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<html>
<head>
<title>
TEST_MIN - Test problems for minimization
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TEST_MIN <br> Test problems for minimization
</h1>
<hr>
<p>
<b>TEST_MIN</b>
is a C++ library which
defines problems involving the minimization
of a scalar function of a scalar argument.
</p>
<p>
TEST_MIN can be useful for testing algorithms that
attempt to minimize a scalar function of a scalar argument.
Each problem has an index number, and there are a corresponding
set of routines, with names beginning with the index number, to:
<ul>
<li>
evaluate f(x);
</li>
<li>
evaluate f'(x);
</li>
<li>
evaluate f"(x);
</li>
<li>
return the title of the problem;
</li>
<li>
return a starting point;
</li>
<li>
return a starting search interval;
</li>
<li>
return the exact solution;
</li>
</ul>
</p>
<p>
There is also a "generic" problem interface, whose routines all
begin with "P00". This allows the user to call all possible
problems in a single simple loop, by passing the desired index
number through the generic interface.
</p>
<p>
The functions can be invoked by an index number, and include:
<ol>
<li>
f(x) = ( x - 2 )^2 + 1;<br>
<a href = "p01_f.png">a PNG image</a>;
</li>
<li>
f(x) = x^2 + exp ( -x );<br>
<a href = "p02_f.png">a PNG image</a>;
</li>
<li>
f(x) = x^4 + 2x^2 + x + 3;<br>
<a href = "p03_f.png">a PNG image</a>;
</li>
<li>
f(x) = exp ( x ) + 0.01 / x;<br>
<a href = "p04_f.png">a PNG image</a>;
</li>
<li>
f(x) = exp ( x ) - 2 * x + 0.01 / x - 0.000001 / x^2;<br>
<a href = "p05_f.png">a PNG image</a>;
</li>
<li>
f(x) = 2 - x;<br>
<a href = "p06_f.png">a PNG image</a>;
</li>
<li>
f(x) = ( x + sin ( x ) ) * exp ( -x^2 );<br>
<a href = "p07_f.png">a PNG image</a>;
</li>
<li>
f(x) = 3 * x^2 + 1 + ( log ( ( x - pi )^2 ) ) / pi^4;<br>
<a href = "p08_f.png">a PNG image</a>;
</li>
<li>
f(x) = x^2 - 10 sin ( x^2 - 3x + 2);<br>
<a href = "p09_f.png">a PNG image</a>;
</li>
<li>
f(x) = cos(x)+5*cos(1.6*x)-2*cos(2*x)+5*cos(4.5*x)+7*cos(9*x);<br>
<a href = "p10_f.png">a PNG image</a>;
</li>
<li>
f(x) = 1+|3x-1|;<br>
<a href = "p11_f.png">a PNG image</a>;
</li>
</ol>
</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_MIN</b> is available in
<a href = "../../c_src/test_min/test_min.html">a C version</a> and
<a href = "../../cpp_src/test_min/test_min.html">a C++ version</a> and
<a href = "../../f77_src/test_min/test_min.html">a FORTRAN77 version</a> and
<a href = "../../f_src/test_min/test_min.html">a FORTRAN90 version</a> and
<a href = "../../m_src/test_min/test_min.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/brent/brent.html">
BRENT</a>,
a C++ library which
contains Richard Brent's routines for finding the zero, local minimizer,
or global minimizer of a scalar function of a scalar argument, without
the use of derivative information.
</p>
<p>
<a href = "../../f_src/nms/nms.html">
NMS</a>,
a FORTRAN90 library which
includes a routine called <b>FMIN</b> which seeks the minimizer of a scalar function
of a scalar argument.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Isabel Beichl, Dianne O'Leary, Francis Sullivan,<br>
Monte Carlo Minimization and Counting: One, Two, Too Many,<br>
Computing in Science and Engineering,<br>
Volume 9, Number 1, January/February 2007.
</li>
<li>
Richard Brent,<br>
Algorithms for Minimization without Derivatives,<br>
Dover, 2002,<br>
ISBN: 0-486-41998-3,<br>
LC: QA402.5.B74.
</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>
Arnold Krommer, Christoph Ueberhuber,<br>
Numerical Integration on Advanced Computer Systems,<br>
Springer, 1994,<br>
ISBN: 3540584102,<br>
LC: QA299.3.K76.
