# Boost C++ Libraries

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### boost/math/tools/polynomial.hpp

```//  (C) Copyright John Maddock 2006.
//  Use, modification and distribution are subject to the
//  Boost Software License, Version 1.0. (See accompanying file

#ifndef BOOST_MATH_TOOLS_POLYNOMIAL_HPP
#define BOOST_MATH_TOOLS_POLYNOMIAL_HPP

#ifdef _MSC_VER
#pragma once
#endif

#include <boost/assert.hpp>
#include <boost/math/tools/rational.hpp>
#include <boost/math/tools/real_cast.hpp>
#include <boost/math/special_functions/binomial.hpp>

#include <vector>
#include <ostream>
#include <algorithm>

namespace boost{ namespace math{ namespace tools{

template <class T>
T chebyshev_coefficient(unsigned n, unsigned m)
{
BOOST_MATH_STD_USING
if(m > n)
return 0;
if((n & 1) != (m & 1))
return 0;
if(n == 0)
return 1;
T result = T(n) / 2;
unsigned r = n - m;
r /= 2;

BOOST_ASSERT(n - 2 * r == m);

if(r & 1)
result = -result;
result /= n - r;
result *= boost::math::binomial_coefficient<T>(n - r, r);
result *= ldexp(1.0f, m);
return result;
}

template <class Seq>
Seq polynomial_to_chebyshev(const Seq& s)
{
// Converts a Polynomial into Chebyshev form:
typedef typename Seq::value_type value_type;
typedef typename Seq::difference_type difference_type;
Seq result(s);
difference_type order = s.size() - 1;
difference_type even_order = order & 1 ? order - 1 : order;
difference_type odd_order = order & 1 ? order : order - 1;

for(difference_type i = even_order; i >= 0; i -= 2)
{
value_type val = s[i];
for(difference_type k = even_order; k > i; k -= 2)
{
val -= result[k] * chebyshev_coefficient<value_type>(static_cast<unsigned>(k), static_cast<unsigned>(i));
}
val /= chebyshev_coefficient<value_type>(static_cast<unsigned>(i), static_cast<unsigned>(i));
result[i] = val;
}
result[0] *= 2;

for(difference_type i = odd_order; i >= 0; i -= 2)
{
value_type val = s[i];
for(difference_type k = odd_order; k > i; k -= 2)
{
val -= result[k] * chebyshev_coefficient<value_type>(static_cast<unsigned>(k), static_cast<unsigned>(i));
}
val /= chebyshev_coefficient<value_type>(static_cast<unsigned>(i), static_cast<unsigned>(i));
result[i] = val;
}
return result;
}

template <class Seq, class T>
T evaluate_chebyshev(const Seq& a, const T& x)
{
// Clenshaw's formula:
typedef typename Seq::difference_type difference_type;
T yk2 = 0;
T yk1 = 0;
T yk = 0;
for(difference_type i = a.size() - 1; i >= 1; --i)
{
yk2 = yk1;
yk1 = yk;
yk = 2 * x * yk1 - yk2 + a[i];
}
return a[0] / 2 + yk * x - yk1;
}

template <class T>
class polynomial
{
public:
// typedefs:
typedef typename std::vector<T>::value_type value_type;
typedef typename std::vector<T>::size_type size_type;

// construct:
polynomial(){}
template <class U>
polynomial(const U* data, unsigned order)
: m_data(data, data + order + 1)
{
}
template <class U>
polynomial(const U& point)
{
m_data.push_back(point);
}

// copy:
polynomial(const polynomial& p)
: m_data(p.m_data) { }

template <class U>
polynomial(const polynomial<U>& p)
{
for(unsigned i = 0; i < p.size(); ++i)
{
m_data.push_back(boost::math::tools::real_cast<T>(p[i]));
}
}

// access:
size_type size()const { return m_data.size(); }
size_type degree()const { return m_data.size() - 1; }
value_type& operator[](size_type i)
{
return m_data[i];
}
const value_type& operator[](size_type i)const
{
return m_data[i];
}
T evaluate(T z)const
{
