laguerre_integrands

laguerre_integrands, a MATLAB 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. 1 / ( x * log(x)^2 );
2. 1 / ( x * log(x)^(3/2) );
3. 1 / ( x^1.01 );
4. Sine integral;
5. Fresnel integral;
6. Complementary error function;
7. Bessel function;
8. Debye function;
9. Gamma(Z=4) function;
10. 1/(1+x*x); (Stroud)
11. 1 / ( (1+x) * sqrt(x) ); (Stroud)
12. exp ( - x ) * cos ( x ); (Stroud)
13. sin(x) / x; (Stroud)
14. sin ( exp(-x) + exp(-4x) ); (Stroud)
15. log(x) / ( 1+100*x*x); (QUADPACK)
17. exp ( - x / 2^beta ) * cos ( x ) / sqrt ( x )
18. x^2 * exp ( - x / 2^beta )
19. x^(beta-1) / ( 1 + 10 x )^2
20. 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.

Languages:

laguerre_integrands is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

test_int, a MATLAB code which defines some test integration problems over finite intervals.

test_int_2d, a MATLAB code which defines test integrands for 2D quadrature rules.

Reference:

1. Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
2. Robert Piessens, Elise deDoncker-Kapenga, Christian Ueberhuber, David Kahaner,
QUADPACK: A Subroutine Package for Automatic Integration,
Springer, 1983,
ISBN: 3540125531,
LC: QA299.3.Q36.
3. Arthur Stroud, Don Secrest,