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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.

** Next:** Newton method
** Up:** Laguerre second method
** Previous:** Mathematical background
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Jean-Pierre Merlet
2012-12-20