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<html>
<head>
<title>
TEST_NINT - Multi-dimensional Integration Test Functions
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TEST_NINT <br> Multi-dimensional Integration <br> Test Functions.
</h1>
<hr>
<p>
<b>TEST_NINT</b>
is a FORTRAN90 library which
defines test problems for the approximate computation of integrals
over multi-dimensional regions.
</p>
<p>
Routines are available to evaluate the integrand, return the exact
value of the integral, report the name of the problem,
report the integration limits, get, set or modify a base point.
</p>
<p>
The integrands is assigned an index. The integrands can be invoked
by index. Most integrands may be defined for any value of the
spatial dimension, which we denote here by <b>m</b>. Most integrands
are defined on the unit <b>m</b>-dimensional hypercube. Some
integrands include one or more parameters. These generally have
default values, which can be altered by the user.
</p>
<p>
For each problem, a set of routines are available with a standard
interface, for manipulating and evaluating the problem. For a
problem with index "87", for instance, we might have the following
set of routines. The most important is <b>P87_F</b> which evaluates
the integrand. We probably also need <b>P87_LIM</b> to determine
the limits of integration, and <b>P87_EXACT</b> to get the exact
value of the integral (if known). A number of routines are available
to set, get, or randomize parameters associated with the problem.
<ul>
<li>
<b>P87_DEFAULT</b> sets default values for problem 87.
</li>
<li>
<b>P87_EXACT</b> returns the exact integral for problem 87.
</li>
<li>
<b>P87_F</b> evaluates the integrand for problem 87.
</li>
<li>
<b>P87_I4</b> sets or gets integer scalar parameters for problem 87.
</li>
<li>
<b>P87_I4VEC</b> sets or gets integer vector parameters for problem 87.
</li>
<li>
<b>P87_LIM</b> returns the integration limits for problem 87.
</li>
<li>
<b>P87_NAME</b> returns the name of problem 87.
</li>
<li>
<b>P87_R8</b> sets or gets real scalar parameters for problem 87.
</li>
<li>
<b>P87_R8VEC</b> sets or gets real vector parameters for problem 87.
</li>
<li>
<b>P87_REGION</b> returns the name of the integration region
for problem 87.
</li>
<li>
<b>P87_TITLE</b> prints a title for problem 87.
</li>
</ul>
</p>
<p>
The list of integrand functions includes:
<ol>
<li>
f(x) = ( sum ( x(1:m) ) )^2;
</li>
<li>
f(x) = ( sum ( 2 * x(1:m) - 1 ) )^4;
</li>
<li>
f(x) = ( sum ( x(1:m) ) )^5;
</li>
<li>
f(x) = ( sum ( 2 * x(1:m) - 1 ) )^6;
</li>
<li>
f(x) = 1 / ( 1 + sum ( 2 * x(1:m) ) );
</li>
<li>
f(x) = product ( 2 * abs ( 2 * x(1:m) - 1 ) );
</li>
<li>
f(x) = product ( pi / 2 ) * sin ( pi * x(1:m) );
</li>
<li>
f(x) = ( sin ( (pi/4) * sum ( x(1:m) ) ) )^2;
</li>
<li>
f(x) = exp ( sum ( c(1:m) * x(1:m) ) );
</li>
<li>
f(x) = sum ( abs ( x(1:m) - 0.5 ) );
</li>
<li>
f(x) = exp ( sum ( abs ( 2 * x(1:m) - 1 ) ) );
</li>
<li>
f(x) = product ( 1 <= i <= m ) ( i * cos ( i * x(i) ) );
</li>
<li>
f(x) = product ( 1 <= i <= m ) t(n(i))(x(i)), t(n(i))
is a Chebyshev polynomial;
</li>
<li>
f(x) = sum ( 1 <= i <= m ) (-1)^i * product ( 1 <= j <= i ) x(j);
</li>
<li>
f(x) = product ( 1 <= i <= order ) x(mod(i-1,m)+1);
</li>
<li>
