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<html>
<head>
<title>
LAGUERRE_TEST_INT - Quadrature Tests for Semi-Infinite Intervals
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
LAGUERRE_TEST_INT <br> Quadrature Tests for Semi-Infinite Intervals
</h1>
<hr>
<p>
<b>LAGUERRE_TEST_INT</b>
is a MATLAB library which
defines integration problems over
semi-infinite intervals of the form [ALPHA,+oo).
</p>
<p>
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.
</p>
<p>
The test integrands include:
<ol>
<li>
1 / ( x * log(x)^2 );
</li>
<li>
1 / ( x * log(x)^(3/2) );
</li>
<li>
1 / ( x^1.01 );
</li>
<li>
Sine integral;
</li>
<li>
Fresnel integral;
</li>
<li>
Complementary error function;
</li>
<li>
Bessel function;
</li>
<li>
Debye function;
</li>
<li>
Gamma(Z=4) function;
</li>
<li>
1/(1+x*x); (Stroud)
</li>
<li>
1 / ( (1+x) * sqrt(x) ); (Stroud)
</li>
<li>
exp ( - x ) * cos ( x ); (Stroud)
</li>
<li>
sin(x) / x; (Stroud)
</li>
<li>
sin ( exp(-x) + exp(-4x) ); (Stroud)
</li>
<li>
log(x) / ( 1+100*x*x); (QUADPACK)
</li>
<li>
cos(0.5*pi*x) / sqrt(x); (QUADPACK)
</li>
<li>
exp ( - x / 2^beta ) * cos ( x ) / sqrt ( x )
</li>
<li>
x^2 * exp ( - x / 2^beta )
</li>
<li>
x^(beta-1) / ( 1 + 10 x )^2
</li>
<li>
1 / ( 2^beta * ( ( x - 1 )^2 + (1/4)^beta ) * ( x - 2 ) )
</li>
</ol>
</p>
<p>
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:
<ul>
<li>
<b>P01_FUN</b> evaluates the integrand for problem 1.
</li>
<li>
<b>P01_ALPHA</b> returns the value of ALPHA for problem 1.
</li>
<li>
<b>P01_EXACT</b> returns the estimated integral for problem 1.
</li>
<li>
<b>P01_TITLE</b> returns a title for problem 1.
</li>
</ul>
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.
</p>
<p>
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.
</p>
<p>
If you wish to call <i>all</i> 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:
<ul>
<li>
<b>P00_FUN</b> evaluates the integrand for any problem.
</li>
<li>
<b>P00_ALPHA</b> returns the value of ALPHA for any problem.
</li>
<li>
<b>P00_EXACT</b> returns the exact integral for any problem.
</li>
<li>
<b>P00_TITLE</b> returns a title for any problem.
</li>
</ul>
</p>
<p>
Finally, some demonstration routines are built in for
simple quadrature methods. These routines include
<ul>
<li>
<b>P00_EXP_TRANSFORM</b> applies an exponential change of
variables, and then uses a Gauss-Legendre quadrature formula
to estimate the integral for any problem.
</li>
<li>
<b>P00_GAUSS_LAGUERRE</b> uses a Gauss-Laguerre quadrature formula
to estimate the integral for any problem.
</li>
<li>
<b>P00_RAT_TRANSFORM</b> applies a rational change of
variables, and then uses a Gauss-Legendre quadrature formula
to estimate the integral for any problem.
</li>
</ul>
and can be used with any of the sample integrands.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>LAGUERRE_TEST_INT</b> is available in
<a href = "../../c_src/laguerre_test_int/laguerre_test_int.html">a C version</a> and
<a href = "../../cpp_src/laguerre_test_int/laguerre_test_int.html">a C++ version</a> and
<a href = "../../f77_src/laguerre_test_int/laguerre_test_int.html">a FORTRAN77 version</a> and
<a href = "../../f_src/laguerre_test_int/laguerre_test_int.html">a FORTRAN90 version</a> and
<a href = "../../m_src/laguerre_test_int/laguerre_test_int.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/quadpack/quadpack.html">
QUADPACK</a>,
a FORTRAN90 library for estimating integrals of
functions in one dimension.
</p>
<p>
<a href = "../../m_src/quadrule/quadrule.html">
QUADRULE</a>,
a MATLAB library which
defines various quadrature rules.
