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This procedure is used to determine the number of real roots in a
given interval, up to an even number.
The syntax of the procedure is:
INT Budan_Fourier_Interval(int Degree,VECTOR &Coeff,INTERVAL In)
INT Budan_Fourier_Interval(int Degree,INTEGER_VECTOR &Coeff,INTERVAL In)

with:
`Degree`: degree of the polynomial
`Coeff`: the `Degree+1` coefficients of the
polynomial in increasing degree, which may be `REAL` or `INT`
`In`: the interval in which we are looking for the number of
roots

If this procedure returns the integer , then the number of real
roots in `In` is with .
A negative returns code indicate a failure of the algorithm:
- -1:
- -2: , the polynomial may be factored
- -3:

This procedure may be used with polynomial whose coefficients are
intervals. The syntax is:
INT Budan_Fourier_Interval(int Degree,INTERVAL_VECTOR &Coeff,INTERVAL In,int *Confidence)

where `Confidence` is a quality index for the result:
- 1: the result is exact, so if is the return code of the
algorithm the number of reals roots of the interval polynomial is
with
- : if is the return code of the
algorithm the number of reals roots of the interval polynomial is

Due to rounding errors incorrect results may be returned by the
previous procedures. A safer procedure is:
INT Budan_Fourier_Safe_Interval(int Degree,VECTOR &Coeff,
INTERVAL In,INTERVAL &NbRoot);

The procedure returns 1 in case of success and an interval for the
number of roots. If `NbRoot`=[a,b], then if a =b the number of
roots is either a,a-2, and if a b the number of roots
is lower than b.
If "safe" value of the coefficients have been pre-computed you may
use:
INT Budan_Fourier_Fast_Safe_Interval(int Degree,INTERVAL_VECTOR &Coeff,
INTERVAL In,INTERVAL &NbRoot);

Another safe procedure is:
INT Budan_Fourier_Interval(int Degree,INTEGER_VECTOR &Coeff,int Inf,int Sup)

where the coefficients and the bounds are integers.

** Next:** Example
** Up:** Budan-Fourier method
** Previous:** Mathematical background
** Contents**
Jean-Pierre Merlet
2012-12-20