f(x) = sum ( abs ( x(1:m) - x0(1:m) ) );
</li>
<li>
f(x) = sum ( ( x(1:m) - x0(1:m) )^2 );
</li>
<li>
f(x) = 1 inside an m-dimensional sphere around x0(1:m), 0 outside;
</li>
<li>
f(x) = product ( sqrt ( abs ( x(1:m) - x0(1:m) ) ) );
</li>
<li>
f(x) = ( sum ( x(1:m) ) ^power;
</li>
<li>
f(x) = c * product ( x(1:m)^e(1:m) ) on the surface of
an m-dimensional unit sphere;
</li>
<li>
f(x) = c * product ( x(1:m)^e(1:m) ) in an m-dimensional ball;
</li>
<li>
f(x) = c * product ( x(1:m)^e(1:m) ) in the unit m-dimensional simplex;
</li>
<li>
f(x) = product ( abs ( 4 * x(1:m) - 2 ) + c(1:m) )
/ ( 1 + c(1:m) ) );
</li>
<li>
f(x) = exp ( c * product ( x(1:m) ) );
</li>
<li>
f(x) = product ( c(1:m) * exp ( - c(1:m) * x(1:m) ) );
</li>
<li>
f(x) = cos ( 2 * pi * r + sum ( c(1:m) * x(1:m) ) ), <br>
Genz "Oscillatory";
</li>
<li>
f(x) = 1 / product ( c(1:m)^2 + (x(1:m) - x0(1:m))^2),<br>
Genz "Product Peak";
</li>
<li>
f(x) = 1 / ( 1 + sum ( c(1:m) * x(1:m) ) )^(m+r),<br>
Genz "Corner Peak";
</li>
<li>
f(x) = exp(-sum(c(1:m)^2 * ( x(1:m) - x0(1:m))^2 ) ),<br>
Genz "Gaussian";
</li>
<li>
f(x) = exp ( - sum ( c(1:m) * abs ( x(1:m) - x0(1:m) ) ) ),
Genz "Continuous";
</li>
<li>
f(x) = exp(sum(c(1:m)*x(1:m)) for x(1:m) <= x0(1:m), 0 otherwise,<br>
Genz "Discontinuous";
</li>
<li>
f(x) = sum ( x(1:n)^2 )<br>
Ball R^2;
</li>
</ol>
</p>
<h3 align = "center">
An Important Quote:
</h3>
<p>
<blockquote><i>
"When good results are obtained in integrating a high-dimensional
function, we should conclude first of all that an especially tractable
integrand was tried and not that a generally successful method has
been found. A secondary conclusion is that we might have made a
very good choice in selecting an integration method to exploit whatever
features of f made it tractable."
</i></blockquote>
Art Owen,<br>
Latin Supercube Sampling for Very High Dimensional Simulation,<br>
ACM Transactions on Modeling and Computer Simulations,<br>
Volume 8, Number 1, January 1998, pages 71-102.
</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_NINT</b> is available in
<a href = "../../cpp_src/test_nint/test_nint.html">a C++ version</a> and
<a href = "../../f_src/test_nint/test_nint.html">a FORTRAN90 version</a> and
<a href = "../../m_src/test_nint/test_nint.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/ball_volume_monte_carlo/ball_volume_monte_carlo.html">
BALL_VOLUME_MONTE_CARLO</a>,
a FORTRAN90 program which
applies a Monte Carlo method to estimate the volume of the unit 6D ball;
</p>
<p>
<a href = "../../f_src/cubpack/cubpack.html">
CUBPACK</a>,
a FORTRAN90 library which
estimates the integral of a function over a collection of N-dimensional
hyperrectangles and simplices.
</p>
<p>
<a href = "../../cpp_src/gsl/gsl.html">
GSL</a>,
a C++ library which
includes routines for estimating multidimensional integrals.
</p>
<p>
<a href = "../../f_src/integral_test/integral_test.html">
INTEGRAL_TEST</a>,
a FORTRAN90 program which
tests the suitability of a set of N points for use in an equal-weight quadrature rule over
the multi-dimensional unit hypercube.
</p>
<p>
<a href = "../../f_src/nint_exactness/nint_exactness.html">
NINT_EXACTNESS</a>,
a FORTRAN90 program which
measures the polynomial exactness of a multidimensional quadrature rule.
