test_con

test_con, a Fortran90 code which defines tests for the continuation problem, which considers a function F(X) with N-dimensional argument X and N-1 dimensional result. In general, this defines an implicit one-dimensional curve of solutions X(LAMBDA). A continuation code starts from a single point on this curve and tries to compute a sequence of solutions that form a path.

A continuation code might carry out the following steps:

1. choose a problem by picking an index number.
2. for problems with several options, pick an option index.
3. find out the number of variables.
4. get a starting point X0.
5. get a suggested stepsize H.
6. get the tangent vector T at X0.
7. use the estimate X1=X0+H*T as a starting point for a new point on the curve; use Newton method to refine the estimate.
8. If the Newton iteration failed, reduce H and try again.
9. If the new point was computed "easily", increase H.
10. Go back to step 6 if another point is desired.

The code includes routines to

• return the number of problems available (p00_problem_num);
• return the number of different "options" for each problem (p00_option_num);
• return the problem size NVAR (p00_nvar);
• provide a starting point X0 (p00_start);
• provide a suggested stepsize H (p00_stepsize);
• determine the tangent vector T(X) (p00_tan);
• apply Newton's method to an approximate solution (p00_newton);
• take a single continuation step (compute the "next" point) (p00_step);
• compute a target point, for which one component has a selected value (p00_target);
• choose the continuation parameter index (p00_par_index);
• evaluate the function F(X) (p00_fun);
• evaluate the jacobian J(X) (p00_problem_jac);
• return the problem title (p00_title);

The list of problems includes:

1. The Freudenstein-Roth function
2. The Boggs function
3. The Powell function
4. The Broyden function
5. The Wacker function
6. The Aircraft stability function
7. The Cell kinetic function
8. The Riks mechanical problem
9. The Oden mechanical problem
10. Torsion of a square rod, finite difference solution
11. Torsion of a square rod, finite element solution
12. The materially nonlinear problem
13. Simpson's mildly nonlinear boundary value problem
14. Keller's boundary value problem
15. The Trigger Circuit
16. The Moore-Spence Chemical Reaction Integral Equation
17. The Bremermann Propane Combustion System
18. The semiconductor problem
19. The Nitric acid absorption flash
20. The Buckling Spring

Licensing:

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

Languages:

test_con is available in a Fortran77 version and a Fortran90 version and a MATLAB version.

Related Data and Programs:

test_con_test

continuation, a MATLAB library which implements the continuation method for a simple 2D problem, which involves finding a point on the unit circle, and then finding a sequence of nearby points which trace out the full curve, using only the information available in the implicit definition of the curve from the function f(x,y)=x^2+y^2-1.

pitcon66, a Fortran77 library which seeks to produce a sequence of points that satisfy a set of nonlinear equations with one degree of freedom; this is version 6.6 of ACM TOMS algorithm 596.

pitcon7, a Fortran90 library which seeks to produce a sequence of points that satisfy a set of nonlinear equations with one degree of freedom; this is version 7.0 of ACM TOMS algorithm 596.

TEST_CON, a dataset directory which contains sequences of points that lie on multidimensional curves defined by sets of nonlinear equations;

TOMS502, a Fortran77 library which seeks to produce a sequence of points that satisfy a set of nonlinear equations with one degree of freedom; this library is commonly called DERPAR;
this is ACM TOMS algorithm 502.

TOMS596, a Fortran77 library which seeks to produce a sequence of points that satisfy a set of nonlinear equations with one degree of freedom; this library is commonly called PITCON;
this is ACM TOMS algorithm 596.

Reference:

