Continuation with initially known starting points

If the starting point of branches is known it is possible to follow the branches without initially solving the system by using the procedure

 
StartContinuation(Func,Vars,Par,Init,RPar,DPar,MinD,MinI,Sens,Base,P)

where

• Func: the list of equations
• Vars: a list with the name of variables
• Par: name of the variable that will be used a parameter
• Init: a list of intervals ranges for the unknowns
• RPar: a list which define the minimal and maximal value of Par
• DPar a point is added to the plot each time the current value of Par is at DPar from the previous one
• MinD a parameter that is used to determine singular points. Its value should be small
• MinI: the accuracy on the parameter with which the initial starting point of the branch is determined: if the parameter is z and the curve start at z0, then if Sens=1 the system has no solution for z=z0-MinI
• Sens 1 is the curve is drawn from Min(Par) to Max(Par) or -1 if drawn from Max(Par) to Min(Par).
• Base a string that indicate the base-name for the files that will be created
• P: a list of starting points for the branches. For two unknowns this list has the form [[[a1,a1],[a2,a2]],...]. These points must be a solution of the system in the Kantorovitch sense and be obtained using of the solving procedure using the Hessian

A mechanism allows the user to stop the determination of a branch. For that purpose the user has to write a C++ program such as

 
 int End_Cont(MATRIX &BRANCH,int nb,int Sens, double z, INTERVAL &LIMITS)

This procedure receives the current value of the parameter (z), possible limits that have been defined for this parameters (LIMITS) and the direction (Sens) in which the value of the parameter is changed. The values of the other variables are stored in BRANCH, while their value for the current z is BRANCH(nb,...). If this procedure returns -1, then the branch will be no more followed, otherwise the procedure must return 0.

The user may indicate that such procedure is available by setting the variable `ALIAS/user_stop_continuation` to a string which indicates the name of the C++ procedure, being understood that the file that includes the code is `ALIAS/user_stop_continuation`.C

A specific procedure exists for distance equations:

 
StartContinuationDistance(Func,Vars,Par,Init,RPar,DPar,MinD,MinI,Sens,Base,P)