</li>
<li>
Dianne O'Leary,<br>
Scientific Computing with Case Studies,<br>
SIAM, 2008,<br>
ISBN13: 978-0-898716-66-5,<br>
LC: QA401.O44.
</li>
<li>
LE Scales,<br>
Introduction to Non-Linear Optimization,<br>
Springer, 1985,<br>
ISBN: 0-387-91252-5,<br>
LC: QA402.5.S33.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "test_min.cpp">test_min.cpp</a>, the source code.
</li>
<li>
<a href = "test_min.hpp">test_min.hpp</a>, the include file.
</li>
<li>
<a href = "test_min.sh">test_min.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "test_min_prb.cpp">test_min_prb.cpp</a>, a sample problem.
</li>
<li>
<a href = "test_min_prb.sh">test_min_prb.sh</a>,
commands to compile, link and run the sample problem.
</li>
<li>
<a href = "test_min_prb_output.txt">test_min_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>P00_F</b> evaluates the function for any problem.
</li>
<li>
<b>P00_F1</b> evaluates the first derivative for any problem.
</li>
<li>
<b>P00_F1_DIF</b> approximates the first derivative via finite differences.
</li>
<li>
<b>P00_F2</b> evaluates the second derivative for any problem.
</li>
<li>
<b>P00_F2_DIF</b> approximates the second derivative via finite differences.
</li>
<li>
<b>P00_FMIN</b> seeks a minimizer of a scalar function of a scalar variable.
</li>
<li>
<b>P00_INTERVAL</b> returns a bracketing interval for any problem.
</li>
<li>
<b>P00_PROBLEM_NUM</b> returns the number of problems available.
</li>
<li>
<b>P00_SOL</b> returns the solution for any problem.
</li>
<li>
<b>P00_START</b> returns a starting point for optimization for any problem.
</li>
<li>
<b>P00_TITLE</b> returns a title for any problem.
</li>
<li>
<b>P01_F</b> evaluates the objective function for problem 1.
</li>
<li>
<b>P01_F1</b> evaluates the first derivative for problem 1.
</li>
<li>
<b>P01_F2</b> evaluates the second derivative for problem 1.
</li>
<li>
<b>P01_INTERVAL</b> returns a starting interval for optimization for problem 1.
</li>
<li>
<b>P01_SOL</b> returns the solution for problem 1.
</li>
<li>
<b>P01_START</b> returns a starting point for optimization for problem 1.
</li>
<li>
<b>P01_TITLE</b> returns a title for problem 1.
</li>
<li>
<b>P02_F</b> evaluates the objective function for problem 2.
</li>
<li>
<b>P02_F1</b> evaluates the first derivative for problem 2.
</li>
<li>
<b>P02_F2</b> evaluates the second derivative for problem 2.
</li>
<li>
<b>P02_INTERVAL</b> returns a starting interval for optimization for problem 2.
</li>
<li>
<b>P02_SOL</b> returns the solution for problem 2.
</li>
<li>
<b>P02_START</b> returns a starting point for optimization for problem 2.
</li>
<li>
<b>P02_TITLE</b> returns a title for problem 2.
</li>
<li>
<b>P03_F</b> evaluates the objective function for problem 3.
</li>
<li>
<b>P03_F1</b> evaluates the first derivative for problem 3.
</li>
<li>
<b>P03_F2</b> evaluates the second derivative for problem 3.
</li>
<li>
<b>P03_INTERVAL</b> returns a starting interval for optimization for problem 3.
</li>
<li>
<b>P03_SOL</b> returns the solution for problem 3.
</li>
<li>
<b>P03_START</b> returns a starting point for optimization for problem 3.
</li>
<li>
<b>P03_TITLE</b> returns a title for problem 3.
</li>
<li>
<b>P04_F</b> evaluates the objective function for problem 4.
</li>
<li>
<b>P04_F1</b> evaluates the first derivative for problem 4.
</li>
<li>
<b>P04_F2</b> evaluates the second derivative for problem 4.
</li>
<li>
<b>P04_INTERVAL</b> returns a starting interval for optimization for problem 4.
</li>
<li>
<b>P04_SOL</b> returns the solution for problem 4.
</li>
<li>
<b>P04_START</b> returns a starting point for optimization for problem 4.
</li>
<li>
<b>P04_TITLE</b> returns a title for problem 4.
</li>
<li>
<b>P05_F</b> evaluates the objective function for problem 5.
