TEST_CON
Continuation Tests

TEST_CON is a FORTRAN77 library which defines test functions for continuation codes.

A simple continuation code is an algorithm for producing a sequence of solutions of the system of equations F(X) = 0, where there are fewer equations F than variables X. Commonly, there is one more variable than equation, resulting in one degree of freedom. The set of solutions will then generally describe a curve.

The full version of this program is not runnable (there are a few missing routines.) However, the problem definitions are generally complete, and usable.

TEST_CON includes routines to

• return the problem title;
• return the number of different "options" for parameter settings;
• return the problem size;
• provide a starting point;
• provide a reasonable initial stepsize H, and stepsize limits HMIN and HMAX;
• evaluate the function F;
• evaluate the jacobian J;

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 kinetics 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. Rabinowitz's symmetric integration formula function.
21. Porsching's tube flow function.
22. The arch limit on lambda.
23. The arch limit run on mu after lambda.
24. The arch limit run on nu after mu after lambda.
25. The three-element arch problem
26. The circle-line problem
27. The finite difference 2D Poisson problem.
28. Walker's arch
29. The trussed dome problem

Licensing:

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

Languages:

TEST_CON is available in a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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. 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.
17. Werner Rheinboldt,
    Computation of Critical Boundaries on Equilibrium Manifolds,
    SIAM Journal on Numerical analysis,
    Volume 19, Number 3, June 1982, pages 653-669.
18. Werner Rheinboldt, John Burkardt,
    A Locally Parameterized Continuation Process,
    ACM Transactions on Mathematical Software,
    Volume 9, Number 2, June 1983, pages 215-235.
19. 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.
20. Werner Rheinboldt,
    Numerical Analysis of Parameterized Nonlinear Equations,
    Wiley, 1986,
    ISBN: 0-471-88814-1,
    LC: QA372.R54.
21. 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.
22. Werner Rheinboldt,
    Solution Fields of Nonlinear Equations and Continuation Methods,
    SIAM Journal on Numerical Analysis,
    Volume 17, Number 2, April 1980, pages 221-237.
23. 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.
24. 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.
25. 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.
26. 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.
27. 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.
28. 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.f, the source code.
• test_con.sh, commands to compile the source code.

Examples and Tests:

• test_con_prb.f, a sample calling program.
• test_con_prb.sh, commands to compile and run the sample program.
• test_con_prb_output.txt, the output file.

List of Routines:

