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<html>
<head>
<title>
TEST_INT_2D - Quadrature Tests for 2D Finite Intervals
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TEST_INT_2D <br> Quadrature Tests for 2D Finite Intervals
</h1>
<hr>
<p>
<b>TEST_INT_2D</b>
is a FORTRAN90 program which
evaluates test integrands of a 2D argument.
</p>
<p>
The test integrands would normally be used to testing 2D
quadrature software. It is possible to invoke a
particular function by number, or to try out all available functions,
as demonstrated in the sample calling program.
</p>
<p>
The current set of problems is:
<ol>
<li>
integral on [0,1]x[0,1] of f(x,y) = 1 / ( 1 - x * y );
singular at [1,1].
</li>
<li>
integral on [-1,1]x[-1,1] of f(x,y) = 1 / sqrt ( 1 - x * x * y * y );
singular at [1,1], [1,-1], [-1,1], [-1,-1];
</li>
<li>
integral on [-1,1]x[-1,1] of f(x,y) = 1 / sqrt ( 2 - x - y );
singular at [1,1];
</li>
<li>
integral on [-1,1]x[-1,1] of f(x,y) = 1 / sqrt ( 3 - x - 2 * y );
singular along the line y = ( 3 - x ) / 2.
</li>
<li>
integral on [0,1]x[0,1] of f(x,y) = sqrt ( x * y );
singular along the lines y = 0 and x = 0.
</li>
<li>
integral on [-1,1]x[-1,1] of f(x,y) = abs ( x * x + y * y - 1/4 );
nondifferentiable along x*x+y*y=1/4.
</li>
<li>
integral on [0,1]x[0,1] of f(x,y) = sqrt ( abs ( x - y ) );
nondifferentiable along y = x.
</li>
<li>
integral on [0,5]x[0,5] of f(x,y) = exp ( - ( (x-4)^2 + (y-1)^2 ) ),
highly localized near (4,1).
</li>
</ol>
</p>
<p>
The library includes not just the integrand, but also the interval
of integration, and the exact value of the integral.
Thus, for each integrand function, three subroutines are supplied. For
instance, for function #5, we have the routines:
<ul>
<li>
<b>P05_FUN</b> evaluates the integrand for problem 5.
</li>
<li>
<b>P05_LIM</b> returns the integration limits for problem 5.
</li>
<li>
<b>P05_EXACT</b> returns the exact integral for problem 5.
</li>
</ul>
So once you have the calling sequences for these routines, you
can easily evaluate the function, or integrate it between the
appropriate limits, 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 2 instead, you simply change
the "05" to "02" 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_LIM</b> returns the integration limits for any problem.
</li>
<li>
<b>P00_EXACT</b> returns the exact integral for any problem.
</li>
</ul>
</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>TEST_INT_2D</b> is available in
<a href = "../../c_src/test_int_2d/test_int_2d.html">a C version</a> and
<a href = "../../cpp_src/test_int_2d/test_int_2d.html">a C++ version</a> and
<a href = "../../f_src/test_int_2d/test_int_2d.html">a FORTRAN90 version</a> and
<a href = "../../f77_src/test_int_2d/test_int_2d.html">a FORTRAN77 version</a> and
<a href = "../../m_src/test_int_2d/test_int_2d.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/test_int/test_int.html">
TEST_INT</a>,
a FORTRAN90 library which
defines some test integration problems over 1D finite intervals.
</p>
<p>
<a href = "../../f_src/test_int_hermite/test_int_hermite.html">
TEST_INT_HERMITE</a>,
a FORTRAN90 library which
defines some test integration problems over infinite intervals.
</p>
<p>
<a href = "../../f_src/test_int_laguerre/test_int_laguerre.html">
TEST_INT_LAGUERRE</a>,
a FORTRAN90 library which
defines test integrands for integration over [ALPHA,+oo).
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Gwynne Evans,<br>
Practical Numerical Integration,<br>
Wiley, 1993,<br>
ISBN: 047193898X,<br>
LC: QA299.3E93.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "test_int_2d.f90">test_int_2d.f90</a>, the source code.
</li>
<li>
<a href = "test_int_2d.sh">test_int_2d.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "test_int_2d_prb.f90">test_int_2d_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "test_int_2d_prb.sh">test_int_2d_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "test_int_2d_prb_output.txt">test_int_2d_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>LEGENDRE_SET</b> sets abscissas and weights for Gauss-Legendre quadrature.
</li>
<li>
<b>P00_EXACT</b> returns the exact integral for any problem.
</li>
<li>
<b>P00_FUN</b> evaluates the integrand for any problem.
</li>
<li>
<b>P00_LIM</b> returns the integration limits for any problem.
</li>
<li>
<b>P00_PROBLEM_NUM</b> returns the number of test integration problems.
</li>
<li>
<b>P01_EXACT</b> returns the exact integral for problem 1.
</li>
<li>
<b>P01_FUN</b> evaluates the integrand for problem 1.
</li>
<li>
<b>P01_LIM</b> returns the integration limits for problem 1.
</li>
<li>
<b>P02_EXACT</b> returns the exact integral for problem 2.
</li>
<li>
<b>P02_FUN</b> evaluates the integrand for problem 2.
</li>
<li>
<b>P02_LIM</b> returns the integration limits for problem 2.
</li>
<li>
<b>P03_EXACT</b> returns the exact integral for problem 3.
</li>
<li>
<b>P03_FUN</b> evaluates the integrand for problem 3.
</li>
<li>
<b>P03_LIM</b> returns the integration limits for problem 3.
</li>
<li>
<b>P04_EXACT</b> returns the exact integral for problem 4.
</li>
<li>
<b>P04_FUN</b> evaluates the integrand for problem 4.
</li>
<li>
<b>P04_LIM</b> returns the integration limits for problem 4.
</li>
<li>
<b>P05_EXACT</b> returns the exact integral for problem 5.
</li>
<li>
<b>P05_FUN</b> evaluates the integrand for problem 5.
</li>
<li>
<b>P05_LIM</b> returns the integration limits for problem 5.
</li>
<li>
<b>P06_EXACT</b> returns the exact integral for problem 6.
</li>
<li>
<b>P06_FUN</b> evaluates the integrand for problem 6.
</li>
<li>
<b>P06_LIM</b> returns the integration limits for problem 6.
</li>
<li>
<b>P07_EXACT</b> returns the exact integral for problem 7.
</li>
<li>
<b>P07_FUN</b> evaluates the integrand for problem 7.
</li>
<li>
<b>P07_LIM</b> returns the integration limits for problem 7.
</li>
<li>
<b>P08_EXACT</b> returns the exact integral for problem 8.
</li>
<li>
<b>P08_FUN</b> evaluates the integrand for problem 8.
</li>
<li>
<b>P08_LIM</b> returns the integration limits for problem 8.
</li>
<li>
<b>R8_CSEVL</b> evaluates a Chebyshev series.
</li>
<li>
<b>R8_ERF</b> evaluates the error function of an R8 argument.
</li>
<li>
<b>R8_ERFC</b> evaluates the co-error function of an R8 argument.
</li>
<li>
<b>R8_INITS</b> initializes a Chebyshev series.
</li>
<li>
<b>R8_MACH</b> returns real ( kind = 8 ) real machine-dependent constants.
</li>
<li>
<b>R8MAT_UNIFORM_01</b> returns a unit pseudorandom R8MAT.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 18 September 2011.
</i>
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