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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 MATLAB library which
defines a set of 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>
</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/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
uses some of
these test integrals to evaluate sets of quadrature points.
</p>
<p>
<a href = "../../m_src/nint_exactness/nint_exactness.html">
NINT_EXACTNESS</a>,
a MATLAB program which
demonstrates how to measure the
polynomial exactness of a multidimensional quadrature rule.
</p>
<p>
<a href = "../../m_src/nintlib/nintlib.html">
NINTLIB</a>,
a MATLAB library which
numerically estimates integrals in multiple dimensions.
</p>
<p>
<a href = "../../m_src/product_rule/product_rule.html">
PRODUCT_RULE</a>,
a MATLAB program which
can create a multidimensional quadrature rule as a product of
one dimensional rules.
</p>
<p>
<a href = "../../datasets/quadrature_rules/quadrature_rules.html">
QUADRATURE_RULES</a>,
a dataset directory which
contains a description and examples of quadrature rules defined
by a set of "X", "W" and "R" files.
</p>
<p>
<a href = "../../m_src/quadrature_test/quadrature_test.html">
QUADRATURE_TEST</a>,
a MATLAB program which
reads the definition of a
multidimensional quadrature rule from three files, applies
the rule to a number of test integrals, and prints the
results.
</p>
<p>
<a href = "../../m_src/sandia_sparse/sandia_sparse.html">
SANDIA_SPARSE</a>,
a MATLAB 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 = "../../m_src/sparse_grid_cc/sparse_grid_cc.html">
SPARSE_GRID_CC</a>,
a MATLAB library which
creates sparse grids based on Clenshaw-Curtis rules.
</p>
<p>
<a href = "../../m_src/stroud/stroud.html">
STROUD</a>,
a MATLAB library which
contains quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
</p>
<p>
<a href = "../../m_src/testpack/testpack.html">
TESTPACK</a>,
a MATLAB 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 Compuer 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.
</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 = "csevl.m">csevl.m</a>,
evaluates an N-term Chebyshev series.
</li>
<li>
<a href = "error_f.m">error_f.m</a>,
evaluates the Error function.
</li>
<li>
<a href = "error_fc.m">error_fc.m</a>,
evaluates the complementary Error function.
</li>
<li>
<a href = "get_problem_num.m">get_problem_num.m</a>,
returns the number of test integration problems.
</li>
<li>
<a href = "inits.m">inits.m</a>,
estimates the order of an orthogonal series for a given accuracy.
</li>
<li>
<a href = "normal_01_cdf_inv.m">normal_01_cdf_inv.m</a>,
inverts the Normal 01 CDF.
</li>
<li>
<a href = "p00_box_gl05.m">p00_box_gl05.m</a>,
uses Gauss-Legendre quadrature in an N-dimensional box.
</li>
<li>
<a href = "p00_box_mc.m">p00_box_mc.m</a>,
integrates over an multi-dimensional box using Monte Carlo.
</li>
<li>
<a href = "p00_default.m">p00_default.m</a>,
sets a problem to a default state.
</li>
<li>
<a href = "p00_exact.m">p00_exact.m</a>,
returns the exact integral for any problem.
</li>
<li>
<a href = "p00_f.m">p00_f.m</a>,
evaluates the integrand for any problem.
</li>
<li>
<a href = "p00i4.m">p00_i4.m</a>,
sets or gets integer scalar parameters for any problem.
</li>
<li>
<a href = "p00_lim.m">p00_lim.m</a>,
returns the integration limits for any problem.
</li>
<li>
<a href = "p00_name.m">p00_name.m</a>,
returns the name of any problem.
</li>
<li>
<a href = "p00_r8vec.m">p00_r8vec.m</a>,
sets or gets real vector parameters for any problem.
</li>
<li>
<a href = "p00_region.m">p00_region.m</a>,
returns the name of the integration region for any problem.