</p>
<p>
<a href = "../../m_src/test_int/test_int.html">
TEST_INT</a>,
a MATLAB library which
defines some test integration problems over finite intervals.
</p>
<p>
<a href = "../../m_src/test_int_2d/test_int_2d.html">
TEST_INT_2D</a>,
a MATLAB library which
defines test integrands for 2D quadrature rules.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Philip Davis, Philip Rabinowitz,<br>
Methods of Numerical Integration,<br>
Second Edition,<br>
Dover, 2007,<br>
ISBN: 0486453391,<br>
LC: QA299.3.D28.
</li>
<li>
Robert Piessens, Elise deDoncker-Kapenga,
Christian Ueberhuber, David Kahaner,<br>
QUADPACK: A Subroutine Package for Automatic Integration,<br>
Springer, 1983,<br>
ISBN: 3540125531,<br>
LC: QA299.3.Q36.
</li>
<li>
Arthur Stroud, Don Secrest,<br>
Gaussian Quadrature Formulas,<br>
Prentice Hall, 1966,<br>
LC: QA299.4G3S7.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "laguerre_compute.m">laguerre_compute.m</a>
computes a Gauss-Laguerre quadrature rule.
</li>
<li>
<a href = "laguerre_recur.m">laguerre_recur.m</a>
finds the value and derivative of a Laguerre polynomial.
</li>
<li>
<a href = "laguerre_root.m">laguerre_root.m</a>
improves an approximate root of a Laguerre polynomial.
</li>
<li>
<a href = "legendre_compute.m">legendre_compute.m</a>
computes a Gauss-Legendre quadrature rule.
</li>
<li>
<a href = "p00_alpha.m">p00_alpha.m</a>
returns the value of ALPHA for any problem.
</li>
<li>
<a href = "p00_exact.m">p00_exact.m</a>
returns the exact integral for any problem.
</li>
<li>
<a href = "p00_exp_transform.m">p00_exp_transform.m</a>
applies an exponential transform and Gauss-Legendre rule.
</li>
<li>
<a href = "p00_fun.m">p00_fun.m</a>
evaluates the integrand for any problem.
</li>
<li>
<a href = "p00_gauss_laguerre.m">p00_gauss_laguerre.m</a>
applies a Gauss-Laguerre rule.
</li>
<li>
<a href = "p00_problem_num.m">p00_problem_num.m</a>
returns the number of test integration problems.
</li>
<li>
<a href = "p00_rat_transform.m">p00_rat_transform.m</a>
applies an exponential transform and Gauss-Legendre rule.
</li>
<li>
<a href = "p00_title.m">p00_title.m</a>
returns the title for any problem.
</li>
<li>
<a href = "p01_alpha.m">p01_alpha.m</a>
returns ALPHA for problem 1.
</li>
<li>
<a href = "p01_exact.m">p01_exact.m</a>
returns the exact integral for problem 1.
</li>
<li>
<a href = "p01_fun.m">p01_fun.m</a>
evaluates the integrand for problem 1.
</li>
<li>
<a href = "p01_title.m">p01_title.m</a>
returns the title for problem 1.
</li>
<li>
<a href = "p02_alpha.m">p02_alpha.m</a>
returns ALPHA for problem 2.
</li>
<li>
<a href = "p02_exact.m">p02_exact.m</a>
returns the exact integral for problem 2.
</li>
<li>
<a href = "p02_fun.m">p02_fun.m</a>
evaluates the integrand for problem 2.
</li>
<li>
<a href = "p02_title.m">p02_title.m</a>
returns the title for problem 2.
</li>
<li>
<a href = "p03_alpha.m">p03_alpha.m</a>
returns ALPHA for problem 3.
</li>
<li>
<a href = "p03_exact.m">p03_exact.m</a>
returns the exact integral for problem 3.
</li>
<li>
<a href = "p03_fun.m">p03_fun.m</a>
evaluates the integrand for problem 3.
</li>
<li>
<a href = "p03_title.m">p03_title.m</a>
returns the title for problem 3.
</li>
<li>
<a href = "p04_alpha.m">p04_alpha.m</a>
returns ALPHA for problem 4.
</li>
<li>
<a href = "p04_exact.m">p04_exact.m</a>
returns the estimated integral for problem 4.
</li>
<li>
<a href = "p04_fun.m">p04_fun.m</a>
evaluates the integrand for problem 4.
