TEST_INT_LAGUERRE
Quadrature Tests for Semi-Infinite Intervals

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

Languages:

TEST_INT_LAGUERRE 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.

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:

laguerre_compute.m computes a Gauss-Laguerre quadrature rule.
laguerre_recur.m finds the value and derivative of a Laguerre polynomial.
laguerre_root.m improves an approximate root of a Laguerre polynomial.
legendre_compute.m computes a Gauss-Legendre quadrature rule.
p00_alpha.m returns the value of ALPHA for any problem.
p00_exact.m returns the exact integral for any problem.
p00_exp_transform.m applies an exponential transform and Gauss-Legendre rule.
p00_fun.m evaluates the integrand for any problem.
p00_gauss_laguerre.m applies a Gauss-Laguerre rule.
p00_problem_num.m returns the number of test integration problems.
p00_rat_transform.m applies an exponential transform and Gauss-Legendre rule.
p00_title.m returns the title for any problem.
p01_alpha.m returns ALPHA for problem 1.
p01_exact.m returns the exact integral for problem 1.
p01_fun.m evaluates the integrand for problem 1.
p01_title.m returns the title for problem 1.
p02_alpha.m returns ALPHA for problem 2.
p02_exact.m returns the exact integral for problem 2.
p02_fun.m evaluates the integrand for problem 2.
p02_title.m returns the title for problem 2.
p03_alpha.m returns ALPHA for problem 3.
p03_exact.m returns the exact integral for problem 3.
p03_fun.m evaluates the integrand for problem 3.
p03_title.m returns the title for problem 3.
p04_alpha.m returns ALPHA for problem 4.
p04_exact.m returns the estimated integral for problem 4.
p04_fun.m evaluates the integrand for problem 4.
p04_title.m returns the title for problem 4.
p05_alpha.m returns ALPHA for problem 5.
p05_exact.m returns the estimated integral for problem 5.
p05_fun.m evaluates the integrand for problem 5.
p05_title.m returns the title for problem 5.
p06_alpha.m returns ALPHA for problem 6.
p06_exact.m returns the exact integral for problem 6.
p06_fun.m evaluates the integrand for problem 6.
p06_title.m returns the title for problem 6.
p07_alpha.m returns ALPHA for problem 7.
p07_exact.m returns the exact integral for problem 7.
p07_fun.m evaluates the integrand for problem 7.
p07_title.m returns the title for problem 7.
p08_alpha.m returns ALPHA for problem 8.
p08_exact.m returns the estimated integral for problem 8.
p08_fun.m evaluates the integrand for problem 8.
p08_title.m returns the title for problem 8.
p09_alpha.m returns ALPHA for problem 9.
p09_exact.m returns the estimated integral for problem 9.
p09_fun.m evaluates the integrand for problem 9.
p09_title.m returns the title for problem 9.
p10_alpha.m returns ALPHA for problem 10.
p10_exact.m returns the estimated integral for problem 10.
p10_fun.m evaluates the integrand for problem 10.
p10_title.m returns the title for problem 10.
p11_alpha.m returns ALPHA for problem 11.
p11_exact.m returns the exact integral for problem 11.
p11_fun.m evaluates the integrand for problem 11.
p11_title.m returns the title for problem 11.
p12_alpha.m returns ALPHA for problem 12.
p12_exact.m returns the exact integral for problem 12.
p12_fun.m evaluates the integrand for problem 12.
p12_title.m returns the title for problem 12.
p13_alpha.m returns ALPHA for problem 13.
p13_exact.m returns the exact integral for problem 13.
p13_fun.m evaluates the integrand for problem 13.
p13_title.m returns the title for problem 13.
p14_alpha.m returns ALPHA for problem 14.
p14_exact.m returns the estimated integral for problem 14.
p14_fun.m evaluates the integrand for problem 14.
p14_title.m returns the title for problem 14.
p15_alpha.m returns ALPHA for problem 15.
p15_exact.m returns the estimated integral for problem 15.
p15_fun.m evaluates the integrand for problem 15.
p15_title.m returns the title for problem 15.
p16_alpha.m returns ALPHA for problem 16.
p16_exact.m returns the estimated integral for problem 16.
p16_fun.m evaluates the integrand for problem 16.
p16_title.m returns the title for problem 16.
p17_alpha.m returns ALPHA for problem 17.
p17_exact.m returns the estimated integral for problem 17.
p17_fun.m evaluates the integrand for problem 17.
p17_title.m returns the title for problem 17.
p18_alpha.m returns ALPHA for problem 18.
p18_exact.m returns the estimated integral for problem 18.
p18_fun.m evaluates the integrand for problem 18.
p18_title.m returns the title for problem 18.
p19_alpha.m returns ALPHA for problem 19.
p19_exact.m returns the estimated integral for problem 19.
p19_fun.m evaluates the integrand for problem 19.
p19_title.m returns the title for problem 19.
p20_alpha.m returns ALPHA for problem 20.
p20_exact.m returns the estimated integral for problem 20.
p20_fun.m evaluates the integrand for problem 20.
p20_title.m returns the title for problem 20.
r8_epsilon.m returns the R8 machine epsilon.
timestamp.m prints the current YMDHMS date as a time stamp.

Examples and Tests:

test_int_laguerre_test.m, calls all the tests
test_int_laguerre_test01.m, demonstrates P00_PROBLEM_NUM and P00_TITLE.
test_int_laguerre_test02.m, demonstrates P00_ALPHA and P00_EXACT.
test_int_laguerre_test03.m, demonstrates P00_GAUSS_LAGUERRE.
test_int_laguerre_test04.m, demonstrates P00_EXP_TRANSFORM.
test_int_laguerre_test05.m, demonstrates P00_RAT_TRANSFORM.
test_int_laguerre_test_output.txt, the output file.

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

Last revised on 28 December 2011.