</li>
<li>
<a href = "p00_remap01.m">p00_remap01.m</a>,
remaps points in [0,1]^DIM_NUM into [A(1:DIM_NUM),B(1:DIM_NUM)].
</li>
<li>
<a href = "p00_title.m">p00_title.m</a>,
prints a title for any problem.
</li>
<li>
<a href = "p00_volume.m">p00_volume.m</a>,
returns the volume of the integration region.
</li>
<li>
<a href = "p01_default.m">p01_default.m</a>,
sets default values for problem 01.
</li>
<li>
<a href = "p01_exact.m">p01_exact.m</a>,
sets the exact integral for problem 01.
</li>
<li>
<a href = "p01_f.m">p01_f.m</a>,
evaluates the integrand for problem 01.
</li>
<li>
<a href = "p01_i4.m">p01_i4.m</a>,
sets or gets I4 parameters for problem 01.
</li>
<li>
<a href = "p01_lim.m">p01_lim.m</a>,
returns the integration limits for problem 01.
</li>
<li>
<a href = "p01_name.m">p01_name.m</a>,
returns a name for problem 01.
</li>
<li>
<a href = "p01_region.m">p01_region.m</a>,
returns the region type for problem 01.
</li>
<li>
<a href = "p01_title.m">p01_title.m</a>,
prints a title for problem 01.
</li>
<li>
<a href = "p02_default.m">p02_default.m</a>,
sets default values for problem 02.
</li>
<li>
<a href = "p02_exact.m">p02_exact.m</a>,
sets the exact integral for problem 02.
</li>
<li>
<a href = "p02_f.m">p02_f.m</a>,
evaluates the integrand for problem 02.
</li>
<li>
<a href = "p02_i4.m">p02_i4.m</a>,
sets or gets I4 parameters for problem 02.
</li>
<li>
<a href = "p02_lim.m">p02_lim.m</a>,
returns the integration limits for problem 02.
</li>
<li>
<a href = "p02_name.m">p02_name.m</a>,
returns a name for problem 02.
</li>
<li>
<a href = "p02_region.m">p02_region.m</a>,
returns the region type for problem 02.
</li>
<li>
<a href = "p02_title.m">p02_title.m</a>,
prints a title for problem 02.
</li>
<li>
<a href = "p03_default.m">p03_default.m</a>,
sets default values for problem 03.
</li>
<li>
<a href = "p03_exact.m">p03_exact.m</a>,
sets the exact integral for problem 03.
</li>
<li>
<a href = "p03_f.m">p03_f.m</a>,
evaluates the integrand for problem 03.
</li>
<li>
<a href = "p03_i4.m">p03_i4.m</a>,
sets or gets I4 parameters for problem 03.
</li>
<li>
<a href = "p03_lim.m">p03_lim.m</a>,
returns the integration limits for problem 03.
</li>
<li>
<a href = "p03_name.m">p03_name.m</a>,
returns a name for problem 03.
</li>
<li>
<a href = "p03_region.m">p03_region.m</a>,
returns the region type for problem 03.
</li>
<li>
<a href = "p03_title.m">p03_title.m</a>,
prints a title for problem 03.
</li>
<li>
<a href = "p04_default.m">p04_default.m</a>,
sets default values for problem 04.
</li>
<li>
<a href = "p04_exact.m">p04_exact.m</a>,
sets the exact integral for problem 04.
</li>
<li>
<a href = "p04_f.m">p04_f.m</a>,
evaluates the integrand for problem 04.
</li>
<li>
<a href = "p04_i4.m">p04_i4.m</a>,
sets or gets I4 parameters for problem 04.
</li>
<li>
<a href = "p04_lim.m">p04_lim.m</a>,
returns the integration limits for problem 04.
</li>
<li>
<a href = "p04_name.m">p04_name.m</a>,
returns a name for problem 04.
</li>
<li>
<a href = "p04_region.m">p04_region.m</a>,
returns the region type for problem 04.
</li>
<li>
<a href = "p04_title.m">p04_title.m</a>,
prints a title for problem 04.
