Analyzing the real part of the roots

For some applications it may be interesting to determine if a polynomial in a set defined by a parametric polynomial has the real part of one of its roots equal to a pre-defined value. This may be done by using the procedure

 
int ALIAS_Is_Root_RealPart(int Degree,  
           int NbParameter,
           int HasInterval,
           INTERVAL_VECTOR (* TheCoeff)(INTERVAL_VECTOR &), 
           INTERVAL_VECTOR (* TheCoeffCentered)(INTERVAL_VECTOR&,double a), 
           int Iteration,
           INTERVAL_VECTOR &Par,
           double *Root,
           INTERVAL_VECTOR (* EvaluateComplex)(int,int,INTERVAL_VECTOR &),
           int (* Simp)(INTERVAL_VECTOR &))

where

• Degree: the degree of the polynomial
• NbParameter: the number of parameters that appear in the coefficient of the polynomial
• HasInterval: 1 if the coefficient include intervals not defined by parameters, 0 otherwise
• TheCoeff: a procedure to calculate the coefficients of the polynomial being given range for the parameters
• TheCoeffCentered: a procedure to calculate the coefficients of the polynomial $P(x+a)$
• Iteration: the maximum number of box that may be used by the algorithm
• Par: the ranges for the parameters
• Root: we look for polynomial whose real part of the root is either Root[0] or Root[1]
• EvaluateComplex: a procedure that returns 4 intervals. The input interval vector has dimension NbParameter+1 elements, the first one being the parameters, the last one being $b$. The procedure should evaluate $U=P(Root[0]+I b), V=P(Root[1]+I b)$ and should return the real part of $U$, the complex part of $U$, the real part of $V$ and the complex part of $V$
• Simp: an optional simplification procedure that returns -1 if the polynomials in a set cannot roots with real part equal to Root[0] or Root[1]