TEST_NONLIN
Nonlinear Equation Tests

TEST_NONLIN is a FORTRAN90 library which defines a set of test problems for nonlinear equation system solvers.

A few of the problems are small (2, 3, or 4 equations in 4 unknowns), but most of the problems may be set to any size whatsoever. The software includes routines defining the initial approximation to the solution of the system, the N function values at any point, and the N by N jacobian matrix at any point.

The list of problems includes:

1. Generalized Rosenbrock function, 1 < N.
2. Powell singular function, N = 4.
3. Powell badly scaled function, N = 2.
4. Wood function, N = 4.
5. Helical valley function, N = 3.
6. Watson function, 1 < N.
7. Chebyquad function, N arbitrary.
8. Brown almost linear function, N arbitrary.
9. Discrete boundary value function, N arbitrary.
10. Discrete integral equation function, N arbitrary.
11. Trigonometric function, N arbitrary.
12. Variably dimensioned function, N arbitrary.
13. Broyden tridiagonal function, N arbitrary.
14. Broyden banded function, N arbitrary.
15. Hammarling 2 by 2 matrix square root problem, N = 4.
16. Hammarling 3 by 3 matrix square root problem, N = 9.
17. Dennis and Schnabel example, N = 2.
18. Sample problem 18, N = 2.
19. Sample problem 19, N = 2.
20. Scalar problem, N = 1.
21. Freudenstein-Roth function, N = 2.
22. Boggs function, N = 2.
23. Chandrasekhar function, N arbitrary.

Licensing:

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

Languages:

TEST_NONLIN is available in a FORTRAN90 version.

Related Data and Programs:

GSL, a C++ library which can perform multidimensional root-finding.

KELLEY, a MATLAB library which can seek solutions of systems of nonlinear equations.

MINPACK, a FORTRAN90 library which is a minimization package for which most of these problems were used as tests, as part of ACM TOMS algorithm 566.

Reference:

1. Subramanyan Chandrasekhar,
   Radiative Transfer,
   Dover, 1960,
   ISBN13: 978-0486605906,
   LC: QB461.C46.
2. John Dennis, David Gay, Phuong Vu,
   A new nonlinear equations test problem,
   Technical Report 83-16,
   Mathematical Sciences Department,
   Rice University, 1983.
3. John Dennis, Robert Schnabel,
   Numerical Methods for Unconstrained Optimization and Nonlinear Equations,
   SIAM, 1996,
   ISBN13: 978-0-898713-64-0,
   LC: QA402.5.D44.
4. Noel deVilliers, David Glasser,
   A continuation method for nonlinear regression,
   SIAM Journal on Numerical Analysis,
   Volume 18, Number 6, December 1981, pages 1139-1154.
5. Chris Fraley,
   Solution of nonlinear least-squares problems,
   Technical Report STAN-CS-1165,
   Computer Science Department,
   Stanford University, 1987.
6. Chris Fraley, Software performance on nonlinear least-squares problems,
   Technical Report SOL 88-17,
   Systems Optimization Laboratory,
   Department of Operations Research,
   Stanford University, 1988.
7. JJ McKeown,
   Specialized versus general-purpose algorithms for functions that are sums of squared terms,
   Mathematical Programming,
   Volume 9, 1975, pages 57-68.
8. JJ McKeown,
   On algorithms for sums of squares problems,
   in Towards Global Optimisation,
   edited by Laurence Dixon, Gabor Szego,
   North-Holland, 1975, pages 229-257,
   ISBN: 0444109552,
   LC: QA402.5.T7.
9. Jorge More, Burton Garbow, Kenneth Hillstrom,
   Testing unconstrained optimization software,
   ACM Transactions on Mathematical Software,
   Volume 7, Number 1, March 1981, pages 17-41.
10. Jorge More, Burton Garbow, Kenneth Hillstrom,
    Algorithm 566: FORTRAN Subroutines for Testing unconstrained optimization software,
    ACM Transactions on Mathematical Software,
    Volume 7, Number 1, March 1981, pages 136-140.
11. James Ortega, Werner Rheinboldt
    Iterative Solution of Nonlinear Equations in Several Variables,
    SIAM, 1987,
    ISBN13: 978-0898714616,
    LC: QA297.8.O77.
12. Werner Rheinboldt,
    Methods for Solving Systems of Nonlinear Equations,
    SIAM, 1998,
    ISBN: 089871415X,
    LC: QA214.R44.
13. Douglas Salane,
    A continuation approach for solving large residual nonlinear least squares problems,
    SIAM Journal on Scientific and Statistical Computing,
    Volume 8, Number 4, July 1987, pages 655-671.