</p>
<p>
<a href = "../../f_src/nintlib/nintlib.html">
NINTLIB</a>,
a FORTRAN90 library which
estimates integrals over multidimensional regions.
</p>
<p>
<a href = "../../f_src/product_rule/product_rule.html">
PRODUCT_RULE</a>,
a FORTRAN90 program which
constructs a product quadrature rule from 1D factor rules.
</p>
<p>
<a href = "../../datasets/quadrature_rules/quadrature_rules.html">
QUADRATURE_RULES</a>,
a dataset directory which
contains sets of files that define quadrature
rules over various 1D intervals or multidimensional hypercubes.
</p>
<p>
<a href = "../../f_src/quadrature_test/quadrature_test.html">
QUADRATURE_TEST</a>,
a FORTRAN90 program which
reads files defining a quadrature rule, and
applies them to all the test integrals defined by <b>TEST_NINT</b>.
</p>
<p>
<a href = "../../f_src/sandia_sparse/sandia_sparse.html">
SANDIA_SPARSE</a>,
a FORTRAN90 library which
can produce a multidimensional sparse grid, based on a variety of 1D
quadrature rules; only isotropic grids are generated, that is, the same
rule is used in each dimension, and the same maximum order is used in
each dimension.
</p>
<p>
<a href = "../../f77_src/simpack/simpack.html">
SIMPACK</a>,
a FORTRAN77 library which
approximates the integral of a function over a multidimensional simplex.
</p>
<p>
<a href = "../../f_src/sparse_grid_cc/sparse_grid_cc.html">
SPARSE_GRID_CC</a>,
a FORTRAN90 library which
creates sparse grids based on Clenshaw-Curtis rules.
</p>
<p>
<a href = "../../f_src/stroud/stroud.html">
STROUD</a>,
a FORTRAN90 library which
defines quadrature rules for a variety of multidimensional reqions.
</p>
<p>
<a href = "../../f_src/test_int_2d/test_int_2d.html">
TEST_INT_2D</a>,
a FORTRAN90 library which
defines test integrands for 2D quadrature rules.
</p>
<p>
<a href = "../../f_src/testpack/testpack.html">
TESTPACK</a>,
a FORTRAN90 library which
defines a set of integrands used to test multidimensional quadrature.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
JD Beasley, SG Springer,<br>
Algorithm AS 111:
The Percentage Points of the Normal Distribution,<br>
Applied Statistics,<br>
Volume 26, 1977, pages 118-121.
</li>
<li>
Paul Bratley, Bennett Fox, Harald Niederreiter,<br>
Implementation and Tests of Low-Discrepancy Sequences,<br>
ACM Transactions on Modeling and Computer Simulation,<br>
Volume 2, Number 3, July 1992, pages 195-213.
</li>
<li>
Roger Broucke,<br>
Algorithm 446:
Ten Subroutines for the Manipulation of Chebyshev Series,<br>
Communications of the ACM,<br>
Volume 16, 1973, pages 254-256.
</li>
<li>
William Cody, Kenneth Hillstrom,<br>
Chebyshev Approximations for the Natural Logarithm of the
Gamma Function,
Mathematics of Computation,<br>
Volume 21, Number 98, April 1967, pages 198-203.
</li>
<li>
Richard Crandall,<br>
Projects in Scientific Computing,<br>
Springer, 2005,<br>
ISBN: 0387950095,<br>
LC: Q183.9.C733.
</li>
<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>
Gerald Folland,<br>
How to Integrate a Polynomial Over a Sphere,<br>
American Mathematical Monthly,<br>
Volume 108, Number 5, May 2001, pages 446-448.
</li>
<li>
Leslie Fox, Ian Parker,<br>
Chebyshev Polynomials in Numerical Analysis,<br>
Oxford Press, 1968,<br>
LC: QA297.F65.
</li>
<li>
Alan Genz,<br>
Testing Multidimensional Integration Routines,<br>
in Tools, Methods, and Languages for Scientific and
Engineering Computation,<br>
edited by B Ford, JC Rault, F Thomasset,<br>
North-Holland, 1984, pages 81-94,<br>
ISBN: 0444875700,<br>
LC: Q183.9.I53.
