The Continuation procedure

The full continuation procedure described in the ALIAS-C++ manual is available in Maple. It enable one to obtain initial points on the different branches of a one dimensional system and then to follow these branches. These points are written in files. The syntax of this procedure is:

 
Continuation([equation set],[variable set],[parameter name],
  [list of ranges for the variable],[range for the parameter],
  dz,epsilon,epsiloni,Sens,file base name);

where dz is the parameter increment for the storage (a point will be stored each time the parameter change value by dz), epsilon has a small value, epsiloni defines the accuracy with which the starting points of the branches are determined (if we change the parameter value of the starting point by epsiloni, then the system has no solution) and Sens is 1 if we compute the branches by increasing values of the parameter (-1 for decreasing values). Note that determining the initial starting points of the branches with an accuracy on the parameter epsiloni smaller than epsilon may a costly process: this will be done only if the flag `ALIAS/allow_backtrack` is set to 1 (its default value is 0). Note also that according to the value of Sens you may not obtain the same number of branches: Sens should be chosen so that at the initial value of the parameter (the first one for which the system has solutions) the number of solutions is maximal. A major change with respect to version 1.2 is that a backtrack mechanism is used to avoid such problem: if at some point the procedure has followed $n$ branches until a problem for Newton iteration has been detected and if the number of solutions for the next value of the parameter is $m$ with $m>n$, then the procedure will store the value of the $m$ solution and, after having completed the computation for the current Sens, will start again a continuation from this point and will follow the $m$ branch using the opposite value of Sens. Note however that only a maximum of 10 such points are stored and hence some branches may still be missing for complex expression. Note also that among the $m$ branches we will have $n$ branches that will be identical to the one that have been computed in the process before the backtrack.

The points on the branches are written in files whose names start with the file base name and is followed by the branch number. Hence if Branch is file base name and the algorithm has found two branches, then the points are written in the files Branch1, Branch2. In these files is written on each line the values of the variables in the order given by variable set followed by the value of the parameter.

By default we use only Kantorovitch theorem to follow the branches but you may also both this test and Moore test: this is done by setting the flag `ALIAS/kraw` to 1.

For this procedure the Newton scheme will be used and will run for a limited number of iteration defined by `ALIAS/newton_iteration` (default value: 100).

Note that the Draw procedure (section 7.5) allows to directly obtain a Maple plot of the branches as soon as there is only one or two unknowns.