Source Code:

• test_nonlin.f90, the source code;

Examples and Tests:

• test_nonlin_test.f90, the calling program;
• test_nonlin_test.txt, the output file.

List of Routines:

• P00_DIF approximates the jacobian via finite differences.
• P00_FX evaluates the function for any problem.
• P00_JAC evaluates the jacobian for any problem.
• P00_N returns the number of equations for a problem.
• P00_PROBLEM_NUM returns the number of problems available.
• P00_SOL returns the solution of any problem.
• P00_START specifies a standard approximate solution.
• P00_TITLE returns the title of the problem.
• P01_FX evaluates the function for problem 1.
• P01_N returns the number of equations for problem 1.
• P01_JAC sets the jacobian for problem 1.
• P01_SOL returns the solution of problem 1.
• P01_START specifies a standard approximate solution for problem 1.
• P01_TITLE returns the title of problem 1.
• P02_FX evaluates the function for problem 2.
• P02_JAC sets the jacobian for problem 2.
• P02_N returns the number of equations for problem 2.
• P02_SOL returns the solution of problem 2.
• P02_START specifies a standard approximate solution for problem 2.
• P02_TITLE returns the title of problem 2.
• P03_FX evaluates the function for problem 3.
• P03_JAC sets the jacobian for problem 3.
• P03_N returns the number of equations for problem 3.
• P03_SOL returns the solution of problem 3.
• P03_START specifies a standard approximate solution for problem 3.
• P03_TITLE returns the title of problem 3.
• P04_FX evaluates the function for problem 4.
• P04_JAC sets the jacobian for problem 4.
• P04_N returns the number of equations for problem 4.
• P04_SOL returns the solution of problem 4.
• P04_START specifies a standard approximate solution for problem 4.
• P04_TITLE returns the title of problem 4.
• P05_FX evaluates the function for problem 4.
• P05_JAC sets the jacobian for problem 5.
• P05_N returns the number of equations for problem 5.
• P05_SOL returns the solution of problem 5.
• P05_START specifies a standard approximate solution for problem 5.
• P05_TITLE returns the title of problem 5.
• P06_FX evaluates the function for problem 6.
• P06_JAC sets the jacobian for problem 6.
• P06_N returns the number of equations for problem 6.
• P06_SOL returns the solution of problem 6.
• P06_START specifies a standard approximate solution for problem 6.
• P06_TITLE returns the title of problem 6.
• P07_FX evaluates the function for problem 7.
• P07_JAC sets the jacobian for problem 7.
• P07_N returns the number of equations for problem 7.
• P07_SOL returns the solution of problem 7.
• P07_START specifies a standard approximate solution for problem 7.
• P07_TITLE returns the title of problem 7.
• P08_FX evaluates the function for problem 8.
• P08_JAC sets the jacobian for problem 8.
• P08_N returns the number of equations for problem 8.
• P08_SOL returns the solution of problem 8.
• P08_START specifies a standard approximate solution for problem 8.
• P08_TITLE returns the title of problem 8.
• P09_FX evaluates the function for problem 9.
• P09_JAC sets the jacobian for problem 9.
• P09_N returns the number of equations for problem 9.
• P09_SOL returns the solution of problem 9.
• P09_START specifies a standard approximate solution for problem 9.
• P09_TITLE returns the title of problem 9.
• P10_FX evaluates the function for problem 10.
• P10_JAC sets the jacobian for problem 10.
• P10_N returns the number of equations for problem 10.
• P10_SOL returns the solution of problem 10.
• P10_START specifies a standard approximate solution for problem 10.
• P10_TITLE returns the title of problem 10.
• P11_FX evaluates the function for problem 11.
• P11_JAC sets the jacobian for problem 11.
• P11_N returns the number of equations for problem 11.
• P11_SOL returns the solution of problem 11.
• P11_START specifies a standard approximate solution for problem 11.
• P11_TITLE returns the title of problem 11.