</li>
<li>
Alan Genz,<br>
A Package for Testing Multiple Integration Subroutines,<br>
in Numerical Integration:
Recent Developments, Software and Applications,<br>
edited by Patrick Keast, Graeme Fairweather,<br>
Reidel, 1987, pages 337-340,<br>
ISBN: 9027725144,<br>
LC: QA299.3.N38.
</li>
<li>
Kenneth Hanson,<br>
Quasi-Monte Carlo: halftoning in high dimensions?<br>
in Computatinal Imaging,<br>
Edited by CA Bouman, RL Stevenson,<br>
Proceedings SPIE,<br>
Volume 5016, 2003, pages 161-172.
</li>
<li>
John Hart, Ward Cheney, Charles Lawson, Hans Maehly,
Charles Mesztenyi, John Rice, Henry Thatcher,
Christoph Witzgall,<br>
Computer Approximations,<br>
Wiley, 1968,<br>
LC: QA297.C64.
</li>
<li>
Claude Itzykson, Jean-Michel Drouffe,<br>
Statistical Field Theory,<br>
Volume 1,<br>
Cambridge, 1991,<br>
ISBN: 0-521-40806-7,<br>
LC: QC174.8.I89.
</li>
<li>
Stephen Joe, Frances Kuo<br>
Remark on Algorithm 659:
Implementing Sobol's Quasirandom Sequence Generator,<br>
ACM Transactions on Mathematical Software,<br>
Volume 29, Number 1, March 2003, pages 49-57.
</li>
<li>
David Kahaner, Cleve Moler, Steven Nash,<br>
Numerical Methods and Software,<br>
Prentice Hall, 1989,<br>
ISBN: 0-13-627258-4,<br>
LC: TA345.K34.
</li>
<li>
Bradley Keister,<br>
Multidimensional Quadrature Algorithms,<br>
Computers in Physics,<br>
Volume 10, Number 2, March/April, 1996, pages 119-122.
</li>
<li>
Arnold Krommer, Christoph Ueberhuber,<br>
Numerical Integration on Advanced Computer Systems,<br>
Springer, 1994,<br>
ISBN: 3540584102,<br>
LC: QA299.3.K76.
</li>
<li>
Anargyros Papageorgiou, Joseph Traub,<br>
Faster Evaluation of Multidimensional Integrals,<br>
Computers in Physics,<br>
Volume 11, Number 6, November/December 1997, pages 574-578.
</li>
<li>
Thomas Patterson,<br>
On the Construction of a Practical Ermakov-Zolotukhin
Multiple Integrator,<br>
in Numerical Integration:
Recent Developments, Software and Applications,<br>
edited by Patrick Keast and Graeme Fairweather,<br>
D. Reidel, 1987, pages 269-290,
LC: QA299.3.N38.
</li>
<li>
Arthur Stroud,<br>
Approximate Calculation of Multiple Integrals,<br>
Prentice Hall, 1971,<br>
ISBN: 0130438936,<br>
LC: QA311.S85.
</li>
<li>
Arthur Stroud, Don Secrest,<br>
Gaussian Quadrature Formulas,<br>
Prentice Hall, 1966,<br>
LC: QA299.4G3S7.
</li>
<li>
Xiaoqun Wang, Kai-Tai Fang,<br>
The Effective Dimension and quasi-Monte Carlo Integration,<br>
Journal of Complexity,<br>
Volume 19, pages 101-124, 2003.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "test_nint.f90">test_nint.f90</a>, the source code.
</li>
<li>
<a href = "test_nint.sh">test_nint.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "test_nint_prb.f90">test_nint_prb.f90</a>, a sample problem.
</li>
<li>
<a href = "test_nint_prb.sh">test_nint_prb.sh</a>,
commands to compile, link and run the sample problem.
</li>
<li>
<a href = "test_nint_prb_output.txt">
test_nint_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CSEVL</b> evaluates an N term Chebyshev series.
</li>
<li>
<b>ERROR_F</b> computes the error function.
</li>
<li>
<b>ERROR_FC</b> computes the complementary error function.
