int Solve_UP_JH_Interval(int Degree,VECTOR Coeff, INTERVAL & TheDomain, int Order,int M,int Stop, double epsilon,double epsilonf, INTERVAL_VECTOR & Solution, INTEGER_VECTOR & IsKanto,int NbSolution);with:

`Degree`: degree of the polynomial`Coeff`: the`Degree+1`coefficients of the polynomial in increasing degree`TheDomain`: the interval in which we are looking for roots`Order`: the type of order which is used to store the intervals created during the bisection process. This order may be either`MAX_FUNCTION_ORDER`or`MAX_MIDDLE_FUNCTION_ORDER`. See the note on the order 2.3.4.4.`M`: the maximum number of boxes which may be stored. See the note 2.5.2.2`Stop`: the possible values are 0,1,2- 0: the algorithm will look for every solution in
`TheDomain` - 1: the algorithm will stop as soon as 1 solution has been found
- 2: the algorithm will stop as soon as
`Nb`solutions have been found

- 0: the algorithm will look for every solution in
`epsilon`: the maximal width of the box, see the note 2.3.4.6`epsilonf`: the maximal width of the equation intervals, see the note 2.3.4.6`Solution`: an interval matrix of size (`Nb`,`m`) which will contained the solution intervals.`IsKanto`: an integer vector of dimension`Nb`. A value of 1 for`IsKanto(i)`indicate that Newton method (see section 2.9) with as estimate the center of some solution interval`Solution(i)`has been used and has converged toward the unique solution`Solution(i)`which lie within this solution intervals. Note that the interval which contain the solution may be retrieved in the interval vector`Interval_Solution_UP`.`NbSolution`: the maximum number of solution we are looking for.

int Solve_UP_JH_Interval(int Degree,VECTOR Coeff, int Order,int M,int Stop, double epsilon,double epsilonf, INTERVAL_VECTOR & Solution, INTEGER_VECTOR & Is_Kanto,int NbSolution);There are two alternate forms of this procedure in the case where we are looking for the positive or negative roots of the polynomial.

int Solve_UP_JH_Positive_Interval int Solve_UP_JH_Negative_IntervalIn the three previous procedures there is no

The previous procedures are numerically safe in the sense that we take
into account rounding errors in the evaluation of the polynomial and
its gradient. For well conditioned polynomials you may use faster
procedures whose name has the prefix `Fast`. For example
`Fast_Solve_UP_JH_Interval` is the general procedure for
finding the roots of a polynomial.

Clearly this procedure is not intended to be used as substitute to more classical algorithms.

It makes use of a specific Krawczyk procedure for polynomials:

int Krawczyk_UP(int Degree,INTERVAL_VECTOR &Coeff, INTERVAL_VECTOR &CoeffG,INTERVAL &Input)