[section Polynomial and Rational Function Evaluation]

[caution __caution ]

[h4 synopsis]

``
#include <boost/math/tools/rational.hpp>
``

   template <class T, class U>
   U evaluate_polynomial(const T* poly, U z, std::size_t count);
   
   template <class T, class U>
   U evaluate_even_polynomial(const T* poly, U z, std::size_t count);
   
   template <class T, class U>
   U evaluate_odd_polynomial(const T* poly, U z, std::size_t count);
   
   template <class T, class U, class V>
   V evaluate_rational(const T* num, const U* denom, V z, unsigned count);

[h4 Description]

   template <class T, class U>
   U evaluate_polynomial(const T* poly, U z, std::size_t count);

Evaluates the polynomial described by
the coefficients stored in /poly/.

The polynomial most have order /count-1/ with /count/ coefficients.

Coefficients should be stored such that the coefficients for the x^i terms
are in poly[i].

The types of the coefficients and of variable
/z/ may differ as long as /*poly/ is convertible to type /U/.
This allows, for example, for the coefficient table
to be a table of integers if this is appropriate.

   template <class T, class U>
   U evaluate_even_polynomial(const T* poly, U z, std::size_t count);

As above, but evaluates an even polynomial: one where all the powers
of /z/ are even numbers.  Equivalent to calling 
`evaluate_polynomial(poly, z*z, count)`.

   template <class T, class U>
   U evaluate_odd_polynomial(const T* poly, U z, std::size_t count);

As above but evaluates a polynomial where all the powers are odd numbers.
Equivalent to `evaluate_polynomial(poly+1, z*z, count-1) * z + poly[0]`.

   template <class T, class U, class V>
   V evaluate_rational(const T* num, const U* denom, V z, unsigned count);
   
Evaluates the rational function (the ratio of two polynomials) described by
the coefficients stored in /num/ and /demom/.

Both polynomials most have order /count-1/ with /count/ coefficients.

Array /num/ describes the numerator, and /demon/ the denominator.

Coefficients should be stored such that the coefficients for the x^i terms
are in num[i] and denom[i].

The types of the coefficients and of variable
/v/ may differ as long as /*num/ and /*denom/ are convertible to type /V/.
This allows, for example, for one or both of the coefficient tables
to be a table of integers if this is appropriate.

[h4 Implementation]

Evaluation is by Horners method: with the two polynomials being evaluated
in parallel to make the most of the processors floating point pipeline.
If /v/ is greater than one, then the polynomials are evaluated in reverse
order as polynomials in ['1\/v]: this avoids unnecessary numerical overflow when the
coefficients are large.

[endsect]