</li>
<li>
<b>GAMMA_LOG</b> calculates the natural logarithm of GAMMA ( X ) for positive X.
</li>
<li>
<b>GET_PROBLEM_NUM</b> returns the number of test integration problems.
</li>
<li>
<b>INITS</b> estimates the order of an orthogonal series for a given accuracy.
</li>
<li>
<b>NORMAL_01_CDF_INV</b> inverts the Normal 01 CDF.
</li>
<li>
<b>P00_BOX_GL05</b> uses Gauss-Legendre quadrature in an N-dimensional box.
</li>
<li>
<b>P00_BOX_MC</b> integrates over an multi-dimensional box using Monte Carlo.
</li>
<li>
<b>P00_DEFAULT</b> sets a problem to a default state.
</li>
<li>
<b>P00_EXACT</b> returns the exact integral for any problem.
</li>
<li>
<b>P00_F</b> evaluates the integrand for any problem.
</li>
<li>
<b>P00_I4</b> sets or gets I4 parameters for any problem.
</li>
<li>
<b>P00_LIM</b> returns the integration limits for any problem.
</li>
<li>
<b>P00_NAME</b> returns the name of the problem.
</li>
<li>
<b>P00_R8VEC</b> sets or gets R8VEC parameters for any problem.
</li>
<li>
<b>P00_REGION</b> returns the name of the integration region for any problem.
</li>
<li>
<b>P00_REMAP01</b> remaps points in [0,1]^DIM_NUM into [A(1:DIM_NUM),B(1:DIM_NUM)].
</li>
<li>
<b>P00_TITLE</b> prints a title for any problem.
</li>
<li>
<b>P00_VOLUME</b> returns the volume of the integration region.
</li>
<li>
<b>P01_DEFAULT</b> sets default values for problem 01.
</li>
<li>
<b>P01_EXACT</b> returns the exact integral for problem 01.
</li>
<li>
<b>P01_F</b> evaluates the integrand for problem 01.
</li>
<li>
<b>P01_I4</b> sets or gets I4 parameters for problem 01.
</li>
<li>
<b>P01_LIM</b> returns the integration limits for problem 01.
</li>
<li>
<b>P01_NAME</b> returns the name of problem 01.
</li>
<li>
<b>P01_REGION</b> returns the name of the integration region for problem 01.
</li>
<li>
<b>P01_TITLE</b> prints a title for problem 01.
</li>
<li>
<b>P02_DEFAULT</b> sets default values for problem 02.
</li>
<li>
<b>P02_EXACT</b> returns the exact integral for problem 02.
</li>
<li>
<b>P02_F</b> evaluates the integrand for problem 02.
</li>
<li>
<b>P02_I4</b> sets or gets I4 parameters for problem 02.
</li>
<li>
<b>P02_LIM</b> returns the integration limits for problem 02.
</li>
<li>
<b>P02_NAME</b> returns the name of problem 02.
</li>
<li>
<b>P02_REGION</b> returns the name of the integration region for problem 02.
</li>
<li>
<b>P02_TITLE</b> prints a title for problem 02.
</li>
<li>
<b>P03_DEFAULT</b> sets default values for problem 03.
</li>
<li>
<b>P03_EXACT</b> returns the exact integral for problem 03.
</li>
<li>
<b>P03_F</b> evaluates the integrand for problem 03.
</li>
<li>
<b>P03_I4</b> sets or gets I4 parameters for problem 03.
</li>
<li>
<b>P03_LIM</b> returns the integration limits for problem 03.
</li>
<li>
<b>P03_NAME</b> returns the name of problem 03.
</li>
<li>
<b>P03_REGION</b> returns the name of the integration region for problem 03.
</li>
<li>
<b>P03_TITLE</b> prints a title for problem 03.
</li>
<li>
<b>P04_DEFAULT</b> sets default values for problem 04.
</li>
<li>
<b>P04_EXACT</b> returns the exact integral for problem 04.
</li>
<li>
<b>P04_F</b> evaluates the integrand for problem 04.
</li>
<li>
<b>P04_I4</b> sets or gets I4 parameters for problem 04.
