Clearly there is a major problem with the method we are proposing: we
may miss some branches. For example imagine that a system has
roots for the initial value of the parameter, but will have more than
solutions for another value of the parameter, meaning that
new branches appear: our algorithm will find only the 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 of the parameter
solutions to the system and that at some point the continuation
method has failed: the solving procedure is called and determine that
the system has now solutions. This mean that for some parameter
value between and we have missed
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 solutions obtained for
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).