TEST_NONLIN
Nonlinear Equation Tests

TEST_NONLIN, an Octave code 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 MIT license

Languages:

test_nonlin is available in a FORTRAN90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

fsolve_test, an Octave code which calls fsolve() which seeks the solution x of one or more nonlinear equations f(x)=0.

test_nonlin_test

test_zero, an Octave code which implements test problems for the solution of a single nonlinear equation in one variable.

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:

• p00_dif.m, for any problem, estimates the jacobian matrix using finite differences.
• p00_fx.m, for any problem, evaluates the function.
• p00_jac.m, for any problem, evaluates the jacobian matrix.
• p00_n.m, for any problem, returns legal problem sizes.
• p00_problem_num.m, returns the number of problems.
• p00_sol.m, for any problem, returns an estimated root, if known.
• p00_start.m, for any problem, returns a starting point.
• p00_title.m, for any problem, returns the title.

• p01_fx.m, for problem 01, evaluates the function.
• p01_jac.m, for problem 01, evaluates the jacobian matrix.
• p01_n.m, for problem 01, returns legal problem sizes.
• p01_sol.m, for problem 01, returns an estimated root, if known.
• p01_start.m, for problem 01, returns a starting point.
• p01_title.m, for problem 01, returns the title.

• p02_fx.m, for problem 02, evaluates the function.
• p02_jac.m, for problem 02, evaluates the jacobian matrix.
• p02_n.m, for problem 02, returns legal problem sizes.
• p02_sol.m, for problem 02, returns an estimated root, if known.
• p02_start.m, for problem 02, returns a starting point.
• p02_title.m, for problem 02, returns the title.

• p03_fx.m, for problem 03, evaluates the function.
• p03_jac.m, for problem 03, evaluates the jacobian matrix.
• p03_n.m, for problem 03, returns legal problem sizes.
• p03_sol.m, for problem 03, returns an estimated root, if known.
• p03_start.m, for problem 03, returns a starting point.
• p03_title.m, for problem 03, returns the title.

• p04_fx.m, for problem 04, evaluates the function.
• p04_jac.m, for problem 04, evaluates the jacobian matrix.
• p04_n.m, for problem 04, returns legal problem sizes.
• p04_sol.m, for problem 04, returns an estimated root, if known.
• p04_start.m, for problem 04, returns a starting point.
• p04_title.m, for problem 04, returns the title.

• p05_fx.m, for problem 05, evaluates the function.
• p05_jac.m, for problem 05, evaluates the jacobian matrix.
• p05_n.m, for problem 05, returns legal problem sizes.
• p05_sol.m, for problem 05, returns an estimated root, if known.
• p05_start.m, for problem 05, returns a starting point.
• p05_title.m, for problem 05, returns the title.

• p06_fx.m, for problem 06, evaluates the function.
• p06_jac.m, for problem 06, evaluates the jacobian matrix.
• p06_n.m, for problem 06, returns legal problem sizes.
• p06_sol.m, for problem 06, returns an estimated root, if known.
• p06_start.m, for problem 06, returns a starting point.
• p06_title.m, for problem 06, returns the title.

• p07_fx.m, for problem 07, evaluates the function.
• p07_jac.m, for problem 07, evaluates the jacobian matrix.
• p07_n.m, for problem 07, returns legal problem sizes.
• p07_sol.m, for problem 07, returns an estimated root, if known.
• p07_start.m, for problem 07, returns a starting point.
• p07_title.m, for problem 07, returns the title.

• p08_fx.m, for problem 08, evaluates the function.
• p08_jac.m, for problem 08, evaluates the jacobian matrix.
• p08_n.m, for problem 08, returns legal problem sizes.
• p08_sol.m, for problem 08, returns an estimated root, if known.
• p08_start.m, for problem 08, returns a starting point.
• p08_title.m, for problem 08, returns the title.

• p09_fx.m, for problem 09, evaluates the function.
• p09_jac.m, for problem 09, evaluates the jacobian matrix.
• p09_n.m, for problem 09, returns legal problem sizes.
• p09_sol.m, for problem 09, returns an estimated root, if known.
• p09_start.m, for problem 09, returns a starting point.
• p09_title.m, for problem 09, returns the title.

• p10_fx.m, for problem 10, evaluates the function.
• p10_jac.m, for problem 10, evaluates the jacobian matrix.
• p10_n.m, for problem 10, returns legal problem sizes.
• p10_sol.m, for problem 10, returns an estimated root, if known.
• p10_start.m, for problem 10, returns a starting point.
• p10_title.m, for problem 10, returns the title.