</li>
<li>
<b>P04_LIM</b> returns the integration limits for problem 04.
</li>
<li>
<b>P04_NAME</b> returns the name of problem 04.
</li>
<li>
<b>P04_REGION</b> returns the name of the integration region for problem 04.
</li>
<li>
<b>P04_TITLE</b> prints a title for problem 04.
</li>
<li>
<b>P05_DEFAULT</b> sets default values for problem 05.
</li>
<li>
<b>P05_EXACT</b> returns the exact integral for problem 05.
</li>
<li>
<b>P05_F</b> evaluates the integrand for problem 05.
</li>
<li>
<b>P05_I4</b> sets or gets I4 parameters for problem 05.
</li>
<li>
<b>P05_LIM</b> returns the integration limits for problem 05.
</li>
<li>
<b>P05_NAME</b> returns the name of problem 05.
</li>
<li>
<b>P05_REGION</b> returns the name of the integration region for problem 05.
</li>
<li>
<b>P05_TITLE</b> prints a title for problem 05.
</li>
<li>
<b>P06_DEFAULT</b> sets default values for problem 06.
</li>
<li>
<b>P06_EXACT</b> returns the exact integral for problem 06.
</li>
<li>
<b>P06_F</b> evaluates the integrand for problem 06.
</li>
<li>
<b>P06_I4</b> sets or gets I4 parameters for problem 06.
</li>
<li>
<b>P06_LIM</b> returns the integration limits for problem 06.
</li>
<li>
<b>P06_NAME</b> returns the name of problem 06.
</li>
<li>
<b>P06_REGION</b> returns the name of the integration region for problem 06.
</li>
<li>
<b>P06_TITLE</b> prints a title for problem 06.
</li>
<li>
<b>P07_DEFAULT</b> sets default values for problem 07.
</li>
<li>
<b>P07_EXACT</b> returns the exact integral for problem 07.
</li>
<li>
<b>P07_F</b> evaluates the integrand for problem 07.
</li>
<li>
<b>P07_I4</b> sets or gets I4 parameters for problem 07.
</li>
<li>
<b>P07_LIM</b> returns the integration limits for problem 07.
</li>
<li>
<b>P07_NAME</b> returns the name of problem 07.
</li>
<li>
<b>P07_REGION</b> returns the name of the integration region for problem 07.
</li>
<li>
<b>P07_TITLE</b> prints a title for problem 07.
</li>
<li>
<b>P08_DEFAULT</b> sets default values for problem 08.
</li>
<li>
<b>P08_EXACT</b> returns the exact integral for problem 08.
</li>
<li>
<b>P08_F</b> evaluates the integrand for problem 08.
</li>
<li>
<b>P08_I4</b> sets or gets I4 parameters for problem 08.
</li>
<li>
<b>P08_LIM</b> returns the integration limits for problem 08.
</li>
<li>
<b>P08_NAME</b> returns the name of problem 08.
</li>
<li>
<b>P08_REGION</b> returns the name of the integration region for problem 08.
</li>
<li>
<b>P08_TITLE</b> prints a title for problem 08.
</li>
<li>
<b>P09_DEFAULT</b> sets default values for problem 09.
</li>
<li>
<b>P09_EXACT</b> returns the exact integral for problem 09.
</li>
<li>
<b>P09_F</b> evaluates the integrand for problem 09.
</li>
<li>
<b>P09_I4</b> sets or gets I4 parameters for problem 09.
</li>
<li>
<b>P09_LIM</b> returns the integration limits for problem 09.
</li>
<li>
<b>P09_NAME</b> returns the name of problem 09.
</li>
<li>
<b>P09_REGION</b> returns the name of the integration region for problem 09.
</li>
<li>
<b>P09_R8VEC</b> sets or gets R8VEC parameters for problem 09.
</li>
<li>
<b>P09_TITLE</b> prints a title for problem 09.
</li>
<li>
<b>P10_DEFAULT</b> sets default values for problem 10.
</li>
<li>
<b>P10_EXACT</b> returns the exact integral for problem 10.
</li>
<li>
<b>P10_F</b> evaluates the integrand for problem 10.
