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Let consider a system of equations in unknowns
and assume that may vary in the range
and that it is considered as the
parameter of this
system. Using the solving algorithms of
`ALIAS` (or any other
method) we are able to determine the real roots of this system, if any, for a
given value of the parameter, for example for
. Let
assume that we have found real roots
. Assume now
that we change the value of to
. Using
Kantorovitch theorem (see section 3.1.2)
we are able to determine if Newton method with as
initial estimate will converge toward a solution of the new
system and the radius
of convergence such that the solution is unique in the ball
(alternatively we may also use
Moore theorem, see section 2.10).
If this is the case and if the
intersection of the is empty, then we are able to compute in a
certified way the new solutions of the new system. Otherwise we
will repeat the procedure with a smaller value for until
the method succeed. A failure case will be when the system become
singular at a point (for example when 2 branches are collapsing). If
this happen we will change the value of
until we find a new set of solutions
which satisfy Kantorovitch theorem and start again the process.
This method enable to follow all the branches for which initial points have
been found when solving the initial system. We may then store these
points in an array, for example at regular step for the value
of the parameter.

By default we use Kantorovitch theorem to follow the branches but by
setting the global variable

`ALIAS_Use_Kraw_Continuation`
to 1 we will use Moore test.

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Jean-Pierre Merlet
2012-12-20