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###

Safe evaluation of a polynomial

Due to the numerical error in the coefficients and in the
evaluation of a polynomial at may be incorrect. The following
procedures enable to compute an interval which is guaranteed to
include the true value:
INTERVAL Evaluate_Polynomial_Safe_Interval(int Degree,VECTOR &Coeff,REAL P);
INTERVAL Evaluate_Polynomial_Safe_Interval(int Degree,VECTOR &Coeff,INTERVAL P);
INTERVAL Evaluate_Polynomial_Safe_Interval(int Degree,INTERVAL_VECTOR &Coeff,REAL P);
INTERVAL Evaluate_Polynomial_Safe_Interval(int Degree,INTERVAL_VECTOR &Coeff,INTERVAL P);

For example for the Wilkinson polynomial at order 15 the
evaluation for 15.1 leads to the interval:

If "safe" intervals have been pre-computed
(for example by using the procedure described in
section 5.9.10) for the coefficients you may
use:
INTERVAL Evaluate_Polynomial_Fast_Safe_Interval(int Degree,INTERVAL_VECTOR &Coeff,REAL P);
INTERVAL Evaluate_Polynomial_Fast_Safe_Interval(int Degree,INTERVAL_VECTOR &Coeff,INTERVAL P);

which are faster.
If you want to use a centered form you may use:
INTERVAL Evaluate_Polynomial_Centered_Safe_Interval(int Degree,VECTOR &Coeff,REAL P);
INTERVAL Evaluate_Polynomial_Centered_Safe_Interval(int Degree,VECTOR &Coeff,INTERVAL P);
INTERVAL Evaluate_Polynomial_Centered_Fast_Safe_Interval(int Degree,
INTERVAL_VECTOR &Coeff,REAL P);
INTERVAL Evaluate_Polynomial_Centered_Fast_Safe_Interval(int Degree,
INTERVAL_VECTOR &Coeff,INTERVAL P);

in which the two last forms assume that you have pre-computed a safe
value of the coefficients of the polynomial.
The evaluation of a polynomial with real coefficients at a complex
point may be performed with:

INTERVAL_VECTOR Evaluate_Complex_Poly(int deg,
INTERVAL_VECTOR &Coeff, INTERVAL &XR,INTERVAL &XI)

where `XR` is the real part of the point and `XI` its
imaginary part. This procedure returns an interval vector whose first
element is the real part of the evaluation and the second element its
imaginary part.

** Next:** Sign of a polynomial
** Up:** Evaluation of a polynomial
** Previous:** Evaluation in centered form
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Jean-Pierre Merlet
2012-12-20