</li>
<li>
<b>P10_I4</b> sets or gets I4 parameters for problem 10.
</li>
<li>
<b>P10_LIM</b> returns the integration limits for problem 10.
</li>
<li>
<b>P10_NAME</b> returns the name of problem 10.
</li>
<li>
<b>P10_REGION</b> returns the name of the integration region for problem 10.
</li>
<li>
<b>P10_TITLE</b> prints a title for problem 10.
</li>
<li>
<b>P11_DEFAULT</b> sets default values for problem 11.
</li>
<li>
<b>P11_EXACT</b> returns the exact integral for problem 11.
</li>
<li>
<b>P11_F</b> evaluates the integrand for problem 11.
</li>
<li>
<b>P11_I4</b> sets or gets I4 parameters for problem 11.
</li>
<li>
<b>P11_LIM</b> returns the integration limits for problem 11.
</li>
<li>
<b>P11_NAME</b> returns the name of problem 11.
</li>
<li>
<b>P11_REGION</b> returns the name of the integration region for problem 11.
</li>
<li>
<b>P11_TITLE</b> prints a title for problem 11.
</li>
<li>
<b>P12_DEFAULT</b> sets default values for problem 12.
</li>
<li>
<b>P12_EXACT</b> returns the exact integral for problem 12.
</li>
<li>
<b>P12_F</b> evaluates the integrand for problem 12.
</li>
<li>
<b>P12_I4</b> sets or gets I4 parameters for problem 12.
</li>
<li>
<b>P12_LIM</b> returns the integration limits for problem 12.
</li>
<li>
<b>P12_NAME</b> returns the name of problem 12.
</li>
<li>
<b>P12_REGION</b> returns the name of the integration region for problem 12.
</li>
<li>
<b>P12_TITLE</b> prints a title for problem 12.
</li>
<li>
<b>P13_DEFAULT</b> sets default values for problem 13.
</li>
<li>
<b>P13_EXACT</b> returns the exact integral for problem 13.
</li>
<li>
<b>P13_F</b> evaluates the integrand for problem 13.
</li>
<li>
<b>P13_I4</b> sets or gets I4 parameters for problem 13.
</li>
<li>
<b>P13_LIM</b> returns the integration limits for problem 13.
</li>
<li>
<b>P13_NAME</b> returns the name of problem 13.
</li>
<li>
<b>P13_REGION</b> returns the name of the integration region for problem 13.
</li>
<li>
<b>P13_TITLE</b> prints a title for problem 13.
</li>
<li>
<b>P14_DEFAULT</b> sets default values for problem 14.
</li>
<li>
<b>P14_EXACT</b> returns the exact integral for problem 14.
</li>
<li>
<b>P14_F</b> evaluates the integrand for problem 14.
</li>
<li>
<b>P14_I4</b> sets or gets I4 parameters for problem 14.
</li>
<li>
<b>P14_LIM</b> returns the integration limits for problem 14.
</li>
<li>
<b>P14_NAME</b> returns the name of problem 14.
</li>
<li>
<b>P14_REGION</b> returns the name of the integration region for problem 14.
</li>
<li>
<b>P14_TITLE</b> prints a title for problem 14.
</li>
<li>
<b>P15_DEFAULT</b> sets default values for problem 15.
</li>
<li>
<b>P15_EXACT</b> returns the exact integral for problem 15.
</li>
<li>
<b>P15_F</b> evaluates the integrand for problem 15.
</li>
<li>
<b>P15_I4</b> sets or gets I4 parameters for problem 15.
</li>
<li>
<b>P15_LIM</b> returns the integration limits for problem 15.
</li>
<li>
<b>P15_NAME</b> returns the name of problem 15.
</li>
<li>
<b>P15_REGION</b> returns the name of the integration region for problem 15.
</li>
<li>
<b>P15_TITLE</b> prints a title for problem 15.
</li>
<li>
<b>P16_DEFAULT</b> sets default values for problem 16.
</li>
<li>
<b>P16_EXACT</b> returns the exact integral for problem 16.
</li>
<li>
<b>P16_F</b> evaluates the integrand for problem 16.
</li>