This procedure takes as input a set of solutions of the system and will return points on the branches. The branches will be followed until a given value for the parameter is reached or if Kantorovitch theorem is no more satisfied for some value of the parameter. It is implemented as:

int ALIAS_Start_Continuation(int m,int n,int NUM, INTERVAL_MATRIX &Solutions, INTERVAL_VECTOR (* IntervalFunction)(int,int,INTERVAL_VECTOR &), INTERVAL_MATRIX (* Gradient)(int, int,INTERVAL_VECTOR &), INTERVAL_MATRIX (* IntervalHessian)(int,int,INTERVAL_VECTOR &), double epsilon,double epsilonf, double *z,double delta,double mindelta, INTERVAL &Rangez, int sens, MATRIX &BRANCH,int *NBBRANCH)the arguments being:

`m`: number of unknowns`n`: number of equations`NUM`: the number of solutions of the system for the current value of the parameter`z``Solutions`: the solutions of the system for the current value of the parameter`z``IntervalFunction`: a function which return the interval vector evaluation of the equations, see note 2.3.4.3.`IntervalGradient`: a function which return the interval matrix of the jacobian of the equations, see note 2.4.2.2`IntervalHessian`: a function which return the interval matrix of the Hessian of the equations, see note 2.5.2.1`epsilon`: the maximal width of the box, see the note 2.3.4.6. If all the variable ranges have a width lower than this value and the interval evaluation of the equations contains all 0, then the set of ranges is considered to be a solution. But they will be not considered as a valid solution as they will not satisfy Kantorovitch theorem. Hence you must put here a very small value`epsilonf`: the maximal width of the equation intervals, see the note 2.3.4.6. This value will be used by the iterative Newton scheme to stop the iteration.`z`: the parameter of the system`delta`: if is the first value of the parameter such that the system has solutions, then we will store in`BRANCH`the solutions for +`delta`, +2`delta`,...`mindelta`: if the algorithm is unable to find a Kantorovitch solution for the parameter value`z+mindelta`while a solution has been found for`z`, then the algorithm will assume that we are close to a singular point. Hence`mindelta`should have a small value`Rangez`: the range for the parameter`z``sens`: either 1 or -1. If 1 the branch will be computed for increasing values of`z`, if -1 for decreasing values of`z`. This parameter should be chosen according to the largest number of solutions of the system for the lower and upper value of`z``BRANCH`: the procedure will return an array of`NBBRANCH`lines and`m+2`columns, which describe the points of the branch. In a line the first`m`+1 elements are the coordinates of the point and the`m`+2 elements is the number of the branch to which belong this point. The points are not ordered with respect to the branch number. The algorithm will take care of resizing`BRANCH`as needed, hence there is no need to give dimension for this parameter`NBBRANCH`: the total number of points in the array`BRANCH`.

- 1: the branches have been successfully determined
- -1, -2: the algorithm has failed to find a Kantorovitch solution for the current value of the parameter
- -3: failure in Newton method (should not occur)
- -10: for the current value of the parameter the jacobian is singular
- -20:
`sens`is not 1 or -1 - -30:
`delta`or`mindelta`is negative