# TOMS502 Dependence of Solution of Nonlinear Systems on a Parameter

TOMS502 is a FORTRAN77 library which investigates the dependence of the solution X of a nonlinear system of equations F(X,ALPHA)=0 on the parameter ALPHA.

The program is an example of the use of the "continuation method". Under reasonable assumptions on the function F, the set of solutions (X,ALPHA) forms a curve, or perhaps a curve with bifurcations.

If one point on the curve is known, or can be found, than further points can be determined by using simple curve-following ideas. One approach is to differentiate the equations with respect to ALPHA, and then apply an ODE solver. However, this approach will fail if the curve has limit points with respect to ALPHA.

Kubicek's approach is to work with a variable that approximates the arc-length of the curve, and to adapt this choice as necessary when the curve changes its direction.

TOMS502 is ACM TOMS Algorithm 502.

The text of many ACM TOMS algorithms is available online through ACM: http://www.acm.org/pubs/calgo or NETLIB: http://www.netlib.org/toms/index.html.

### Languages:

TOMS502 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.

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. Milan Kubicek,
Algorithm 502: Dependence of solution of nonlinear systems on a parameter,
ACM Transactions on Mathematical Software,
Volume 2, Number 1, March 1976, pages 98-107.

### List of Routines:

• DERPAR carries out Kubicek's continuation algorithm.
• ADAMS applies Adams-Bashforth methods to an ODE.
• GAUSE solves a linear system of N equations in N+1 unknowns.

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

Last revised on 02 September 2008.