</li>
<li>
<a href = "p04_title.m">p04_title.m</a>
returns the title for problem 4.
</li>
<li>
<a href = "p05_alpha.m">p05_alpha.m</a>
returns ALPHA for problem 5.
</li>
<li>
<a href = "p05_exact.m">p05_exact.m</a>
returns the estimated integral for problem 5.
</li>
<li>
<a href = "p05_fun.m">p05_fun.m</a>
evaluates the integrand for problem 5.
</li>
<li>
<a href = "p05_title.m">p05_title.m</a>
returns the title for problem 5.
</li>
<li>
<a href = "p06_alpha.m">p06_alpha.m</a>
returns ALPHA for problem 6.
</li>
<li>
<a href = "p06_exact.m">p06_exact.m</a>
returns the exact integral for problem 6.
</li>
<li>
<a href = "p06_fun.m">p06_fun.m</a>
evaluates the integrand for problem 6.
</li>
<li>
<a href = "p06_title.m">p06_title.m</a>
returns the title for problem 6.
</li>
<li>
<a href = "p07_alpha.m">p07_alpha.m</a>
returns ALPHA for problem 7.
</li>
<li>
<a href = "p07_exact.m">p07_exact.m</a>
returns the exact integral for problem 7.
</li>
<li>
<a href = "p07_fun.m">p07_fun.m</a>
evaluates the integrand for problem 7.
</li>
<li>
<a href = "p07_title.m">p07_title.m</a>
returns the title for problem 7.
</li>
<li>
<a href = "p08_alpha.m">p08_alpha.m</a>
returns ALPHA for problem 8.
</li>
<li>
<a href = "p08_exact.m">p08_exact.m</a>
returns the estimated integral for problem 8.
</li>
<li>
<a href = "p08_fun.m">p08_fun.m</a>
evaluates the integrand for problem 8.
</li>
<li>
<a href = "p08_title.m">p08_title.m</a>
returns the title for problem 8.
</li>
<li>
<a href = "p09_alpha.m">p09_alpha.m</a>
returns ALPHA for problem 9.
</li>
<li>
<a href = "p09_exact.m">p09_exact.m</a>
returns the estimated integral for problem 9.
</li>
<li>
<a href = "p09_fun.m">p09_fun.m</a>
evaluates the integrand for problem 9.
</li>
<li>
<a href = "p09_title.m">p09_title.m</a>
returns the title for problem 9.
</li>
<li>
<a href = "p10_alpha.m">p10_alpha.m</a>
returns ALPHA for problem 10.
</li>
<li>
<a href = "p10_exact.m">p10_exact.m</a>
returns the estimated integral for problem 10.
</li>
<li>
<a href = "p10_fun.m">p10_fun.m</a>
evaluates the integrand for problem 10.
</li>
<li>
<a href = "p10_title.m">p10_title.m</a>
returns the title for problem 10.
</li>
<li>
<a href = "p11_alpha.m">p11_alpha.m</a>
returns ALPHA for problem 11.
</li>
<li>
<a href = "p11_exact.m">p11_exact.m</a>
returns the exact integral for problem 11.
</li>
<li>
<a href = "p11_fun.m">p11_fun.m</a>
evaluates the integrand for problem 11.
</li>
<li>
<a href = "p11_title.m">p11_title.m</a>
returns the title for problem 11.
</li>
<li>
<a href = "p12_alpha.m">p12_alpha.m</a>
returns ALPHA for problem 12.
</li>
<li>
<a href = "p12_exact.m">p12_exact.m</a>
returns the exact integral for problem 12.
</li>
<li>
<a href = "p12_fun.m">p12_fun.m</a>
evaluates the integrand for problem 12.
</li>
<li>
<a href = "p12_title.m">p12_title.m</a>
returns the title for problem 12.
</li>
<li>
<a href = "p13_alpha.m">p13_alpha.m</a>
returns ALPHA for problem 13.
</li>
<li>
<a href = "p13_exact.m">p13_exact.m</a>
returns the exact integral for problem 13.
</li>
<li>
<a href = "p13_fun.m">p13_fun.m</a>
evaluates the integrand for problem 13.
</li>
<li>
<a href = "p13_title.m">p13_title.m</a>
returns the title for problem 13.
