TEST_MIN
Test problems for minimization

TEST_MIN is a C++ library which defines problems involving the minimization of a scalar function of a scalar argument.

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:

• 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;

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL 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.

Related Data and Programs:

BRENT, 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.

NMS, a FORTRAN90 library which includes a routine called FMIN which seeks the minimizer of a scalar function of a scalar argument.

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.cpp, the source code.
• test_min.hpp, the include file.
• test_min.sh, commands to compile the source code.

Examples and Tests:

• test_min_prb.cpp, a sample problem.
• test_min_prb.sh, commands to compile, link and run the sample problem.
• test_min_prb_output.txt, the output file.

List of Routines:

• P00_F evaluates the function for any problem.
• P00_F1 evaluates the first derivative for any problem.
• P00_F1_DIF approximates the first derivative via finite differences.
• P00_F2 evaluates the second derivative for any problem.
• P00_F2_DIF approximates the second derivative via finite differences.
• P00_FMIN seeks a minimizer of a scalar function of a scalar variable.
• P00_INTERVAL returns a bracketing interval for any problem.
• P00_PROBLEM_NUM returns the number of problems available.
• P00_SOL returns the solution for any problem.
• P00_START returns a starting point for optimization for any problem.
• P00_TITLE returns a title for any problem.
• P01_F evaluates the objective function for problem 1.
• P01_F1 evaluates the first derivative for problem 1.
• P01_F2 evaluates the second derivative for problem 1.
• P01_INTERVAL returns a starting interval for optimization for problem 1.
• P01_SOL returns the solution for problem 1.
• P01_START returns a starting point for optimization for problem 1.
• P01_TITLE returns a title for problem 1.
• P02_F evaluates the objective function for problem 2.
• P02_F1 evaluates the first derivative for problem 2.
• P02_F2 evaluates the second derivative for problem 2.
• P02_INTERVAL returns a starting interval for optimization for problem 2.
• P02_SOL returns the solution for problem 2.
• P02_START returns a starting point for optimization for problem 2.
• P02_TITLE returns a title for problem 2.
• P03_F evaluates the objective function for problem 3.
• P03_F1 evaluates the first derivative for problem 3.
• P03_F2 evaluates the second derivative for problem 3.
• P03_INTERVAL returns a starting interval for optimization for problem 3.
• P03_SOL returns the solution for problem 3.
• P03_START returns a starting point for optimization for problem 3.
• P03_TITLE returns a title for problem 3.
• P04_F evaluates the objective function for problem 4.
• P04_F1 evaluates the first derivative for problem 4.
• P04_F2 evaluates the second derivative for problem 4.
• P04_INTERVAL returns a starting interval for optimization for problem 4.
• P04_SOL returns the solution for problem 4.
• P04_START returns a starting point for optimization for problem 4.
• P04_TITLE returns a title for problem 4.
• P05_F evaluates the objective function for problem 5.
• P05_F1 evaluates the first derivative for problem 5.
• P05_F2 evaluates the second derivative for problem 5.
• P05_INTERVAL returns a starting interval for optimization for problem 5.
• P05_SOL returns the solution for problem 5.
• P05_START returns a starting point for optimization for problem 5.
• P05_TITLE returns a title for problem 5.
• P06_F evaluates the objective function for problem 6.
• P06_F1 evaluates the first derivative for problem 6.
• P06_F2 evaluates the second derivative for problem 6.
• P06_INTERVAL returns a starting interval for optimization for problem 6.
• P06_SOL returns the solution for problem 6.
• P06_START returns a starting point for optimization for problem 6.
• P06_TITLE returns a title for problem 6.
• P07_F evaluates the objective function for problem 7.
• P07_F1 evaluates the first derivative for problem 7.
• P07_F2 evaluates the second derivative for problem 7.
• P07_INTERVAL returns a starting interval for optimization for problem 7.
• P07_SOL returns the solution for problem 7.
• P07_START returns a starting point for optimization for problem 7.
• P07_TITLE returns a title for problem 7.
• P08_F evaluates the objective function for problem 8.
• P08_F1 evaluates the first derivative for problem 8.
• P08_F2 evaluates the second derivative for problem 8.
• P08_INTERVAL returns a starting interval for optimization for problem 8.
• P08_SOL returns the solution for problem 8.
• P08_START returns a starting point for optimization for problem 8.
• P08_TITLE returns a title for problem 8.
• P09_F evaluates the objective function for problem 9.
• P09_F1 evaluates the first derivative for problem 9.
• P09_F2 evaluates the second derivative for problem 9.
• P09_INTERVAL returns a starting interval for optimization for problem 9.
• P09_SOL returns the solution for problem 9.
• P09_START returns a starting point for optimization for problem 9.
• P09_TITLE returns a title for problem 9.
• P10_F evaluates the objective function for problem 10.
• P10_F1 evaluates the first derivative for problem 10.
• P10_F2 evaluates the second derivative for problem 10.
• P10_INTERVAL returns a starting interval for optimization for problem 10.
• P10_SOL returns the solution for problem 10.
• P10_START returns a starting point for optimization for problem 10.
• P10_TITLE returns a title for problem 10.
• P11_F evaluates the objective function for problem 11.
• P11_F1 evaluates the first derivative for problem 11.
• P11_F2 evaluates the second derivative for problem 11.
• P11_INTERVAL returns a starting interval for optimization for problem 11.
• P11_SOL returns the solution for problem 11.
• P11_START returns a starting point for optimization for problem 11.
• P11_TITLE returns a title for problem 11.
• R8_ABS returns the absolute value of an R8.
• R8_EPSILON returns the R8 roundoff unit.
• R8_SIGN returns the sign of an R8.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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