720 lines
18 KiB
C++
720 lines
18 KiB
C++
|
// (C) Copyright John Maddock 2006.
|
||
|
|
// (C) Copyright Jeremy William Murphy 2015.
|
||
|
|
|
||
|
|
|
||
|
|
// Use, modification and distribution are subject to the
|
||
|
|
// Boost Software License, Version 1.0. (See accompanying file
|
||
|
|
// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
|
||
|
|
|
||
|
|
#ifndef BOOST_MATH_TOOLS_POLYNOMIAL_HPP
|
||
|
|
#define BOOST_MATH_TOOLS_POLYNOMIAL_HPP
|
||
|
|
|
||
|
|
#ifdef _MSC_VER
|
||
|
|
#pragma once
|
||
|
|
#endif
|
||
|
|
|
||
|
|
#include <boost/assert.hpp>
|
||
|
|
#include <boost/config.hpp>
|
||
|
|
#include <boost/config/suffix.hpp>
|
||
|
|
#include <boost/function.hpp>
|
||
|
|
#include <boost/lambda/lambda.hpp>
|
||
|
|
#include <boost/math/tools/rational.hpp>
|
||
|
|
#include <boost/math/tools/real_cast.hpp>
|
||
|
|
#include <boost/math/policies/error_handling.hpp>
|
||
|
|
#include <boost/math/special_functions/binomial.hpp>
|
||
|
|
#include <boost/operators.hpp>
|
||
|
|
|
||
|
|
#include <vector>
|
||
|
|
#include <ostream>
|
||
|
|
#include <algorithm>
|
||
|
|
#ifndef BOOST_NO_CXX11_HDR_INITIALIZER_LIST
|
||
|
|
#include <initializer_list>
|
||
|
|
#endif
|
||
|
|
|
||
|
|
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 <typename T>
|
||
|
|
class polynomial;
|
||
|
|
|
||
|
|
namespace detail {
|
||
|
|
|
||
|
|
/**
|
||
|
|
* Knuth, The Art of Computer Programming: Volume 2, Third edition, 1998
|
||
|
|
* Chapter 4.6.1, Algorithm D: Division of polynomials over a field.
|
||
|
|
*
|
||
|
|
* @tparam T Coefficient type, must be not be an integer.
|
||
|
|
*
|
||
|
|
* Template-parameter T actually must be a field but we don't currently have that
|
||
|
|
* subtlety of distinction.
|
||
|
|
*/
|
||
|
|
template <typename T, typename N>
|
||
|
|
BOOST_DEDUCED_TYPENAME disable_if_c<std::numeric_limits<T>::is_integer, void >::type
|
||
|
|
division_impl(polynomial<T> &q, polynomial<T> &u, const polynomial<T>& v, N n, N k)
|
||
|
|
{
|
||
|
|
q[k] = u[n + k] / v[n];
|
||
|
|
for (N j = n + k; j > k;)
|
||
|
|
{
|
||
|
|
j--;
|
||
|
|
u[j] -= q[k] * v[j - k];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T, class N>
|
||
|
|
T integer_power(T t, N n)
|
||
|
|
{
|
||
|
|
switch(n)
|
||
|
|
{
|
||
|
|
case 0:
|
||
|
|
return static_cast<T>(1u);
|
||
|
|
case 1:
|
||
|
|
return t;
|
||
|
|
case 2:
|
||
|
|
return t * t;
|
||
|
|
case 3:
|
||
|
|
return t * t * t;
|
||
|
|
}
|
||
|
|
T result = integer_power(t, n / 2);
|
||
|
|
result *= result;
|
||
|
|
if(n & 1)
|
||
|
|
result *= t;
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
/**
|
||
|
|
* Knuth, The Art of Computer Programming: Volume 2, Third edition, 1998
|
||
|
|
* Chapter 4.6.1, Algorithm R: Pseudo-division of polynomials.
|
||
|
|
*
|
||
|
|
* @tparam T Coefficient type, must be an integer.
|
||
|
|
*
|
||
|
|
* Template-parameter T actually must be a unique factorization domain but we
|
||
|
|
* don't currently have that subtlety of distinction.
|
||
|
|
*/
|
||
|
|
template <typename T, typename N>
|
||
|
|
BOOST_DEDUCED_TYPENAME enable_if_c<std::numeric_limits<T>::is_integer, void >::type
|
||
|
|
division_impl(polynomial<T> &q, polynomial<T> &u, const polynomial<T>& v, N n, N k)
|
||
|
|
{
|
||
|
|
q[k] = u[n + k] * integer_power(v[n], k);
|
||
|
|
for (N j = n + k; j > 0;)
|
||
|
|
{
|
||
|
|
j--;
|
||
|
|
u[j] = v[n] * u[j] - (j < k ? T(0) : u[n + k] * v[j - k]);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
/**
|
||
|
|
* Knuth, The Art of Computer Programming: Volume 2, Third edition, 1998
|
||
|
|
* Chapter 4.6.1, Algorithm D and R: Main loop.
|
||
|
|
*
|
||
|
|
* @param u Dividend.
|
||
|
|
* @param v Divisor.