</li>
<li>
<a href = "p14_alpha.m">p14_alpha.m</a>
returns ALPHA for problem 14.
</li>
<li>
<a href = "p14_exact.m">p14_exact.m</a>
returns the estimated integral for problem 14.
</li>
<li>
<a href = "p14_fun.m">p14_fun.m</a>
evaluates the integrand for problem 14.
</li>
<li>
<a href = "p14_title.m">p14_title.m</a>
returns the title for problem 14.
</li>
<li>
<a href = "p15_alpha.m">p15_alpha.m</a>
returns ALPHA for problem 15.
</li>
<li>
<a href = "p15_exact.m">p15_exact.m</a>
returns the estimated integral for problem 15.
</li>
<li>
<a href = "p15_fun.m">p15_fun.m</a>
evaluates the integrand for problem 15.
</li>
<li>
<a href = "p15_title.m">p15_title.m</a>
returns the title for problem 15.
</li>
<li>
<a href = "p16_alpha.m">p16_alpha.m</a>
returns ALPHA for problem 16.
</li>
<li>
<a href = "p16_exact.m">p16_exact.m</a>
returns the estimated integral for problem 16.
</li>
<li>
<a href = "p16_fun.m">p16_fun.m</a>
evaluates the integrand for problem 16.
</li>
<li>
<a href = "p16_title.m">p16_title.m</a>
returns the title for problem 16.
</li>
<li>
<a href = "p17_alpha.m">p17_alpha.m</a>
returns ALPHA for problem 17.
</li>
<li>
<a href = "p17_exact.m">p17_exact.m</a>
returns the estimated integral for problem 17.
</li>
<li>
<a href = "p17_fun.m">p17_fun.m</a>
evaluates the integrand for problem 17.
</li>
<li>
<a href = "p17_title.m">p17_title.m</a>
returns the title for problem 17.
</li>
<li>
<a href = "p18_alpha.m">p18_alpha.m</a>
returns ALPHA for problem 18.
</li>
<li>
<a href = "p18_exact.m">p18_exact.m</a>
returns the estimated integral for problem 18.
</li>
<li>
<a href = "p18_fun.m">p18_fun.m</a>
evaluates the integrand for problem 18.
</li>
<li>
<a href = "p18_title.m">p18_title.m</a>
returns the title for problem 18.
</li>
<li>
<a href = "p19_alpha.m">p19_alpha.m</a>
returns ALPHA for problem 19.
</li>
<li>
<a href = "p19_exact.m">p19_exact.m</a>
returns the estimated integral for problem 19.
</li>
<li>
<a href = "p19_fun.m">p19_fun.m</a>
evaluates the integrand for problem 19.
</li>
<li>
<a href = "p19_title.m">p19_title.m</a>
returns the title for problem 19.
</li>
<li>
<a href = "p20_alpha.m">p20_alpha.m</a>
returns ALPHA for problem 20.
</li>
<li>
<a href = "p20_exact.m">p20_exact.m</a>
returns the estimated integral for problem 20.
</li>
<li>
<a href = "p20_fun.m">p20_fun.m</a>
evaluates the integrand for problem 20.
</li>
<li>
<a href = "p20_title.m">p20_title.m</a>
returns the title for problem 20.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>
prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "laguerre_test_int_test.m">laguerre_test_int_test.m</a>,
calls all the tests
</li>
<li>
<a href = "laguerre_test_int_test01.m">laguerre_test_int_test01.m</a>,
demonstrates P00_PROBLEM_NUM and P00_TITLE.
</li>
<li>
<a href = "laguerre_test_int_test02.m">laguerre_test_int_test02.m</a>,
demonstrates P00_ALPHA and P00_EXACT.
</li>
<li>
<a href = "laguerre_test_int_test03.m">laguerre_test_int_test03.m</a>,
demonstrates P00_GAUSS_LAGUERRE.
</li>
<li>
<a href = "laguerre_test_int_test04.m">laguerre_test_int_test04.m</a>,
demonstrates P00_EXP_TRANSFORM.
</li>
<li>
<a href = "laguerre_test_int_test05.m">laguerre_test_int_test05.m</a>,
demonstrates P00_RAT_TRANSFORM.
</li>
<li>
<a href = "laguerre_test_int_test_output.txt">laguerre_test_int_test_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 28 December 2011.
</i>
<!-- John Burkardt -->
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