Missed branches

Clearly there is a major problem with the method we are proposing: we may miss some branches. For example imagine that a system has $n$ roots for the initial value of the parameter, but will have more than $n$ solutions for another value of the parameter, meaning that new branches appear: our algorithm will find only the $n$ initial branches. There are two methods that can be used to find the correct number of branches. The first one is simply to start following the branches with as initial value for the parameter the one among the highest or smallest value having led to the maximum number of solution. There is also another mechanism that enable to avoid missing branches. Assume that the solving procedure has determined for some initial value $\lambda_0$ of the parameter $n$ solutions to the system and that at some point $\lambda_1$ the continuation method has failed: the solving procedure is called and determine that the system has now $m>n$ solutions. This mean that for some parameter value between $\lambda_0$ and $\lambda_1$ we have missed $m-n$ branches. At such point, called problem point it would be interesting to backtrack i.e. to start again a continuation process with as initial point for the branches the $m$ solutions obtained for $\lambda_1$ and a value for Sens which is the opposite of the initial value. This is what is done by the procedure which may store up to 10 problem points. As this process may be computer intensive it is possible to disable it by setting the integer global variable ALIAS_Problem_Continuation to -1 (it's default value is 0, which mean that the process is enabled).