TEST_INT_LAGUERRE
Quadrature Tests for Semi-Infinite Intervals

TEST_INT_LAGUERRE is a library of MATLAB routines, using double precision arithmetic, which define integration problems over semi-infinite intervals of the form [ALPHA,+Infinity).

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);
11. 1 / ( (1+x) * sqrt(x) );
12. exp ( - x ) * cos ( x );
13. sin(x) / x;
14. sin ( exp(-x) + exp(-4x) );
15. log(x) / ( 1+100*x*x);
16. cos(0.5*pi*x) / sqrt(x);

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.

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.

Related Data and Programs:

CLENSHAW_CURTIS is a FORTRAN90 library which can set up a Clenshaw Curtis quadrature grid in multiple dimensions.

INTLIB is a FORTRAN90 library of routines for estimating integrals in one dimension.

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

QUADRULE is a FORTRAN90 library of routines which define various quadrature rules.

TEST_INT is a FORTRAN90 library of routines which defines some test integration problems over finite intervals.

TEST_INT_LAGUERRE is also available in a C++ version and a FORTRAN90 version.

TOMS351 is a FORTRAN77 library which estimates an integral using Romberg integration.

TOMS379 is a FORTRAN77 library which estimates an integral.

TOMS418 is a FORTRAN77 library which estimates the integral of a function with a sine or cosine factor.

TOMS424 is a FORTRAN77 library which estimates the integral of a function using Clenshaw-Curtis quadrature.

TOMS468 is a FORTRAN77 library for the "automatic" integration of a function.

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,
   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.
• r8_epsilon.m returns the R8 machine epsilon.
• r8_gamma.m calculates the Gamma function for a real argument X.
• 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.out, the sample output.

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