TEST_NLS
Nonlinear least squares test problems

TEST_NLS is a FORTRAN90 library which defines test problems for least squares minimization.

The typical problem comprises M functions in N variables, and it is desired to minimize the sum of squares of the values of the functions. Each problem is specified by a starting point, a function and jacobian routines.

The problems include:

1. Linear function, full rank
2. Linear function, rank 1.
3. Linear function, rank 1, zero columns and rows.
4. Rosenbrock function.
5. Helical valley function.
6. Powell singular function.
7. Freudenstein/Roth function.
8. Bard function.
9. Kowalik and Osborne function.
10. Meyer function.
11. Watson function.
12. Box 3-dimensional function.
13. Jennrich and Sampson function.
14. Brown and Dennis function.
15. Chebyquad function.
16. Brown almost-linear function.
17. Osborne function 1.
18. Osborne function 2.
19. Hanson function 1
20. Hanson function 2
21. McKeown function 1
22. McKeown function 2
23. McKeown function 3
24. Devilliers and Glasser function 1
25. Devilliers and Glasser function 2
26. Madsen example

Licensing:

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

Languages:

TEST_NLS is available in a FORTRAN90 version.

Related Data and Programs:

DQED, a FORTRAN90 library which seeks the solution of bounded or constrained minimization problems.

MINPACK, a FORTRAN90 library which seeks the solution of nonlinear equations, or the least squares minimization of the residual.

Reference:

1. John Dennis, David Gay, and Phuong Vu,
   A new nonlinear equations test problem,
   Technical Report 83-16,
   Mathematical Sciences Department,
   Rice University, (1983 - revised 1985).
2. N. de Villiers and D. Glasser,
   A continuation method for nonlinear regression,
   SIAM Journal of Numerical Analysis,
   Volume 18, pages 1139-1154, 1981.
3. C. Fraley,
   Solution of nonlinear least-squares problems,
   Technical Report STAN-CS-1165,
   Computer Science Department,
   Stanford University, 1987.
4. C. Fraley,
   Software performance on nonlinear least-squares problems,
   Technical Report SOL 88-17,
   Systems Optimization Laboratory,
   Department of Operations Research,
   Stanford University, 1988.
5. JJ McKeown,
   Specialized versus general-purpose algorithms for functions that are sums of squared terms,
   Math. Prog.,
   Volume 9, pages 57-68, 1975.
6. JJ McKeown,
   On algorithms for sums of squares problems,
   in Towards Global Optimization,
   L. C. W. Dixon and G. Szego (editors),
   North-Holland, pages 229-257, 1975.
7. Jorge More, Burton Garbow, Kenneth Hillstrom,
   Testing unconstrained optimization software,
   ACM Transactions on Mathematical Software,
   Volume 7, Number 1, March 1981, pages 17-41.
8. 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.
9. Douglas Salane,
   A continuation approach for solving large residual nonlinear least squares problems,
   SIAM Journal of Scientific and Statistical Computing,
   Volume 8, pages 655-671, 1987.

Source Code:

• test_nls.f90, the source code.

Examples and Tests:

• test_nls_test.f90, a sample problem.
• test_nls_test.txt, the output file.

List of Routines:

• P00_F evaluates the M nonlinear functions for any problem.
• P00_G evaluates the least squares gradient for any problem.
• P00_J evaluates the jacobian for any problem.
• P00_MN reports the default sizes M and N for the least squares problems.
• P00_MPROB returns the number of problems available.
• P00_SOL returns the solution of the problem, if known.
• P00_START sets a starting point for any problem.
• P00_title sets the title of any problem.
• P01_F evaluates the M nonlinear functions for problem 1.
• P01_J evaluates the jacobian for problem 1.
• P01_SOL returns the solution of problem 1.
• P01_START sets a starting point for problem 1.
• P01_title specifies the title for problem 1.
• P02_F evaluates the M nonlinear functions for problem 2.
• P02_J evaluates the jacobian for problem 2.
• P02_SOL returns the solution of problem 2.
• P02_START sets a starting point for problem 2.
• P02_title specifies the title for problem 2.
• P03_F evaluates the M nonlinear functions for problem 3.
• P03_J evaluates the jacobian for problem 3.
• P03_START sets a starting point for problem 3.
• P03_SOL returns the solution of problem 3.
• P03_title specifies the title for problem 3.
• P04_F evaluates the M nonlinear functions for problem 4.
• P04_J evaluates the jacobian for problem 4.
• P04_SOL returns the solution of problem 4.
• P04_START sets a starting point for problem 4.
• P04_title specifies the title for problem 4.
• P05_F evaluates the M nonlinear functions for problem 5.
• P05_J evaluates the jacobian for problem 5.
• P05_SOL returns the solution of problem 5.
• P05_START sets a starting point for problem 5.
• P05_title specifies the title for problem 5.
• P06_F evaluates the M nonlinear functions for problem 6.
• P06_J evaluates the jacobian for problem 6.
• P06_SOL returns the solution of problem 6.
• P06_START sets a starting point for problem 6.
• P06_title specifies the title for problem 6.
• P07_F evaluates the M nonlinear functions for problem 7.
• P07_J evaluates the jacobian for problem 7.
• P07_SOL returns the solution of problem 7.
• P07_START sets a starting point for problem 7.
• P07_title specifies the title for problem 7.
• P08_F evaluates the M nonlinear functions for problem 8.
• P08_J evaluates the jacobian for problem 8.
• P08_SOL returns the solution of problem 8.
• P08_START sets a starting point for problem 8.
• P08_title specifies the title for problem 8.
• P09_F evaluates the M nonlinear functions for problem 9.
• P09_J evaluates the jacobian for problem 9.
• P09_SOL returns the solution of problem 9.
• P09_START sets a starting point for problem 9.
• P09_title specifies the title for problem 9.
• P10_F evaluates the M nonlinear functions for problem 10.
• P09_J evaluates the jacobian for problem 9.
• P10_SOL returns the solution of problem 10.
• P10_START sets a starting point for problem 10.
• P10_title specifies the title for problem 10.
• P11_F evaluates the M nonlinear functions for problem 11.
• P11_J evaluates the jacobian for problem 11.
• P11_SOL returns the solution of problem 11.
• P11_START sets a starting point for problem 11.
• P11_title specifies the title for problem 11.
• P12_F evaluates the M nonlinear functions for problem 12.
• P12_J evaluates the jacobian for problem 12.
• P12_SOL returns the solution of problem 12.
• P12_START sets a starting point for problem 12.
• P12_title specifies the title for problem 12.
• P13_F evaluates the M nonlinear functions for problem 13.
• P13_J evaluates the jacobian for problem 13.
• P13_SOL returns the solution of problem 13.
• P13_START sets a starting point for problem 13.
• P13_title specifies the title for problem 13.
• P14_F evaluates the M nonlinear functions for problem 14.
• P14_J evaluates the jacobian for problem 14.
• P14_SOL returns the solution of problem 14.
• P14_START sets a starting point for problem 14.
• P14_title specifies the title for problem 14.
• P15_F evaluates the M nonlinear functions for problem 15.
• P15_J evaluates the jacobian for problem 15.
• P15_SOL returns the solution of problem 15.
• P15_START sets a starting point for problem 15.
• P15_title specifies the title for problem 15.
• P16_F evaluates the M nonlinear functions for problem 16.
• P16_J evaluates the jacobian for problem 16.
• P16_SOL returns the solution of problem 16.
• P16_START sets a starting point for problem 16.
• P16_title specifies the title for problem 16.
• P17_F evaluates the M nonlinear functions for problem 17.
• P17_J evaluates the jacobian for problem 17.
• P17_SOL returns the solution of problem 17.
• P17_START sets a starting point for problem 17.
• P17_title specifies the title for problem 17.
• P18_F evaluates the M nonlinear functions for problem 18.
• P18_J evaluates the jacobian for problem 18.
• P18_SOL returns the solution of problem 18.
• P18_START sets a starting point for problem 18.
• P18_title specifies the title for problem 18.
• P19_F evaluates the M nonlinear functions for problem 19.
• P19_J evaluates the jacobian for problem 19.
• P19_SOL returns the solution of problem 19.
• P19_START sets a starting point for problem 19.
• P19_title specifies the title for problem 19.
• P20_F evaluates the M nonlinear functions for problem 20.
• P20_J evaluates the jacobian for problem 20.
• P20_SOL returns the solution of problem 20.
• P20_START sets a starting point for problem 20.
• P20_title specifies the title for problem 20.
• P21_F evaluates the M nonlinear functions for problem 21.
• P21_J evaluates the jacobian for problem 21.
• P21_SOL returns the solution of problem 21.
• P21_START sets a starting point for problem 21.
• P21_title specifies the title for problem 21.
• P22_F evaluates the M nonlinear functions for problem 22.
• P22_J evaluates the jacobian for problem 22.
• P22_SOL returns the solution of problem 22.
• P22_START sets a starting point for problem 22.
• P22_title specifies the title for problem 22.
• P23_F evaluates the M nonlinear functions for problem 23.
• P23_J evaluates the jacobian for problem 23.
• P23_SOL returns the solution of problem 23.
• P23_START sets a starting point for problem 23.
• P23_title specifies the title for problem 23.
• P24_F evaluates the M nonlinear functions for problem 24.
• P24_J evaluates the jacobian for problem 24.
• P24_SOL returns the solution of problem 24.
• P24_START sets a starting point for problem 24.
• P24_title specifies the title for problem 24.
• P25_F evaluates the M nonlinear functions for problem 25.
• P25_J evaluates the jacobian for problem 25.
• P25_SOL returns the solution of problem 25.
• P25_START sets a starting point for problem 25.
• P25_title specifies the title for problem 25.
• P26_F evaluates the M nonlinear functions for problem 26.
• P26_J evaluates the jacobian for problem 26.
• P26_SOL returns the solution of problem 26.
• P26_START sets a starting point for problem 26.
• P26_title specifies the title for problem 26.
• R_PI returns the value of pi.
• RMAT_PRINT prints a real matrix.
• RVEC_PRINT prints a real vector.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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