• P00_FUN evaluates the function for any problem.
• P00_JAC evaluates the jacobian for any problem.
• P00_NVAR returns the number of variables for any problem.
• P00_OPTION_NUM returns the number of options available for a problem.
• P00_PROBLEM_NUM returns the number of problems available.
• P00_START returns a starting point for any problem.
• P00_STEPSIZE returns step sizes for any problem.
• P00_TITLE sets the title for any problem.
• P01_FUN evaluates the Freudenstein-Roth function.
• P01_GX is an auxilliary routine for the Freudenstein-Roth function.
• P01_JAC evaluates the Freudenstein-Roth jacobian.
• P01_NVAR sets the number of variables for problem 1.
• P01_OPTION_NUM returns the number of options for problem 1.
• P01_START returns a starting point for problem 1.
• P01_STEPSIZE returns step sizes for problem 1.
• P01_TITLE sets the title for problem 1.
• P02_FUN evaluates the function for problem 2.
• P02_GX evaluates the underlying function for problem 2.
• P02_JAC evaluates the jacobian for problem 2.
• P02_NVAR sets the number of variables for problem 2.
• P02_OPTION_NUM returns the number of options for problem 2.
• P02_START returns a starting point for problem 2.
• P02_STEPSIZE returns step sizes for problem 2.
• P02_TITLE sets the title for problem 2.
• P03_DATA sets parameters for the Powell function.
• P03_FUN evaluates the Powell function.
• P03_GX is an auxilliary routine for the Powell function.
• P03_JAC evaluates the Powell jacobian.
• P03_NVAR sets the number of variables for problem 3.
• P03_OPTION_NUM returns the number of options for problem 3.
• P03_START returns a starting point for problem 3.
• P03_STEPSIZE returns step sizes for problem 3.
• P03_TITLE sets the title for problem 3.
• P04_DATA sets parameters for the Broyden function.
• P04_FUN evaluates the Broyden function.
• P04_GX is an auxilliary routine for the Broyden function.
• P04_JAC evaluates the Broyden jacobian.
• P04_NVAR sets the number of variables for problem 4.
• P04_OPTION_NUM returns the number of options for problem 4.
• P04_START returns a starting point for problem 4.
• P04_STEPSIZE returns step sizes for problem 4.
• P04_TITLE sets the title for problem 4.
• P05_DATA sets parameters for the Wacker function.
• P05_FUN evaluates the Wacker function.
• P05_JAC evaluates the Wacker jacobian.
• P05_NVAR sets the number of variables for problem 5.
• P05_OPTION_NUM returns the number of options for problem 5.
• P05_START returns a starting point for problem 5.
• P05_STEPSIZE returns step sizes for problem 5.
• P05_TITLE sets the title for problem 5.
• P06_DATA sets parameters for the Aircraft Stability function.
• P06_FUN evaluates the aircraft stability function.
• P06_JAC evaluates the aircraft stability jacobian.
• P06_NVAR sets the number of variables for problem 6.
• P06_OPTION_NUM returns the number of options for problem 6.
• P06_START returns a starting point for problem 6.
• P06_STEPSIZE returns step sizes for problem 6.
• P06_TITLE sets the title for problem 6.
• P07_DATA sets parameters for the cell kinetics function.
• P07_FUN evaluates the cell kinetics function.
• P07_JAC evaluates the cell kinetics jacobian.
• P07_NVAR sets the number of variables for problem 7.
• P07_OPTION_NUM returns the number of options for problem 7.
• P07_START returns a starting point for problem 7.
• P07_STEPSIZE returns step sizes for problem 7.
• P07_TITLE sets the title for problem 7.
• P08_DATA sets parameters for Riks's mechanical function.
• P08_FUN evaluates Riks's mechanical function.
• P08_JAC evaluates the Riks's mechanical jacobian.
• P08_NVAR sets the number of variables for problem 8.
• P08_OPTION_NUM returns the number of options for problem 8.
• P08_START returns a starting point for problem 8.
• P08_STEPSIZE returns step sizes for problem 8.
• P08_TITLE sets the title for problem 8.
• P09_DATA sets parameters for Oden's mechanical function.
• P09_FUN evaluates Oden's mechanical function.
• P09_JAC evaluates Oden's mechanical jacobian.
• P09_NVAR sets the number of variables for problem 9.
• P09_OPTION_NUM returns the number of options for problem 9.
• P09_START returns a starting point for problem 9.
• P09_STEPSIZE returns step sizes for problem 9.
• P09_TITLE sets the title for problem 9.
