# 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

### 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:

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Reliable Error Estimations and Mesh Adaptation for the Finite Element Method,
in International Conference on Computational Methods in Nonlinear Mechanics,
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The Solution of Nonlinear Systems by A-stable Integration Techniques,
SIAM Journal on Numerical Analysis,
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Calculation of Equilibrium Points for Models of Ecological and Chemical Systems,
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A New Method of Solving Nonlinear Simultaneous Equations,
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Air Products and Chemicals, Inc.
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Allentown, Pennsylvania, 18105.
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Computing,
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Computing,
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A Continuation Method for the Calculation of Electrostatic Potentials in Semiconductors,
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Catastrophe Theory and its Applications,
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Computation of Critical Boundaries on Equilibrium Manifolds,
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A Locally Parameterized Continuation Process,
ACM Transactions on Mathematical Software,
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Algorithm 596: A Program for a Locally Parameterized Continuation Process,
ACM Transactions on Mathematical Software,
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Numerical Analysis of Parameterized Nonlinear Equations,
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Institute for Computational Mathematics and Applications,
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The Application of Newton's Method to the Problem of Elastic Stability,
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Prediction of Jump Phenomena in Roll-coupled Maneuvers of Airplanes,
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A Method for the Numerical Determination of Bifurcation States of Nonlinear Systems of Equations,
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Optimal Stepsize Control for the Globalized Newton Method,
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