Full continuation procedure

int ALIAS_Full_Continuation(int m,int n, INTERVAL_VECTOR (* IntervalFunction)(int,int,INTERVAL_VECTOR &), INTERVAL_MATRIX (* Gradient)(int, int,INTERVAL_VECTOR &), INTERVAL_MATRIX (* IntervalHessian)(int,int,INTERVAL_VECTOR &), INTERVAL_VECTOR &Domain, int M, double epsilon,double epsilonf, double *z,double delta,double mindelta,double mindz, INTERVAL &Rangez, int sens, MATRIX &BRANCH,int *NBBRANCH)The arguments are the same than for the previous procedure except for:

`Domain`: the range for the`m`variable in which we will look for solutions of the system,`M`is the maximum number of boxes which may be stored (see the note 2.3.4.5)`mindz`: the accuracy with which the starting point for the branches will be determined: if the value of the parameter at which the branches will start is , then the system has no solution for -`mindz`. This is done by using a bisection on the parameter: if at the system has no solution and has a positive number of solution at +`delta`, then we will solve the system for +`delta`/2. At each time we will store the value of the parameter for which there is no solution of the system and the value for which we have solutions and we will stop the bisection on the parameter as soon as`mindz`.

There is also another version of this program where you indicate just
before `Domain` the solutions which have already been found. The
syntax is

INTERVAL_MATRIX (* IntervalHessian)(int,int,INTERVAL_VECTOR &), int NUM, INTERVAL_MATRIX &Solutions, INTERVAL_VECTOR &Domain

The return code for these procedures are:

- : the number of branches found by the algorithm
- -1: no initial point have been found
- -10: Newton algorithm has failed (should not occur)
- -20:
`sens`is not 1 or -1 - -30:
`delta`or`mindelta`is negative

Finding the initial starting point with the accuracy
`mindz` may be computer intensive.
Hence the integer global variable
`ALIAS_Allow_Backtrack`
enable to disable this process if it is set to 0 (its default value is
1): in that case as soon as starting points have been found (hence at
+ `delta`) we will start following the branches.

In fact these procedures are special occurrences of another `ALIAS`
procedure which has another argument right after the `Hessian`
argument. Assume for example that you are considering a system which
has one equation written as:

where is the parameter of the system and the unknowns. When using the continuation method we have to define ranges for these unknowns in order to be able to solve the system of equations. Up to now we have indicated bounds that are constants but for the equation example it will be interesting to be able to specify that these bounds may change according to the value for the parameter using a

INTERVAL_VECTOR Range(double z, INTERVAL_VECTOR &Variable)where

Note also that the `ALIAS`-Maple package offers a procedure that
uses the method described in section 2.14 for finding
the initial starting points of the branches: this method is efficient
if the equations include linear terms in the unknowns.