• P10_DATA sets parameters for the finite difference rod torsion function.
• P10_FUN evaluates the finite difference rod torsion function.
• P10_GX is an auxilliary function for the rod torsion problem.
• P10_JAC evaluates the finite difference rod torsion jacobian.
• P10_GP is an auxilliary function for the rod torsion problem.
• P10_NVAR sets the number of variables for the finite difference rod torsion function.
• P10_OPTION_NUM returns the number of options for the finite difference rod torsion function.
• P10_START returns a starting point for the finite difference rod torsion function.
• P10_STEPSIZE returns step sizes for the finite difference rod torsion function.
• P10_TITLE sets the title for the finite difference rod torsion function.
• P11_DATA sets parameters for the finite element rod torsion function.
• P11_FUN evaluates the finite element rod torsion function.
• P11_FLAM is an auxilliary function for the finite element rod torsion function.
• P11_PHI is an auxilliary function for the finite element rod torsion function.
• P11_UVAL is an auxilliary function for the finite element rod torsion function.
• P11_SH is an auxilliary function for the finite element rod torsion function.
• P11_GAUS returns Gauss points for the finite element rod torsion problem.
• P11_JAC evaluates the finite element rod torsion jacobian.
• P11_NVAR sets the number of variables for the finite element rod torsion function.
• P11_OPTION_NUM returns the number of options for the finite element rod torsion function.
• P11_START returns a starting point for the finite element rod torsion function.
• P11_STEPSIZE returns step sizes for the finite element rod torsion function.
• P11_TITLE sets the title for the finite element rod torsion function.
• P12_DATA sets parameters for the materially nonlinear function.
• P12_BD sets boundary values for the materially nonlinear problem.
• P12_FUN evaluates the materially nonlinear function.
• P12_GX is an auxilliary routine for the materially nonlinear problem.
• P12_JAC evaluates the materially nonlinear jacobian.
• P12_VL is an auxilliary routine for the materially nonlinear problem.
• P12_GP is an auxilliary function for the materially nonlinear problem.
• P12_NVAR sets the number of variables for the materially nonlinear function.
• P12_OPTION_NUM returns the number of options for the materially nonlinear function.
• P12_START returns a starting point for the materially nonlinear function.
• P12_STEPSIZE returns step sizes for the materially nonlinear function.
• P12_TITLE sets the title for the materially nonlinear function.
• P13_DATA sets parameters for Simpson's boundary value function.
• P13_FUN evaluates Simpson's mildly nonlinear boundary value function.
• P13_GP is an auxilliary function for simpsons mildly nonlinear boundary function.
• P13_GX is an auxilliary function for simpsons mildly nonlinear boundary function.
• P13_JAC evaluates the Simpson's boundary value jacobian.
• P13_NVAR sets the number of variables for problem 13.
• P13_OPTION_NUM returns the number of options for problem 13.
• P13_START returns a starting point for problem 13.
• P13_STEPSIZE returns step sizes for problem 13.
• P13_TITLE sets the title for problem 13.
• P14_DATA sets parameters for Keller's boundary value function.
• P14_FUN evaluates Keller's boundary value function.
• P14_GX is an auxilliary function for Keller's boundary value problem.
• P14_JAC evaluates the Keller boundary value jacobian.
• P14_GP is an auxilliary function for Keller's boundary value problem.
• P14_NVAR sets the number of variables for problem 14.
• P14_OPTION_NUM returns the number of options for problem 14.
• P14_START returns a starting point for problem 14.
• P14_STEPSIZE returns step sizes for problem 14.
• P14_TITLE sets the title for problem 14.
• P15_DATA sets parameters for the trigger circuit function.
• P15_FUN evaluates the trigger circuit function.
• P15_JAC evaluates the trigger circuit jacobian.
• P15_NVAR sets the number of variables for the trigger circuit function.
• P15_OPTION_NUM returns the number of options for the trigger circuit function.
• P15_START returns a starting point for the trigger circuit function.
• P15_STEPSIZE returns step sizes for the trigger circuit function.
