** Next:** Safe evaluation of a
** Up:** Evaluation of a polynomial
** Previous:** Evaluation of a polynomial
** Contents**

###

Evaluation in centered form

The previous procedures may yield sometime to a bad evaluation of the
polynomial. For example for the Wilkinson polynomial at order 15 the
evaluation for 15.1 leads to 11977410100.791748046875, the correct
value being 11977396665.006561033796551. The evaluation obtained when
considering that the coefficients and the unknown are intervals is
evidently correct and leads to the interval:
[11977379510.705810546875,11977410100.791748046875]

A
better evaluation may be obtained if we use a centered form of the
polynomial. Consider the polynomials:

where is a real. For an appropriate choice of the we may
have for all .
The procedures:
VECTOR Coeff_Polynomial_Centered_Interval(int Degree,VECTOR &Coeff,REAL P);
INTERVAL_VECTOR Coeff_Polynomial_Centered_Interval(int Degree,
INTERVAL_VECTOR &Coeff,REAL P);

enable to compute the for the centered form at `P` either
for polynomial or interval polynomial.
To take into account numerical errors you may use:
INTERVAL_VECTOR Coeff_Polynomial_Centered_Safe_Interval(int Degree,VECTOR &Coeff,REAL P);
INTERVAL_VECTOR Coeff_Polynomial_Centered_Fast_Safe_Interval(int Degree,
INTERVAL_VECTOR &Coeff,REAL P);

which return safe value for the coefficients (in the second form we
assume that you have pre-computed safe value for the coefficients of
the polynomial using the procedure described in
section 5.9.10).
Then we may use the procedures:

REAL Evaluate_Polynomial_Centered_Interval(int Degree,VECTOR &Coeff,REAL Center,REAL P);
INTERVAL Evaluate_Polynomial_Centered_Interval(int Degree,VECTOR &Coeff,INTERVAL P);
INTERVAL Evaluate_Polynomial_Centered_Interval(int Degree,INTERVAL_VECTOR &Coeff,
REAL Center,REAL P);
INTERVAL Evaluate_Polynomial_Centered_Interval(int Degree,INTERVAL_VECTOR &Coeff,INTERVAL P);

These procedures return the evaluation of the polynomial at `P` using the
centered form at `Center` or at the middle point of `P` if is
an interval. For example for the Wilkinson polynomial at order 15 the
evaluation for 15.1 using the centered form at 15 leads to
11977396665.00650787353516 which is largely better than the previous
evaluation.

** Next:** Safe evaluation of a
** Up:** Evaluation of a polynomial
** Previous:** Evaluation of a polynomial
** Contents**
Jean-Pierre Merlet
2012-12-20