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# ifndef CPPAD_POLY_INCLUDED
# define CPPAD_POLY_INCLUDED

/* --------------------------------------------------------------------------
CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-06 Bradley M. Bell

CppAD is distributed under multiple licenses. This distribution is under
the terms of the 
                    Common Public License Version 1.0.

A copy of this license is included in the COPYING file of this distribution.
Please visit http://www.coin-or.org/CppAD/ for information on other licenses.
-------------------------------------------------------------------------- */
/*
$begin Poly$$
$spell
        cppad.hpp
        CppAD
        namespace
        cstddef
        ifndef
        endif
        deg
        const
        std
        da
$$

$index Poly$$
$index polynomial$$
$index derivative, polynomial template$$
$index template, polynomial derivative$$

$section Evaluate a Polynomial or its Derivative$$

$head Syntax$$
$code # include <cppad/poly.hpp>$$
$pre
$$
$syntax%%p% = Poly(%k%, %a%, %z%)%$$


$head Description$$
Computes the $th k$$ derivative of the polynomial 
$latex \[
        P(z) = a_0 + a_1 z^1 + \cdots + a_d z^d
\] $$
If $italic k$$ is equal to zero, the return value is $latex P(z)$$.

$head Include$$
The file $code cppad/poly.hpp$$ is included by $code cppad/cppad.hpp$$
but it can also be included separately with out the rest of 
the $code CppAD$$ routines.
Including this file defines
$code Poly$$ within the $code CppAD$$ namespace.

$head k$$
The argument $italic k$$ has prototype
$syntax%
        size_t %k%
%$$
It specifies the order of the derivative to calculate.

$head a$$
The argument $italic a$$ has prototype
$syntax%
        const %Vector% &%a%
%$$
(see $xref/Poly/Vector/Vector/$$ below).
It specifies the vector corresponding to the polynomial $latex P(z)$$.

$head z$$
The argument $italic z$$ has prototype
$syntax%
        const %Type% &%z%
%$$
(see $italic Type$$ below).
It specifies the point at which to evaluate the polynomial

$head p$$
The result $italic p$$  has prototype
$syntax%
        %Type% %p%
%$$
(see $xref/Poly/Type/Type/$$ below)
and it is equal to the $th k$$ derivative of $latex P(z)$$; i.e., 
$latex \[
p = \frac{k !}{0 !} a_k 
  + \frac{(k+1) !}{1 !} a_{k+1} z^1 
  + \ldots
  + \frac{d !}{(d - k) !} a_d z^{d - k}
\]
$$
If $latex k > d$$, $syntax%%p% = %Type%(0)%$$.

$head Type$$
The type $italic Type$$ is determined by the argument $italic z$$.
It is assumed that
multiplication and addition of $italic Type$$ objects
are commutative.

$subhead Casting$$
The syntax $syntax%%Type%(%i%)%$$ 
must convert from a $code size_t$$ value $italic i$$
to a $italic Type$$ object. 

$subhead Operations$$
The following operations must be supported by $italic Type$$:
$table
$code  =$$    $cnext assignment     $rnext
$code  *=$$   $cnext multiplication computed assignment $rnext
$code  +=$$   $cnext addition computed assignment 

$tend


$head Vector$$
The type $italic Vector$$ must be a $xref/SimpleVector/$$ class with
$xref/SimpleVector/Elements of Specified Type/elements of type/$$
$italic Type$$.
The routine $xref/CheckSimpleVector/$$ will generate an error message
if this is not the case.

$head Operation Sequence$$
The $italic Type$$ operation sequence used to calculate $italic p$$ is 
$xref/glossary/Operation/Independent/independent/1/$$
of $italic z$$ and the elements of $italic a$$
(it does depend on the size of the vector $italic a$$).


$children%
        example/poly.cpp%
        omh/poly_hpp.omh
%$$

$head Example$$
The file
$xref/Poly.cpp/$$
contains an example and test of this routine.
It returns true if it succeeds and false otherwise.

$head Source$$
The file $cref/poly.hpp/$$ contains the 
current source code that implements these specifications.

$end
------------------------------------------------------------------------------
*/
// BEGIN PROGRAM
# include <cstddef>  // used to defined size_t
# include <cppad/check_simple_vector.hpp>

namespace CppAD {    // BEGIN CppAD namespace

template <class Type, class Vector>
Type Poly(size_t k, const Vector &a, const Type &z)
{       size_t i;
        size_t d = a.size() - 1;

        // check Vector is Simple Vector class with Type elements
        CheckSimpleVector<Type, Vector>();

        // case where derivative order greater than degree of polynomial
        if( k > d )
                return Type(0);
        // case where we are evaluating a derivative
        if( k > 0 )
        {       // initialize factor as (k-1) !
                size_t factor = 1;
                for(i = 2; i < k; i++)
                        factor *= i;

                // set b to coefficient vector corresponding to derivative
                Vector b(d - k + 1);
                for(i = k; i <= d; i++)
                {       factor   *= i;
                        b[i - k]  = a[i] * Type(factor); 
                        factor   /= (i - k + 1);
                }
                // value of derivative polynomial
                return Poly(0, b, z);
        }
        // case where we are evaluating the original polynomial
        Type sum = a[d];
        i        = d;
        while(i > 0)
        {       sum *= z;
                sum += a[--i];
        }
        return sum;
}
} // END CppAD namespace
// END PROGRAM
# endif

---