Quadrature Tests for Semi-Infinite Intervals

LAGUERRE_TEST_INT is a MATLAB library 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)
  16. cos(0.5*pi*x) / sqrt(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:

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:

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

and can be used with any of the sample integrands.


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


LAGUERRE_TEST_INT is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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

QUADRULE, a MATLAB library which defines various quadrature rules.

TEST_INT, a MATLAB library which defines some test integration problems over finite intervals.

TEST_INT_2D, a MATLAB library which defines test integrands for 2D quadrature rules.


  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,
    Gaussian Quadrature Formulas,
    Prentice Hall, 1966,
    LC: QA299.4G3S7.

Source Code:

Examples and Tests:

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

Last revised on 28 December 2011.