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This procedure is able to give upper and lower bound on the value of
the roots.
In the implementation we define as the smallest real which satisfy
. If we find such that
, then we
increase by a given positive value `sens` and start again. We
limit the number of iteration of the scheme by giving a maximal value
for the number of iteration:

int Newton_Bound_Interval(int Degree,VECTOR &Coeff1,double amp_sens,int MaxIter,double *bound);

with:
`Degree`: degree of the polynomial
`Coeff`: the `Degree+1` coefficients of the
polynomial in increasing degree
`sens`: a positive real indicating the increase of in the
scheme
`MaxIter`: the maximum number of iteration. If this number
is exceeded the procedure returns 0.
`bound`: the upper value of the real root

This procedure fail and returns 0 if `Degree`=0,
`Coeff(1)`=0, if `Coeff(Degree+1)`=0 and if the number of
iteration exceed `MaxIter`.
On success the return code is 1. Note that the bound given by Newton
(assuming that `Coeff(Degree+1)` is positive) cannot be lower
than -`Coeff(Degree)/(Degree Coeff(Degree+1))`.
The lower bound of the root may be determined by:
int Newton_Bound_Inverse_Interval(int Degree,VECTOR &Coeff1,double amp_sens,
int MaxIter,double *bound);

We have also a procedure which determine upper and lower bound for the
real roots:
int Newton_Bound_Interval(int Degree,VECTOR &Coeff1,double amp_sens,
int MaxIter,INTERVAL &Bound);

All the real roots lie within `Bound`. We may also use this
procedure for interval polynomial:
int Newton_Bound_Interval(int Degree,INTERVAL_VECTOR &Coeff,double sens,
int MaxIter,INTERVAL &Bound);

This procedure fail and returns 0 if `Degree`=0,
`Coeff(1)`, if `Coeff(Degree+1)`
and if the number of
iteration exceed `MaxIter`.
If `Bound`=[a,b], then the real roots of all the polynomial in the
set are lower than b and for some polynomial in the set the roots may
be lower than a. A similar procedure exists for upper and lower bound.
int Newton_Bound_Interval(int Degree,INTERVAL_VECTOR &Coeff,double sens,
int MaxIter,INTERVAL &Lower,INTERVAL &Upper);

In that case `Upper`=[a,b] will be such that
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
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 theorem
** Up:** Newton method
** Previous:** Mathematical background
** Contents**
Jean-Pierre Merlet
2012-12-20