</li>
<li>
<b>P05_F1</b> evaluates the first derivative for problem 5.
</li>
<li>
<b>P05_F2</b> evaluates the second derivative for problem 5.
</li>
<li>
<b>P05_INTERVAL</b> returns a starting interval for optimization for problem 5.
</li>
<li>
<b>P05_SOL</b> returns the solution for problem 5.
</li>
<li>
<b>P05_START</b> returns a starting point for optimization for problem 5.
</li>
<li>
<b>P05_TITLE</b> returns a title for problem 5.
</li>
<li>
<b>P06_F</b> evaluates the objective function for problem 6.
</li>
<li>
<b>P06_F1</b> evaluates the first derivative for problem 6.
</li>
<li>
<b>P06_F2</b> evaluates the second derivative for problem 6.
</li>
<li>
<b>P06_INTERVAL</b> returns a starting interval for optimization for problem 6.
</li>
<li>
<b>P06_SOL</b> returns the solution for problem 6.
</li>
<li>
<b>P06_START</b> returns a starting point for optimization for problem 6.
</li>
<li>
<b>P06_TITLE</b> returns a title for problem 6.
</li>
<li>
<b>P07_F</b> evaluates the objective function for problem 7.
</li>
<li>
<b>P07_F1</b> evaluates the first derivative for problem 7.
</li>
<li>
<b>P07_F2</b> evaluates the second derivative for problem 7.
</li>
<li>
<b>P07_INTERVAL</b> returns a starting interval for optimization for problem 7.
</li>
<li>
<b>P07_SOL</b> returns the solution for problem 7.
</li>
<li>
<b>P07_START</b> returns a starting point for optimization for problem 7.
</li>
<li>
<b>P07_TITLE</b> returns a title for problem 7.
</li>
<li>
<b>P08_F</b> evaluates the objective function for problem 8.
</li>
<li>
<b>P08_F1</b> evaluates the first derivative for problem 8.
</li>
<li>
<b>P08_F2</b> evaluates the second derivative for problem 8.
</li>
<li>
<b>P08_INTERVAL</b> returns a starting interval for optimization for problem 8.
</li>
<li>
<b>P08_SOL</b> returns the solution for problem 8.
</li>
<li>
<b>P08_START</b> returns a starting point for optimization for problem 8.
</li>
<li>
<b>P08_TITLE</b> returns a title for problem 8.
</li>
<li>
<b>P09_F</b> evaluates the objective function for problem 9.
</li>
<li>
<b>P09_F1</b> evaluates the first derivative for problem 9.
</li>
<li>
<b>P09_F2</b> evaluates the second derivative for problem 9.
</li>
<li>
<b>P09_INTERVAL</b> returns a starting interval for optimization for problem 9.
</li>
<li>
<b>P09_SOL</b> returns the solution for problem 9.
</li>
<li>
<b>P09_START</b> returns a starting point for optimization for problem 9.
</li>
<li>
<b>P09_TITLE</b> returns a title for problem 9.
</li>
<li>
<b>P10_F</b> evaluates the objective function for problem 10.
</li>
<li>
<b>P10_F1</b> evaluates the first derivative for problem 10.
</li>
<li>
<b>P10_F2</b> evaluates the second derivative for problem 10.
</li>
<li>
<b>P10_INTERVAL</b> returns a starting interval for optimization for problem 10.
</li>
<li>
<b>P10_SOL</b> returns the solution for problem 10.
</li>
<li>
<b>P10_START</b> returns a starting point for optimization for problem 10.
</li>
<li>
<b>P10_TITLE</b> returns a title for problem 10.
</li>
<li>
<b>P11_F</b> evaluates the objective function for problem 11.
</li>
<li>
<b>P11_F1</b> evaluates the first derivative for problem 11.
</li>
<li>
<b>P11_F2</b> evaluates the second derivative for problem 11.
</li>
<li>
<b>P11_INTERVAL</b> returns a starting interval for optimization for problem 11.
</li>
<li>
<b>P11_SOL</b> returns the solution for problem 11.
</li>
<li>
<b>P11_START</b> returns a starting point for optimization for problem 11.
</li>
<li>
<b>P11_TITLE</b> returns a title for problem 11.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8_SIGN</b> returns the sign of an R8.
</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 = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 11 February 2009
</i>
<!-- John Burkardt -->
</body>
</html>