test_min

test_min, a Python code which defines problems involving the minimization of a scalar function of a scalar argument.

The code 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:

• evaluate f(x);
• evaluate f'(x);
• evaluate f"(x);
• return the title of the problem;
• return a starting point;
• return a starting search interval;
• return the exact solution;

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.

The functions can be invoked by an index number, and include:

1. f(x) = ( x - 2 )^2 + 1;
   a PNG image;
2. f(x) = x^2 + exp ( -x );
   a PNG image;
3. f(x) = x^4 + 2x^2 + x + 3;
   a PNG image;
4. f(x) = exp ( x ) + 0.01 / x;
   a PNG image;
5. f(x) = exp ( x ) - 2 * x + 0.01 / x - 0.000001 / x^2;
   a PNG image;
6. f(x) = 2 - x;
   a PNG image;
7. f(x) = ( x + sin ( x ) ) * exp ( -x^2 );
   a PNG image;
8. f(x) = 3 * x^2 + 1 + ( log ( ( x - pi )^2 ) ) / pi^4;
   a PNG image;
9. f(x) = x^2 - 10 sin ( x^2 - 3x + 2);
   a PNG image;
10. f(x) = cos(x)+5*cos(1.6*x)-2*cos(2*x)+5*cos(4.5*x)+7*cos(9*x);
    a PNG image;
11. f(x) = 1+|3x-1|;
    a PNG image;
12. f(x) = x^2+sin(53*x);
    a PNG image;
13. f(x) = 2*x^4-7*x^2+3*x+5;
    a PNG image;
14. f(x) = 1.0 / ( ( x - 0.3 )^2 + 0.01 ) + 1.0 / ( ( x - 0.9 )^2 + 0.04 ) - 6.0;
    a PNG image;

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

test_min is available in a C version and a C++ version and a Fortran77 version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

glomin, a Python code which finds a global minimum of a scalar function of a scalar argument, without the use of derivative information, by Richard Brent.

golden_section, a Python code which estimates a minimizer of a function f(x), assuming f(x) is unimodular ("U-shaped") over [a,b].

local_min, a Python code which finds a local minimum of a scalar function of a scalar variable, without the use of derivative information, by Richard Brent.

local_min_rc, a Python code which finds a local minimum of a scalar function of a scalar variable, without the use of derivative information, using reverse communication (RC), by Richard Brent.

test_uni, a Python code which defines a number of unimodal functions, each one a scalar valued function of a scalar argument. Over a specified interval [a,b], each function decreases to a minimum value and then increases. These functions are designed to test the efficiency of algorithms for locating the minimizing argument, such as bisection, golden search, or Brent's method.

Reference:

1. Isabel Beichl, Dianne O'Leary, Francis Sullivan,
   Monte Carlo Minimization and Counting: One, Two, Too Many,
   Computing in Science and Engineering,
   Volume 9, Number 1, January/February 2007.
2. Richard Brent,
   Algorithms for Minimization without Derivatives,
   Dover, 2002,
   ISBN: 0-486-41998-3,
   LC: QA402.5.B74.
3. David Kahaner, Cleve Moler, Steven Nash,
   Numerical Methods and Software,
   Prentice Hall, 1989,
   ISBN: 0-13-627258-4,
   LC: TA345.K34.
4. Arnold Krommer, Christoph Ueberhuber,
   Numerical Integration on Advanced Computer Systems,
   Springer, 1994,
   ISBN: 3540584102,
   LC: QA299.3.K76.
5. Dianne O'Leary,
   Scientific Computing with Case Studies,
   SIAM, 2008,
   ISBN13: 978-0-898716-66-5,
   LC: QA401.O44.
6. LE Scales,
   Introduction to Non-Linear Optimization,
   Springer, 1985,
   ISBN: 0-387-91252-5,
   LC: QA402.5.S33.

Source Code:

• test_min.py, the source code.
• test_min.sh, runs all the tests.
• test_min.txt, the output file.