• P12_FX evaluates the function for problem 12.
• P12_JAC sets the jacobian for problem 12.
• P12_N returns the number of equations for problem 12.
• P12_SOL returns the solution of problem 12.
• P12_START specifies a standard approximate solution for problem 12.
• P12_TITLE returns the title of problem 12.
• P13_FX evaluates the function for problem 13.
• P13_JAC sets the jacobian for problem 13.
• P13_N returns the number of equations for problem 13.
• P13_SOL returns the solution of problem 13.
• P13_START specifies a standard approximate solution for problem 13.
• P13_TITLE returns the title of problem 13.
• P14_FX evaluates the function for problem 14.
• P14_JAC sets the jacobian for problem 14.
• P14_N returns the number of equations for problem 14.
• P14_SOL returns the solution of problem 14.
• P14_START specifies a standard approximate solution for problem 14.
• P14_TITLE returns the title of problem 14.
• P15_FX evaluates the function for problem 15.
• P15_JAC sets the jacobian for problem 15.
• P15_N returns the number of equations for problem 15.
• P15_SOL returns the solution of problem 15.
• P15_START specifies a standard approximate solution for problem 15.
• P15_TITLE returns the title of problem 15.
• P16_FX evaluates the function for problem 16.
• P16_JAC sets the jacobian for problem 16.
• P16_N returns the number of equations for problem 16.
• P16_SOL returns the solution of problem 16.
• P16_START specifies a standard approximate solution for problem 16.
• P16_TITLE returns the title of problem 16.
• P17_FX evaluates the function for problem 17.
• P17_JAC sets the jacobian for problem 17.
• P17_N returns the number of equations for problem 17.
• P17_SOL returns the solution of problem 17.
• P17_START specifies a standard approximate solution for problem 17.
• P17_TITLE returns the title of problem 17.
• P18_FX evaluates the function for problem 18.
• P18_JAC sets the jacobian for problem 18.
• P18_N returns the number of equations for problem 18.
• P18_SOL returns the solution of problem 18.
• P18_START specifies a standard approximate solution for problem 18.
• P18_TITLE returns the title of problem 18.
• P19_FX evaluates the function for problem 19.
• P19_JAC sets the jacobian for problem 19.
• P19_N returns the number of equations for problem 19.
• P19_SOL returns the solution of problem 19.
• P19_START specifies a standard approximate solution for problem 19.
• P19_TITLE returns the title of problem 19.
• P20_FX evaluates the function for problem 20.
• P20_JAC sets the jacobian for problem 20.
• P20_N returns the number of equations for problem 20.
• P20_SOL returns the solution of problem 20.
• P20_START specifies a standard approximate solution for problem 20.
• P20_TITLE returns the title of problem 20.
• P21_FX evaluates the function for problem 21.
• P21_JAC sets the jacobian for problem 21
• P21_N returns the number of equations for problem 21.
• P21_SOL returns the solution of problem 21.
• P21_START specifies a standard approximate solution for problem 21.
• P21_TITLE returns the title of problem 21.
• P22_FX evaluates the function for problem 22.
• P22_JAC sets the jacobian for problem 22.
• P22_N returns the number of equations for problem 22.
• P22_SOL returns the solution of problem 22.
• P22_START specifies a standard approximate solution for problem 22.
• P22_TITLE returns the title of problem 22.
• P23_FX evaluates the function for problem 23.
• P23_N returns the number of equations for problem 23.
• P23_JAC sets the jacobian for problem 23.
• P23_SOL returns the solution of problem 23.
• P23_START specifies a standard approximate solution for problem 23.
• P23_TITLE returns the title of problem 23.
• R8_SWAP switches two R8's.
• R8VEC_NORM2 returns the 2-norm of a vector.
• R8GE_FA factors a general matrix.
• R8GE_SL solves a system factored by SGE_FA.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.