1. Ivo Babuska, Werner Rheinboldt,
   Reliable Error Estimations and Mesh Adaptation for the Finite Element Method,
   in International Conference on Computational Methods in Nonlinear Mechanics,
   edited by John Oden,
   Elsevier, 1980,
   ISBN: 0444853820,
   LC: QA808.I57.
2. Paul Boggs,
   The Solution of Nonlinear Systems by A-stable Integration Techniques,
   SIAM Journal on Numerical Analysis,
   Volume 8, Number 4, December 1971, pages 767-785.
3. Hans Bremermann,
   Calculation of Equilibrium Points for Models of Ecological and Chemical Systems,
   in Proceedings of a Conference on the Applications of Undergraduate Mathematics in the Engineering, Life, Managerial and Social Sciences,
   Georgia Institute of Technology, June 1973, pages 198-217.
4. Charles Broyden,
   A New Method of Solving Nonlinear Simultaneous Equations,
   The Computer Journal,
   Volume 12, 1969, pages 94-99.
5. Tama Copeman,
   Air Products and Chemicals, Inc.
   Box 538,
   Allentown, Pennsylvania, 18105.
6. Cor denHeijer, Werner Rheinboldt,
   On Steplength Algorithms for a Class of Continuation Methods,
   SIAM Journal on Numerical Analysis,
   Volume 18, Number 5, October 1981, pages 925-947.
7. Ferdinand Freudenstein, Bernhard Roth,
   Numerical Solutions of Nonlinear Equations,
   Journal of the ACM,
   Volume 10, Number 4, October 1963, pages 550-556.
8. Kathie Hiebert,
   A Comparison of Software Which Solves Systems of Nonlinear Equations,
   Technical Report SAND-80-0181,
   Sandia National Laboratory, 1980.
9. Herbert Keller,
   Numerical Methods for Two-point Boundary Value Problems,
   Dover, 1992,
   ISBN: 0486669254,
   LC: QA372.K42.
10. Raman Mehra, William Kessel, James Carroll,
    Global stability and contral analysis of aircraft at high angles of attack,
    Technical Report CR-215-248-1, -2, -3,
    Office of Naval Research, June 1977.
11. Rami Melhem, Werner Rheinboldt,
    A Comparison of Methods for Determining Turning Points of Nonlinear Equations,
    Computing,
    Volume 29, Number 3, September 1982, pages 201-226.
12. Gerald Moore, Alastair Spence,
    The Calculation of Turning Points of Nonlinear Equations,
    SIAM Journal on Numerical Analysis,
    Volume 17, Number 4, August 1980, pages 567-576.
13. John Oden,
    Finite Elements of Nonlinear Continua,
    Dover, 2006,
    ISBN: 0486449734,
    LC: QA808.2.O33.
14. Gerd Poenisch, Hubert Schwetlick,
    Computing Turning Points of Curves Implicitly Defined by Nonlinear Equations Depending on a Parameter,
    Computing,
    Volume 26, Number 2, June 1981, pages 107-121.
15. SJ Polak, A Wachten, H Vaes, A deBeer, Cor denHeijer,
    A Continuation Method for the Calculation of Electrostatic Potentials in Semiconductors,
    Technical Report ISA-TIS/CARD,
    NV Philips Gloeilampen-Fabrieken, 1979.
16. Tim Poston, Ian Stewart,
    Catastrophe Theory and its Applications,
    Dover, 1996,
    ISBN13: 978-0486692715,
    LC: QA614.58.P66.
17. Michael Powell,
    A Fortran Subroutine for Solving Systems of Nonlinear Algebraic Equations,
    in Numerical Methods for Nonlinear Algebraic Equations,
    edited by Philip Rabinowitz,
    Gordon and Breach, 1970,
    ISBN13: 978-0677142302,
    LC: QA218.N85.
18. Werner Rheinboldt,
    Computation of Critical Boundaries on Equilibrium Manifolds,
    SIAM Journal on Numerical analysis,
    Volume 19, Number 3, June 1982, pages 653-669.
19. Werner Rheinboldt, John Burkardt,
    A Locally Parameterized Continuation Process,
    ACM Transactions on Mathematical Software,
    Volume 9, Number 2, June 1983, pages 215-235.
20. Werner Rheinboldt, John Burkardt,
    Algorithm 596: A Program for a Locally Parameterized Continuation Process,
    ACM Transactions on Mathematical Software,
    Volume 9, Number 2, June 1983, pages 236-241.
21. Werner Rheinboldt,
    Numerical Analysis of Parameterized Nonlinear Equations,
    Wiley, 1986,
    ISBN: 0-471-88814-1,
    LC: QA372.R54.
22. Werner Rheinboldt,
    Sample Problems for Continuation Processes,
    Technical Report ICMA-80-?,
    Institute for Computational Mathematics and Applications,
    Department of Mathematics,
    University of Pittsburgh, November 1980.
23. Werner Rheinboldt,
    Solution Fields of Nonlinear Equations and Continuation Methods,
    SIAM Journal on Numerical Analysis,
    Volume 17, Number 2, April 1980, pages 221-237.
24. Werner Rheinboldt,
    On the Solution of Some Nonlinear Equations Arising in the Application of Finite Element Methods,
    in The Mathematics of Finite Elements and Applications II,
    edited by John Whiteman,
    Academic Press, 1976,
    LC: TA347.F5.M37.
25. E Riks,
    The Application of Newton's Method to the Problem of Elastic Stability,
    Transactions of the ASME, Journal of Applied Mechanics,
    December 1972, pages 1060-1065.
26. Albert Schy, Margery Hannah,
    Prediction of Jump Phenomena in Roll-coupled Maneuvers of Airplanes,
    Journal of Aircraft,
    Volume 14, Number 4, 1977, pages 375-382.
27. Bruce Simpson,
    A Method for the Numerical Determination of Bifurcation States of Nonlinear Systems of Equations,
    SIAM Journal on Numerical Analysis,
    Volume 12, Number 3, June 1975, pages 439-451.
28. Hans-Joerg Wacker, Erich Zarzer, Werner Zulehner,
    Optimal Stepsize Control for the Globalized Newton Method,
    in Continuation Methods,
    edited by Hans-Joerg Wacker,
    Academic Press, 1978,
    ISBN: 0127292500,
    LC: QA1.S899.
29. John Young, Albert Schy, Katherine Johnson,
    Prediction of Jump Phenomena in Aircraft Maneuvers, Including Nonlinear Aerodynamic Effects,
    Journal of Guidance and Control,
    Volume 1, Number 1, 1978, pages 26-31.

Source Code:

• test_con.f90, the source code;
• test_con.sh, compiles the source code;