• P15_TITLE sets the title for the trigger circuit function.
• P16_DATA sets parameters for the Moore-Spence chemical reaction function.
• P16_FUN evaluates the Moore-Spence chemical reaction function.
• P16_JAC evaluates the Moore-Spence chemical reaction jacobian.
• P16_NVAR sets the number of variables for the Moore-Spence chemical reaction function.
• P16_OPTION_NUM returns the number of options for the Moore-Spence chemical reaction function.
• P16_START returns a starting point for the Moore-Spence chemical reaction function.
• P16_STEPSIZE returns step sizes for the Moore-Spence chemical reaction function.
• P16_TITLE sets the title for the Moore-Spence chemical reaction function.
• P17_DATA sets parameters for the Bremerman propane combustion system.
• P17_FUN evaluates the Bremerman propane combustion function.
• P17_JAC evaluates the Bremerman propane combustion jacobian.
• P17_NVAR sets the number of variables for the Bremerman propane combustion system.
• P17_OPTION_NUM returns the number of options for the Bremerman propane combustion system.
• P17_START returns a starting point for the Bremerman propane combustion system.
• P17_STEPSIZE returns step sizes for the Bremerman propane combustion system.
• P17_TITLE sets the title for the Bremerman propane combustion system.
• P18_DATA sets parameters for the semiconductor function.
• P18_FUN evaluates the semiconductor function.
• P18_JAC evaluates the semiconductor jacobian.
• P18_NVAR sets the number of variables for the semiconductor function.
• P18_OPTION_NUM returns the number of options for the semiconductor function.
• P18_START returns a starting point for the semiconductor function.
• P18_STEPSIZE returns step sizes for the semiconductor function.
• P18_TITLE sets the title for the semiconductor function.
• P19_DATA sets parameters for the nitric absorption problem.
• P19_EDIT is an edit routine for the nitric absorption problem.
• P19_FUN evaluates the nitric absorption function.
• P19_CN evaluates constitutive relations for the nitric absorption function.
• P19_JAC evaluates the nitric absorption jacobian.
• P19_SC scales and unscales data for the nitric absorption problem.
• P19_NVAR sets the number of variables for the nitric absorption problem.
• P19_OPTION_NUM returns the number of options for the nitric absorption problem.
• P19_START returns a starting point for the nitric absorption problem.
• P19_STEPSIZE returns step sizes for the nitric absorption problem.
• P19_TITLE sets the title for the nitric absorption problem.
• P20_DATA sets parameters for Rabinowitz's symmetric integration rule function.
• P20_FUN evaluates the Rabinowitz symmetric integration formula function.
• P20_JAC evaluates the Rabinowitz symmetric integration formula jacobian.
• P20_NVAR sets the number of variables for problem 20.
• P20_OPTION_NUM returns the number of options for problem 20.
• P20_START returns a starting point for problem 20.
• P20_STEPSIZE returns step sizes for problem 20.
• P20_TITLE sets the title for problem 20.
• P21_DATA sets parameters for Porsching's tube flow.
• P21_FUN evaluates the Porsching tube flow function.
• P21_JAC evaluates the Porsching tube flow jacobian.
• P21_CN evaluates constitutive relations for the Porsching tube flow function.
• P21_CP differentiates constitutive relations for the Porsching tube flow function.
• P21_NVAR sets the number of variables for problem 21.
• P21_OPTION_NUM returns the number of options for problem 21.
• P21_START returns a starting point for problem 21.
• P21_STEPSIZE returns step sizes for problem 21.
• P21_TITLE sets the title for problem 21.
• P22_DATA sets parameters for the arch limit run on lambda.
• P22_FUN evaluates the arch function for limit run on lambda.
• P22_JAC evaluates the arch limit run on lambda jacobian.
• P22_EDIT is an edit routine for problem 22.
• P22_BS evaluates basis functions for the arch limit run.
• P23_BS is a new version of BS0022 with a different formula.
• P22_VL evaluates finite element functions for the arch limit run.
• P22_LD evaluates the load function for the arch limit run.
• P22_LEXP evaluates load and load derivatives for the arch limit run.
• P22_LCON evaluates the load or load derivatives for the arch limit run.
