Implementation

This procedure enable to determine upper and lower bound on the roots of a polynomial if all the roots are real. In the implementation we check if all the roots of the polynomial are real using Huat theorem (see section 5.5.4) and if the answer is positive we determine the bounds. The syntax is:

 
int Laguerre_Second_Bound_Interval(int Degree,VECTOR &Coeff,INTERVAL &Bound);

with:

• Degree: degree of the polynomial
• Coeff: the Degree+1 coefficients of the polynomial in increasing degree
• Bound: upper and lower bound on the roots

On success the return code is 1 while the return code is 0 if the Degree is lower than 2. A similar algorithm exists for interval polynomial:

 
int Laguerre_Second_Bound_Interval(int Degree,
      INTERVAL_VECTOR &Coeff,INTERVAL &Lower,INTERVAL &Upper);

In that case Upper=[a,b] will be such that the maximal value of all roots of any polynomial within the set is lower than b, while for some polynomial they will be lower than a. On the other hand Lower=[a,b] will be such that the minimal value of all roots of any polynomial within the set is greater than a, while for some polynomial they will be greater than b.