TOMS596
Continuation Code for Parameterized Nonlinear Equations

TOMS596 is a FORTRAN77 library which carries out the continuation method for producing a series of solutions of a set of nonlinear equations with one degree of freedom.

The program is designed for problems in which N variables X are constrained by N-1 nonlinear equations F(X)=0. Generally, there is an entire family of solutions to such a problem, which can be thought of as a curve in N-dimensional space. We can imagine this curve parameterized by a variable LAMBDA. Given one solution (X,LAMBDA0), the program attempts to determine more points on the curve of solutions.

The copy of the program distributed through NETLIB is the original version. However, the original version was written with the assumption that local memory in subroutines was saved in between calls. This assumption is not generally true. The version of the code presented here has been modified by the insertion of the appropriate SAVE statements to guarantee that local variable values are preserved.

TOMS596 is ACM TOMS Algorithm 596. The text of the original FORTRAN77 version is available online through ACM: http://www.acm.org/pubs/calgo or NETLIB: http://www.netlib.org/toms/index.html.

Languages:

TOMS596 is available in a FORTRAN77 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 FORTRAN90 library which implements test problems for numerical continuation.

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.

Reference:

1. Cor den Heijer, 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.
2. Werner Rheinboldt,
   Solution Field of Nonlinear Equations and Continuation Methods,
   SIAM Journal on Numerical Analysis,
   Volume 17, Number 2, April 1980, pages 221-237.
3. Werner Rheinboldt,
   Numerical Analysis of Continuation Methods for Nonlinear Structural Problems,
   Computers and Structures,
   Volume 13, 1981, pages 103-114.
4. Werner Rheinboldt, John Burkardt,
   A Locally Parameterized Continuation Process,
   ACM Transactions on Mathematical Software,
   Volume 9, Number 2, June 1983, pages 215-235.
5. 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.

Source Code:

• toms596.f, the source code.
• toms596.sh, commands to compile the source code.

Examples and Tests:

• toms596_prb.f, a sample calling program.
• toms596_prb.sh, commands to compile and run the sample program.
• toms596_prb_output.txt, the output file.
• toms596_prb_output_saved.txt, the output file, as recorded on the NETLIB site.

List of Routines:

• PITCON is the driver routine for the continuation package.
• CORECT corrects an approximate solution.
• TANGNT computes a unit tangent vector at a point on the solution curve.
• ROOT seeks a root of a scalar nonlinear equation in a change-of-sign interval.
• FSOLVE solves a linear system related to the continuation process.
• ISAMAX returns the index of the vector element of maximum absolute value.
• SASUM returns the sum of the absolute values of the entries of a vector.
• SAXPY adds a multiple of one vector to another.
• SCOPY copies a vector.
• SDOT computes the dot product of two vectors.
• SNRM2 computes the Euclidean norm of a vector.
• SSCAL multiplies a vector by a scale factor.
• SGEFA factors a real matrix by Gaussian elimination.
• SGESL solves a linear system that was factored by SGEFA.

You can go up one level to the FORTRAN77 source codes.