• p11_fx.m, for problem 11, evaluates the function.
• p11_jac.m, for problem 11, evaluates the jacobian matrix.
• p11_n.m, for problem 11, returns legal problem sizes.
• p11_sol.m, for problem 11, returns an estimated root, if known.
• p11_start.m, for problem 11, returns a starting point.
• p11_title.m, for problem 11, returns the title.

• p12_fx.m, for problem 12, evaluates the function.
• p12_jac.m, for problem 12, evaluates the jacobian matrix.
• p12_n.m, for problem 12, returns legal problem sizes.
• p12_sol.m, for problem 12, returns an estimated root, if known.
• p12_start.m, for problem 12, returns a starting point.
• p12_title.m, for problem 12, returns the title.

• p13_fx.m, for problem 13, evaluates the function.
• p13_jac.m, for problem 13, evaluates the jacobian matrix.
• p13_n.m, for problem 13, returns legal problem sizes.
• p13_sol.m, for problem 13, returns an estimated root, if known.
• p13_start.m, for problem 13, returns a starting point.
• p13_title.m, for problem 13, returns the title.

• p14_fx.m, for problem 14, evaluates the function.
• p14_jac.m, for problem 14, evaluates the jacobian matrix.
• p14_n.m, for problem 14, returns legal problem sizes.
• p14_sol.m, for problem 14, returns an estimated root, if known.
• p14_start.m, for problem 14, returns a starting point.
• p14_title.m, for problem 14, returns the title.

• p15_fx.m, for problem 15, evaluates the function.
• p15_jac.m, for problem 15, evaluates the jacobian matrix.
• p15_n.m, for problem 15, returns legal problem sizes.
• p15_sol.m, for problem 15, returns an estimated root, if known.
• p15_start.m, for problem 15, returns a starting point.
• p15_title.m, for problem 15, returns the title.

• p16_fx.m, for problem 16, evaluates the function.
• p16_jac.m, for problem 16, evaluates the jacobian matrix.
• p16_n.m, for problem 16, returns legal problem sizes.
• p16_sol.m, for problem 16, returns an estimated root, if known.
• p16_start.m, for problem 16, returns a starting point.
• p16_title.m, for problem 16, returns the title.

• p17_fx.m, for problem 17, evaluates the function.
• p17_jac.m, for problem 17, evaluates the jacobian matrix.
• p17_n.m, for problem 17, returns legal problem sizes.
• p17_sol.m, for problem 17, returns an estimated root, if known.
• p17_start.m, for problem 17, returns a starting point.
• p17_title.m, for problem 17, returns the title.

• p18_fx.m, for problem 18, evaluates the function.
• p18_jac.m, for problem 18, evaluates the jacobian matrix.
• p18_n.m, for problem 18, returns legal problem sizes.
• p18_sol.m, for problem 18, returns an estimated root, if known.
• p18_start.m, for problem 18, returns a starting point.
• p18_title.m, for problem 18, returns the title.

• p19_fx.m, for problem 19, evaluates the function.
• p19_jac.m, for problem 19, evaluates the jacobian matrix.
• p19_n.m, for problem 19, returns legal problem sizes.
• p19_sol.m, for problem 19, returns an estimated root, if known.
• p19_start.m, for problem 19, returns a starting point.
• p19_title.m, for problem 19, returns the title.

• p20_fx.m, for problem 20, evaluates the function.
• p20_jac.m, for problem 20, evaluates the jacobian matrix.
• p20_n.m, for problem 20, returns legal problem sizes.
• p20_sol.m, for problem 20, returns an estimated root, if known.
• p20_start.m, for problem 20, returns a starting point.
• p20_title.m, for problem 20, returns the title.

• p21_fx.m, for problem 21, evaluates the function.
• p21_jac.m, for problem 21, evaluates the jacobian matrix.
• p21_n.m, for problem 21, returns legal problem sizes.
• p21_sol.m, for problem 21, returns an estimated root, if known.
• p21_start.m, for problem 21, returns a starting point.
• p21_title.m, for problem 21, returns the title.

• p22_fx.m, for problem 22, evaluates the function.
• p22_jac.m, for problem 22, evaluates the jacobian matrix.
• p22_n.m, for problem 22, returns legal problem sizes.
• p22_sol.m, for problem 22, returns an estimated root, if known.
• p22_start.m, for problem 22, returns a starting point.
• p22_title.m, for problem 22, returns the title.

• p23_fx.m, for problem 23, evaluates the function.
• p23_jac.m, for problem 23, evaluates the jacobian matrix.
• p23_n.m, for problem 23, returns legal problem sizes.
• p23_sol.m, for problem 23, returns an estimated root, if known.
• p23_start.m, for problem 23, returns a starting point.
• p23_title.m, for problem 23, returns the title.

• timestamp.m, prints the YMDHMS date as a timestamp.