</li>
<li>
<a href = "p05_default.m">p05_default.m</a>,
sets default values for problem 05.
</li>
<li>
<a href = "p05_exact.m">p05_exact.m</a>,
sets the exact integral for problem 05.
</li>
<li>
<a href = "p05_f.m">p05_f.m</a>,
evaluates the integrand for problem 05.
</li>
<li>
<a href = "p05_i4.m">p05_i4.m</a>,
sets or gets I4 parameters for problem 05
</li>
<li>
<a href = "p05_lim.m">p05_lim.m</a>,
returns the integration limits for problem 05.
</li>
<li>
<a href = "p05_name.m">p05_name.m</a>,
returns a name for problem 05.
</li>
<li>
<a href = "p05_region.m">p05_region.m</a>,
returns the region type for problem 05.
</li>
<li>
<a href = "p05_title.m">p05_title.m</a>,
prints a title for problem 05.
</li>
<li>
<a href = "p06_default.m">p06_default.m</a>,
sets default values for problem 06.
</li>
<li>
<a href = "p06_exact.m">p06_exact.m</a>,
sets the exact integral for problem 06.
</li>
<li>
<a href = "p06_f.m">p06_f.m</a>,
evaluates the integrand for problem 06.
</li>
<li>
<a href = "p06_i4.m">p06_i4.m</a>,
sets or gets I4 parameters for problem 06.
</li>
<li>
<a href = "p06_lim.m">p06_lim.m</a>,
returns the integration limits for problem 06.
</li>
<li>
<a href = "p06_name.m">p06_name.m</a>,
returns a name for problem 06.
</li>
<li>
<a href = "p06_region.m">p06_region.m</a>,
returns the region type for problem 06.
</li>
<li>
<a href = "p06_title.m">p06_title.m</a>,
prints a title for problem 06.
</li>
<li>
<a href = "p07_default.m">p07_default.m</a>,
sets default values for problem 07.
</li>
<li>
<a href = "p07_exact.m">p07_exact.m</a>,
sets the exact integral for problem 07.
</li>
<li>
<a href = "p07_f.m">p07_f.m</a>,
evaluates the integrand for problem 07.
</li>
<li>
<a href = "p07_i4.m">p07_i4.m</a>,
sets or gets I4 parameters for problem 07.
</li>
<li>
<a href = "p07_lim.m">p07_lim.m</a>,
returns the integration limits for problem 07.
</li>
<li>
<a href = "p07_name.m">p07_name.m</a>,
returns a name for problem 07.
</li>
<li>
<a href = "p07_region.m">p07_region.m</a>,
returns the region type for problem 07.
</li>
<li>
<a href = "p07_title.m">p07_title.m</a>,
prints a title for problem 07.
</li>
<li>
<a href = "p08_default.m">p08_default.m</a>,
sets default values for problem 08.
</li>
<li>
<a href = "p08_exact.m">p08_exact.m</a>,
sets the exact integral for problem 08.
</li>
<li>
<a href = "p08_f.m">p08_f.m</a>,
evaluates the integrand for problem 08.
</li>
<li>
<a href = "p08_i4.m">p08_i4.m</a>,
sets or gets I4 parameters for problem 08.
</li>
<li>
<a href = "p08_lim.m">p08_lim.m</a>,
returns the integration limits for problem 08.
</li>
<li>
<a href = "p08_name.m">p08_name.m</a>,
returns a name for problem 08.
</li>
<li>
<a href = "p08_region.m">p08_region.m</a>,
returns the region type for problem 08.
</li>
<li>
<a href = "p08_title.m">p08_title.m</a>,
prints a title for problem 08.
</li>
<li>
<a href = "p09_default.m">p09_default.m</a>,
sets default values for problem 09.
</li>
<li>
<a href = "p09_exact.m">p09_exact.m</a>,
sets the exact integral for problem 09.
</li>
<li>
<a href = "p09_f.m">p09_f.m</a>,
evaluates the integrand for problem 09.
