hermite_integrands
hermite_integrands,
a MATLAB code which
defines integration problems over infinite intervals of the form (oo,+oo).
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.
For a given integrand function f(x), the problem is to estimate
I(f) = integral ( oo < x < +oo ) w(x) * f(x) dx
We consider three variations of the problem, depending on the
form of the weight factor w(x):

option = 0, the unweighted integral:
Integral ( oo < x < +oo ) f(x) dx

option = 1, the physicist weighted integral:
Integral ( oo < x < +oo ) exp(x*x) f(x) dx

option = 2, the probabilist weighted integral:
Integral ( oo < x < +oo ) exp(x*x/2) f(x) dx
For option 0, the test integrands have the form:

f1(x) = exp(x*x) * cos(2*omega*x);

f2(x) = exp(x*x);

f3(x) = exp(px)/(1+exp(qx));

f4(x) = sin ( x^2 );

f5(x) = 1 / (1+x^2) sqrt (4+3x^2) );

f6(x) = exp(x*x) * x^m;

f7(x) = x^2 cos(x) exp(x*x);

f8(x) = sqrt ( 1 + x * x / 2 ) * exp(x*x/2);
For option 1, the test integrands have the form:

f1(x) = cos(2*omega*x);

f2(x) = 1

f3(x) = exp(x*x) * exp(px)/(1+exp(qx));

f4(x) = exp(x*x) * sin ( x^2 );

f5(x) = exp(x*x) * 1 / (1+x^2) sqrt (4+3x^2) );

f6(x) = x^m;

f7(x) = x^2 cos(x);

f8(x) = sqrt ( 1 + x * x / 2 ) * exp(+x*x/2);
For option 2, the test integrands have the form:

f1(x) = exp(x*x/2) * cos(2*omega*x);

f2(x) = exp(x*x/2);

f3(x) = exp(+x*x/2) * exp(px)/(1+exp(qx));

f4(x) = exp(+x*x/2) * sin ( x^2 );

f5(x) = exp(+x*x/2) * 1 / (1+x^2) sqrt (4+3x^2) );

f6(x) = exp(x*x/2) * x^m;

f7(x) = x^2 cos(x) exp(x*x/2);

f8(x) = sqrt ( 1 + x * x / 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.
Some demonstration routines are built in for simple quadrature methods:

P00_GAUSS_HERMITE uses a GaussHermite quadrature formula;

P00_MONTE_CARLO uses a Monte Carlo scheme, with
sample points selected according to the standard
normal probability distribution;

P00_TURING applies a simple equally spaced method of
Turing.
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
Languages:
hermite_integrands is available in
a C version and
a C++ version and
a FORTRAN90 version and
a MATLAB version.
Related Data and Programs:
hermite_exactness,
a MATLAB code which
tests the polynomial exactness of GaussHermite quadrature rules.
hermite_integrands_test
hermite_rule,
a MATLAB code which
can compute and print a GaussHermite quadrature rule.
quadrature_rules_hermite_physicist,
a dataset directory which
contains GaussHermite quadrature rules, for integration
on the interval (oo,+oo), with weight function exp(x^2).
quadrature_rules_hermite_probabilist,
a dataset directory which
contains GaussHermite quadrature rules, for integration
on the interval (oo,+oo), with weight function exp(x^2/2).
quadrature_rules_hermite_unweighted,
a dataset directory which
contains GaussHermite quadrature rules, for integration
on the interval (oo,+oo), with weight function 1.
test_int,
a MATLAB code which
defines some test integration problems over finite intervals.
Reference:

Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.

Prem Kythe, Michael Schaeferkotter,
Handbook of Computational Methods for Integration,
Chapman and Hall, 2004,
ISBN: 1584884282,
LC: QA299.3.K98.

Robert Piessens, Elise deDonckerKapenga,
Christian Ueberhuber, David Kahaner,
QUADPACK: A Subroutine Package for Automatic Integration,
Springer, 1983,
ISBN: 3540125531,
LC: QA299.3.Q36.

William Squire,
Comparison of GaussHermite 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 337340,
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 180197.
Source Code:
Last revised on 12 January 2021.