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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 branches until a problem for Newton iteration has been
detected and if the number of solutions for the next value of the
parameter is with , then the procedure will store the value
of the solution and, after having completed the computation for
the current `Sens`, will start again a continuation from this
point and will follow the 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 branches we will have
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.

**Subsections**

** Next:** Optional arguments
** Up:** Continuation for one-dimensional system
** Previous:** Introduction
** Contents**
Jean-Pierre Merlet
2012-12-20