• P22_LLIN evaluates the load and load derivatives for the arch limit run.
• P22_NVAR sets the number of variables for problem 22.
• P22_OPTION_NUM returns the number of options for problem 22.
• P22_START returns a starting point for problem 22.
• P22_STEPSIZE returns step sizes for problem 22.
• P22_TITLE sets the title for problem 22.
• P23_DATA sets parameters for the arch limit run on mu after lambda.
• P23_FUN evaluates the arch function for limit run on mu after lambda.
• P23_JAC evaluates the arch limit run on mu after lambda jacobian.
• P23_NVAR sets the number of variables for problem 23.
• P23_OPTION_NUM returns the number of options for problem 23.
• P23_START returns a starting point for problem 23.
• P23_STEPSIZE returns step sizes for problem 23.
• P23_TITLE sets the title for problem 23.
• P24_DATA sets parameters for the arch limit run on nu after lambda and mu.
• P24_FUN evaluates the arch function for limit run on nu after mu after lambda.
• P24_JAC evaluates the arch limit run on nu after mu after lambda jacobian.
• P22_GX is an auxilliary function for the arch limit run.
• P22_GP is an auxilliary function for the arch limit run.
• P22_GPA computes y = J * w for the arch limit run.
• P22_G2A is an auxilliary function for the arch limit run.
• P22_G2AB is an auxilliary function for the arch limit run.
• P22_G3AB is an auxilliary function for the arch limit run.
• P24_NVAR sets the number of variables for problem 24.
• P24_OPTION_NUM returns the number of options for problem 24.
• P24_START returns a starting point for problem 24.
• P24_STEPSIZE returns step sizes for problem 24.
• P24_TITLE sets the title for problem 24.
• P25_DATA sets parameters for the three-element arch.
• P25_FUN evaluates the three-element arch function.
• P25_JAC evaluates the three-element arch jacobian.
• P25_NVAR sets the number of variables for problem 25.
• P25_OPTION_NUM returns the number of options for problem 25.
• P25_START returns a starting point for problem 25.
• P25_STEPSIZE returns step sizes for problem 25.
• P25_TITLE sets the title for problem 25.
• P26_DATA sets parameters for the circle-line function.
• P26_FUN evaluates the circle-line function.
• P26_JAC evaluates the circle-line jacobian.
• P26_NVAR sets the number of variables for problem 26.
• P26_OPTION_NUM returns the number of options for problem 26.
• P26_START returns a starting point for problem 26.
• P26_STEPSIZE returns step sizes for problem 26.
• P26_TITLE sets the title for problem 26.
• P27_DATA sets parameters for finite difference 2D Poisson problem.
• P27_FUN evaluates the finite difference 2D Poisson function.
• P27_JAC evaluates the finite difference 2D Poisson jacobian.
• P27_NVAR sets the number of variables for problem 27.
• P27_OPTION_NUM returns the number of options for problem 27.
• P27_START returns a starting point for problem 27.
• P27_STEPSIZE returns step sizes for problem 27.
• P27_TITLE sets the title for problem 27.
• P28_DATA sets parameters for Walker's arch.
• P28_EDIT is an edit routine for Walker's arch.
• P28_FUN evaluates Walker's arch function.
• P28_BS is an auxilliary routine for Walker's arch.
• P28_GX is an auxilliary function for Walker's arch.
• P28_JAC evaluates the Walker's arch jacobian.
• P28_GP is an auxilliary routine for Walker's arch.
• P28_VL is an auxilliary function for Walker's arch.
• P28_NVAR sets the number of variables for Walker's arch.
• P28_OPTION_NUM returns the number of options for Walker's arch.
• P28_START returns a starting point for Walker's arch.
• P28_STEPSIZE returns step sizes for Walker's arch.
• P28_TITLE sets the title for Walker's arch.
• P29_DATA sets parameters for the trussed dome function.
• P29_FUN evaluates the trussed dome function.
• P29_GX is an auxilliary routine for the trussed dome function.
• P29_JAC evaluates the trussed dome jacobian.
• P29_NVAR sets the number of variables for the trussed dome.
• P29_OPTION_NUM returns the number of options for the trussed dome.
• P29_START returns a starting point for the trussed dome.
• P29_STEPSIZE returns step sizes for the trussed dome.
• P29_TITLE sets the title for the trussed dome.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.

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