return boost::math::tools::evaluate_polynomial(&m_data[0], z, m_data.size());;
}
std::vector<T> chebyshev()const
{
return polynomial_to_chebyshev(m_data);
}

// operators:
template <class U>
polynomial& operator +=(const U& value)
{
if(m_data.size() == 0)
m_data.push_back(value);
else
{
m_data[0] += value;
}
return *this;
}
template <class U>
polynomial& operator -=(const U& value)
{
if(m_data.size() == 0)
m_data.push_back(-value);
else
{
m_data[0] -= value;
}
return *this;
}
template <class U>
polynomial& operator *=(const U& value)
{
for(size_type i = 0; i < m_data.size(); ++i)
m_data[i] *= value;
return *this;
}
template <class U>
polynomial& operator +=(const polynomial<U>& value)
{
size_type s1 = (std::min)(m_data.size(), value.size());
for(size_type i = 0; i < s1; ++i)
m_data[i] += value[i];
for(size_type i = s1; i < value.size(); ++i)
m_data.push_back(value[i]);
return *this;
}
template <class U>
polynomial& operator -=(const polynomial<U>& value)
{
size_type s1 = (std::min)(m_data.size(), value.size());
for(size_type i = 0; i < s1; ++i)
m_data[i] -= value[i];
for(size_type i = s1; i < value.size(); ++i)
m_data.push_back(-value[i]);
return *this;
}
template <class U>
polynomial& operator *=(const polynomial<U>& value)
{
// TODO: FIXME: use O(N log(N)) algorithm!!!
BOOST_ASSERT(value.size());
polynomial base(*this);
*this *= value[0];
for(size_type i = 1; i < value.size(); ++i)
{
polynomial t(base);
t *= value[i];
size_type s = size() - i;
for(size_type j = 0; j < s; ++j)
{
m_data[i+j] += t[j];
}
for(size_type j = s; j < t.size(); ++j)
m_data.push_back(t[j]);
}
return *this;
}

private:
std::vector<T> m_data;
};

template <class T>
inline polynomial<T> operator + (const polynomial<T>& a, const polynomial<T>& b)
{
polynomial<T> result(a);
result += b;
return result;
}

template <class T>
inline polynomial<T> operator - (const polynomial<T>& a, const polynomial<T>& b)
{
polynomial<T> result(a);
result -= b;
return result;
}

template <class T>
inline polynomial<T> operator * (const polynomial<T>& a, const polynomial<T>& b)
{
polynomial<T> result(a);
result *= b;
return result;
}

template <class T, class U>
inline polynomial<T> operator + (const polynomial<T>& a, const U& b)
{
polynomial<T> result(a);
result += b;
return result;
}

template <class T, class U>
inline polynomial<T> operator - (const polynomial<T>& a, const U& b)
{
polynomial<T> result(a);
result -= b;
return result;
}

template <class T, class U>
inline polynomial<T> operator * (const polynomial<T>& a, const U& b)
{
polynomial<T> result(a);
result *= b;
return result;
}

template <class U, class T>
inline polynomial<T> operator + (const U& a, const polynomial<T>& b)
{
polynomial<T> result(b);
result += a;
return result;
}

template <class U, class T>
inline polynomial<T> operator - (const U& a, const polynomial<T>& b)
{
polynomial<T> result(a);
result -= b;
return result;
}

template <class U, class T>
inline polynomial<T> operator * (const U& a, const polynomial<T>& b)
{
polynomial<T> result(b);
result *= a;
return result;
}

template <class charT, class traits, class T>
inline std::basic_ostream<charT, traits>& operator << (std::basic_ostream<charT, traits>& os, const polynomial<T>& poly)
{
os << "{ ";
for(unsigned i = 0; i < poly.size(); ++i)
{
if(i) os << ", ";
os << poly[i];
}
os << " }";
return os;
}

} // namespace tools
} // namespace math
} // namespace boost

#endif // BOOST_MATH_TOOLS_POLYNOMIAL_HPP

```