</li>
<li>
<a href = "p09_i4.m">p09_i4.m</a>,
sets or gets I4 parameters for problem 09.
</li>
<li>
<a href = "p09_lim.m">p09_lim.m</a>,
returns the integration limits for problem 09.
</li>
<li>
<a href = "p09_name.m">p09_name.m</a>,
returns a name for problem 09.
</li>
<li>
<a href = "p09_r8vec.m">p09_r8vec.m</a>,
sets or gets R8VEC parameters for problem 09.
</li>
<li>
<a href = "p09_region.m">p09_region.m</a>,
returns the region type for problem 09.
</li>
<li>
<a href = "p09_title.m">p09_title.m</a>,
prints a title for problem 09.
</li>
<li>
<a href = "p10_default.m">p10_default.m</a>,
sets default values for problem 10.
</li>
<li>
<a href = "p10_exact.m">p10_exact.m</a>,
sets the exact integral for problem 10.
</li>
<li>
<a href = "p10_f.m">p10_f.m</a>,
evaluates the integrand for problem 10.
</li>
<li>
<a href = "p10_i4.m">p10_i4.m</a>,
sets or gets I4 parameters for problem 10.
</li>
<li>
<a href = "p10_lim.m">p10_lim.m</a>,
returns the integration limits for problem 10.
</li>
<li>
<a href = "p10_name.m">p10_name.m</a>,
returns a name for problem 10.
</li>
<li>
<a href = "p10_region.m">p10_region.m</a>,
returns the region type for problem 10.
</li>
<li>
<a href = "p10_title.m">p10_title.m</a>,
prints a title for problem 10.
</li>
<li>
<a href = "p11_default.m">p11_default.m</a>,
sets default values for problem 11.
</li>
<li>
<a href = "p11_exact.m">p11_exact.m</a>,
sets the exact integral for problem 11.
</li>
<li>
<a href = "p11_f.m">p11_f.m</a>,
evaluates the integrand for problem 11.
</li>
<li>
<a href = "p11_i4.m">p11_i4.m</a>,
sets or gets I4 parameters for problem 11.
</li>
<li>
<a href = "p11_lim.m">p11_lim.m</a>,
returns the integration limits for problem 11.
</li>
<li>
<a href = "p11_name.m">p11_name.m</a>,
returns a name for problem 11.
</li>
<li>
<a href = "p11_region.m">p11_region.m</a>,
returns the region type for problem 11.
</li>
<li>
<a href = "p11_title.m">p11_title.m</a>,
prints a title for problem 11.
</li>
<li>
<a href = "p12_default.m">p12_default.m</a>,
sets default values for problem 12.
</li>
<li>
<a href = "p12_exact.m">p12_exact.m</a>,
sets the exact integral for problem 12.
</li>
<li>
<a href = "p12_f.m">p12_f.m</a>,
evaluates the integrand for problem 12.
</li>
<li>
<a href = "p12_i4.m">p12_i4.m</a>,
sets or gets I4 parameters for problem 12.
</li>
<li>
<a href = "p12_lim.m">p12_lim.m</a>,
returns the integration limits for problem 12.
</li>
<li>
<a href = "p12_name.m">p12_name.m</a>,
returns a name for problem 12.
</li>
<li>
<li>
<a href = "p12_region.m">p12_region.m</a>,
returns the region type for problem 12.
</li>
<li>
<a href = "p12_title.m">p12_title.m</a>,
prints a title for problem 12.
</li>
<li>
<a href = "p13_default.m">p13_default.m</a>,
sets default values for problem 13.
</li>
<li>
<a href = "p13_exact.m">p13_exact.m</a>,
sets the exact integral for problem 13.
</li>
<li>
<a href = "p13_f.m">p13_f.m</a>,
evaluates the integrand for problem 13.
</li>
<li>
<a href = "p13_i4.m">p13_i4.m</a>,
sets or gets I4 parameters for problem 13.
</li>