laguerre_integrands
laguerre_integrands,
a Fortran90 code which
defines integration problems over
semi-infinite intervals of the form [ALPHA,+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.
The test integrands include:
-
1 / ( x * log(x)^2 );
-
1 / ( x * log(x)^(3/2) );
-
1 / ( x^1.01 );
-
Sine integral;
-
Fresnel integral;
-
Complementary error function;
-
Bessel function;
-
Debye function;
-
Gamma(Z=4) function;
-
1/(1+x*x); (Stroud)
-
1 / ( (1+x) * sqrt(x) ); (Stroud)
-
exp ( - x ) * cos ( x ); (Stroud)
-
sin(x) / x; (Stroud)
-
sin ( exp(-x) + exp(-4x) ); (Stroud)
-
log(x) / ( 1+100*x*x); (QUADPACK)
-
cos(0.5*pi*x) / sqrt(x); (QUADPACK)
-
exp ( - x / 2^beta ) * cos ( x ) / sqrt ( x )
-
x^2 * exp ( - x / 2^beta )
-
x^(beta-1) / ( 1 + 10 x )^2
-
1 / ( 2^beta * ( ( x - 1 )^2 + (1/4)^beta ) * ( x - 2 ) )
The library includes not just the integrand, but also the value of
ALPHA which defines the interval of integration, and the exact value
of the integral (or, typically, an estimate of this value).
Thus, for each integrand function, three subroutines are supplied. For
instance, for function #1, we have the routines:
-
P01_FUN evaluates the integrand for problem 1.
-
P01_ALPHA returns the value of ALPHA 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 again has three
subroutines, but which requires you to specify the problem
number as an extra input argument:
-
P00_FUN evaluates the integrand for any problem.
-
P00_ALPHA returns the value of ALPHA 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_EXP_TRANSFORM applies an exponential change of
variables, and then uses a Gauss-Legendre quadrature formula
to estimate the integral for any problem.
-
P00_GAUSS_LAGUERRE uses a Gauss-Laguerre quadrature formula
to estimate the integral for any problem.
-
P00_RAT_TRANSFORM applies a rational change of
variables, and then uses a Gauss-Legendre quadrature formula
to estimate the integral for any problem.
and can be used with any of the sample integrands.
Licensing:
The information on this web page is distributed under the MIT license.
Languages:
laguerre_integrands is available in
a C version and
a C++ version and
a Fortran90 version and
a MATLAB version and
an Octave version.
Related Data and Programs:
laguerre_integrands_test
laguerre_polynomial,
a Fortran90 code which
which evaluates the Laguerre polynomial, the generalized Laguerre polynomials,
and the Laguerre function.
laguerre_rule,
a Fortran90 code which
generates a Gauss-Laguerre quadrature rule on request.
quadpack,
a Fortran90 code which
estimates integrals of functions in one dimension.
quad_rule,
a Fortran90 code which
defines 1-dimensional quadrature rules.
test_int,
a Fortran90 code which
defines some test integration problems over finite intervals.
test_int_2d,
a Fortran90 code which
defines test integrands for 2D quadrature rules.
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.
-
Arthur Stroud, Don Secrest,
Gaussian Quadrature Formulas,
Prentice Hall, 1966,
LC: QA299.4G3S7.
Source Code:
Last revised on 24 July 2020.