TEST_INT_HERMITE
Quadrature Tests for Infinite Intervals

TEST_INT_HERMITE is a FORTRAN90 library which defines integration problems over infinite intervals of the form (-Infinity,+Infinity).

The test integrands would normally be used to testing one dimensional quadrature software. It is possible to invoke a particular function by index, or to try out all available functions, as demonstrated in the sample calling program.

The test integrands include:

exp(-x*x) * cos(2*omega*t);
exp(-x*x);
exp(-px)/(1+exp(-qx))
sin ( x^2 )
1 / (1+x^2) sqrt (4+3x^2) )

The library includes not just the integrand, but also the exact value of the integral (or, typically, an estimate of this value), and a title for the problem. Thus, for each integrand function, several routines are supplied. For instance, for function #1, we have the routines:

P01_FUN evaluates the integrand for problem 1.
P01_EXACT returns the estimated integral for problem 1.
P01_TITLE returns a title for problem 1.

So once you have the calling sequences for these routines, you can easily evaluate the function, or integrate it on the appropriate interval, or compare your estimate of the integral to the exact value.

Moreover, since the same interface is used for each function, if you wish to work with problem 5 instead, you simply change the "01" to "05" in your routine calls.

If you wish to call all of the functions, then you simply use the generic interface, which requires you to specify the problem number as an extra input argument:

P00_FUN evaluates the integrand for any problem.
P00_EXACT returns the exact integral for any problem.
P00_TITLE returns a title for any problem.

Finally, some demonstration routines are built in for simple quadrature methods. These routines include

P00_GAUSS_HERMITE uses a Gauss-Hermite quadrature formula to estimate the integral for any problem.
P00_TURING applies a simple equally spaced method of Turing to estimate the integral for any problem.

and can be used with any of the sample integrands.

Related Data and Programs:

CLENSHAW_CURTIS is a FORTRAN90 library which can set up a Clenshaw Curtis quadrature grid in multiple dimensions.

INTLIB is a FORTRAN90 library of routines for estimating integrals in one dimension.

QUADPACK is a FORTRAN90 library of routines for estimating integrals of functions in one dimension.

QUADRULE is a FORTRAN90 library of routines which define various quadrature rules.

TEST_INT is a FORTRAN90 library of routines which defines some test integration problems over finite intervals.

TEST_INT_HERMITE is also available in a C++ version and a MATLAB version.

TEST_INT_LAGUERRE is a FORTRAN90 library which defines test integrands for integration over [-ALPHA,+Infinity).

TOMS351 is a FORTRAN77 library which estimates an integral using Romberg integration.

TOMS379 is a FORTRAN77 library which estimates an integral.

TOMS418 is a FORTRAN77 library which estimates the integral of a function with a sine or cosine factor.

TOMS424 is a FORTRAN77 library which estimates the integral of a function using Clenshaw-Curtis quadrature.

TOMS468 is a FORTRAN77 library for the "automatic" integration of a function.

Reference:

Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
Robert Piessens, Elise deDoncker-Kapenga, Christian Ueberhuber, David Kahaner,
QUADPACK: A Subroutine Package for Automatic Integration,
Springer, 1983,
ISBN: 3540125531,
LC: QA299.3.Q36.
William Squire,
Comparison of Gauss-Hermite and Midpoint Quadrature with Application to the Voigt Function,
in Numerical Integration: Recent Developments, Software and Applications,
edited by Patrick Keast, Graeme Fairweather,
Reidel, 1987, pages 337-340,
ISBN: 9027725144,
LC: QA299.3.N38.
Arthur Stroud, Don Secrest,
Gaussian Quadrature Formulas,
Prentice Hall, 1966,
LC: QA299.4G3S7.
Alan Turing,
A Method for the Calculation of the Zeta Function,
Proceedings of the London Mathematical Society,
Volume 48, 1943, pages 180-197.

Source Code:

test_int_hermite.f90, the source code;
test_int_hermite.csh, commands to compile the source code;

Examples and Tests:

test_int_hermite_prb.f90, the calling program;
test_int_hermite_prb.csh, commands to compile, link and run the calling program;
test_int_hermite_prb_output.txt, the sample output.

List of Routines:

HERMITE_COMPUTE computes a Gauss-Hermite quadrature rule.
HERMITE_INTEGRAL returns the value of a Hermite polynomial integral.
HERMITE_RECUR finds the value and derivative of a Hermite polynomial.
HERMITE_ROOT improves an approximate root of a Hermite polynomial.
I4_FACTORIAL2 computes the double factorial function N!!
P00_EXACT returns the exact integral for any problem.
P00_FUN evaluates the integrand for any problem.
P00_GAUSS_HERMITE applies a Gauss-Hermite quadrature rule.
P00_PROBLEM_NUM returns the number of test integration problems.
P00_TITLE returns the title for any problem.
P00_TURING applies the Turing quadrature rule.
P01_EXACT returns the exact integral for problem 1.
P01_FUN evaluates the integrand for problem 1.
P01_TITLE returns the title for problem 1.
P02_EXACT returns the exact integral for problem 2.
P02_FUN evaluates the integrand for problem 2.
P02_TITLE returns the title for problem 2.
P03_EXACT returns the exact integral for problem 3.
P03_FUN evaluates the integrand for problem 3.
P03_TITLE returns the title for problem 3.
P04_EXACT returns the exact integral for problem 4.
P04_FUN evaluates the integrand for problem 4.
P04_TITLE returns the title for problem 4.
P05_EXACT returns the exact integral for problem 5.
P05_FUN evaluates the integrand for problem 5.
P05_TITLE returns the title for problem 5.
R8_GAMMA computes the gamma function.
TIMESTAMP prints the current YMDHMS date as a time stamp.

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