Gauss-Hermite quadrature rule.

JACOBI_RULE, a FORTRAN90 program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, a FORTRAN90 program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, a FORTRAN90 program which computes a Gauss-Legendre quadrature rule.

LEGENDRE_RULE_FAST, a FORTRAN90 program which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.

PATTERSON_RULE, a FORTRAN90 program which returns the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

PATTERSON_RULE_COMPUTE, a FORTRAN90 program which computes the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

QUADRATURE_RULES_HERMITE, a dataset directory which contains triples of files defining standard Hermite quadrature rules.

QUADRULE, a FORTRAN90 library which contains 1-dimensional quadrature rules.

R16_INT_EXACTNESS_GEN_HERMITE, a FORTRAN90 program which tests the polynomial exactness of generalized Gauss-Hermite quadrature rules, using "quadruple precision real" arithmetic.

R16_SUBPAK, a FORTRAN90 library which contains many utility routines;

TANH_SINH_RULE, a FORTRAN90 program which computes and writes out a tanh-sinh quadrature rule of given order.

TRUNCATED_NORMAL_RULE, a FORTRAN90 program which computes a quadrature rule for a normal distribution that has been truncated to [A,+oo), (-oo,B] or [A,B].

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:

• r16_hermite_rule.f90, the source code.
• r16_hermite_rule.csh, commands to compile the source code.

Examples and Tests:

• r16_herm_o4_r.txt, the region file created by the command

  
              r16_hermite_rule 4 0.0 1.0 r16_herm_o4
            

• r16_herm_o4_w.txt, the weight file created by the command

  
              r16_hermite_rule 4 0.0 1.0 r16_herm_o4
            

• r16_herm_o4_x.txt, the abscissa file created by the command

  
              r16_hermite_rule 4 0.0 1.0 r16_herm_o4
            

List of Routines:

• MAIN is the main program for R16_HERMITE_RULE.
• CDGQF computes a Gauss quadrature formula with default A, B and simple knots.
• CGQF computes knots and weights of a Gauss quadrature formula.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• CLASS_MATRIX computes the Jacobi matrix for a quadrature rule.
• GET_UNIT returns a free FORTRAN unit number.
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• PARCHK checks parameters ALPHA and BETA for classical weight functions.
• R16_HUGE returns a very large R16.
• R16_PI returns the value of pi as an R16.
• R16MAT_WRITE writes an R16MAT file.
• RULE_WRITE writes a quadrature rule to a file.
• S_TO_I4 reads an I4 from a string.
• S_TO_R16 reads an R16 from a string.
• SCQF scales a quadrature formula to a nonstandard interval.
• SGQF computes knots and weights of a Gauss Quadrature formula.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.