|
||
|
|
*/
|
||
|
|
template <typename T>
|
||
|
|
std::pair< polynomial<T>, polynomial<T> >
|
||
|
|
division(polynomial<T> u, const polynomial<T>& v)
|
||
|
|
{
|
||
|
|
BOOST_ASSERT(v.size() <= u.size());
|
||
|
|
BOOST_ASSERT(v);
|
||
|
|
BOOST_ASSERT(u);
|
||
|
|
|
||
|
|
typedef typename polynomial<T>::size_type N;
|
||
|
|
|
||
|
|
N const m = u.size() - 1, n = v.size() - 1;
|
||
|
|
N k = m - n;
|
||
|
|
polynomial<T> q;
|
||
|
|
q.data().resize(m - n + 1);
|
||
|
|
|
||
|
|
do
|
||
|
|
{
|
||
|
|
division_impl(q, u, v, n, k);
|
||
|
|
}
|
||
|
|
while (k-- != 0);
|
||
|
|
u.data().resize(n);
|
||
|
|
u.normalize(); // Occasionally, the remainder is zeroes.
|
||
|
|
return std::make_pair(q, u);
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
struct identity
|
||
|
|
{
|
||
|
|
T operator()(T const &x) const
|
||
|
|
{
|
||
|
|
return x;
|
||
|
|
}
|
||
|
|
};
|
||
|
|
|
||
|
|
} // namespace detail
|
||
|
|
|
||
|
|
/**
|
||
|
|
* Returns the zero element for multiplication of polynomials.
|
||
|
|
*/
|
||
|
|
template <class T>
|
||
|
|
polynomial<T> zero_element(std::multiplies< polynomial<T> >)
|
||
|
|
{
|
||
|
|
return polynomial<T>();
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
polynomial<T> identity_element(std::multiplies< polynomial<T> >)
|
||
|
|
{
|
||
|
|
return polynomial<T>(T(1));
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Calculates a / b and a % b, returning the pair (quotient, remainder) together
|
||
|
|
* because the same amount of computation yields both.
|
||
|
|
* This function is not defined for division by zero: user beware.
|
||
|
|
*/
|
||
|
|
template <typename T>
|
||
|
|
std::pair< polynomial<T>, polynomial<T> >
|
||
|
|
quotient_remainder(const polynomial<T>& dividend, const polynomial<T>& divisor)
|
||
|
|
{
|
||
|
|
BOOST_ASSERT(divisor);
|
||
|
|
if (dividend.size() < divisor.size())
|
||
|
|
return std::make_pair(polynomial<T>(), dividend);
|
||
|
|
return detail::division(dividend, divisor);
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
class polynomial :
|
||
|
|
equality_comparable< polynomial<T>,
|
||
|
|
dividable< polynomial<T>,
|
||
|
|
dividable2< polynomial<T>, T,
|
||
|
|
modable< polynomial<T>,
|
||
|
|
modable2< polynomial<T>, T > > > > >
|
||
|
|
{
|
||
|
|
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)
|
||
|
|
{
|
||
|
|
normalize();
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class I>
|
||
|
|
polynomial(I first, I last)
|
||
|
|
: m_data(first, last)
|
||
|
|
{
|
||
|
|
normalize();
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class U>
|
||
|
|
explicit polynomial(const U& point)
|
||
|
|
{
|
||
|
|
if (point != U(0))
|
||
|
|
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]));
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
#if !defined(BOOST_NO_CXX11_HDR_INITIALIZER_LIST) && !BOOST_WORKAROUND(BOOST_GCC_VERSION, < 40500)
|
||
|
|
polynomial(std::initializer_list<T> l) : polynomial(std::begin(l), std::end(l))
|
||
|
|
{
|
||
|
|
}
|
||
|
|
|
||
|
|
polynomial&
|
||
|
|
operator=(std::initializer_list<T> l)
|
||
|
|
{
|
||
|
|
m_data.assign(std::begin(l), std::end(l));
|
||
|
|
normalize();
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
#endif
|
||
|
|
|
||
|
|
|
||
|
|
// access:
|
||
|
|
size_type size()const { return m_data.size(); }
|
||
|
|
size_type degree()const
|
||
|
|
{
|
||
|
|
if (size() == 0)
|
||
|
|
throw std::logic_error("degree() is undefined for the zero polynomial.");
|
||
|
|
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 m_data.size() > 0 ? boost::math::tools::evaluate_polynomial(&m_data[0], z, m_data.size()) : 0;
|
||
|
|
}
|
||
|
|
std::vector<T> chebyshev()const
|
||
|
|
{
|
||
|
|
return polynomial_to_chebyshev(m_data);
|
||
|
|
}
|
||
|
|
|
||
|
|
std::vector<T> const& data() const
|
||
|
|
{
|
||
|
|
return m_data;
|
||
|
|
}
|
||
|
|
|
||
|
|
std::vector<T> & data()
|
||
|
|
{
|
||
|
|
return m_data;
|
||
|
|
}
|
||
|
|
|
||
|
|
// operators:
|
||
|
|
template <class U>
|
||
|
|
polynomial& operator +=(const U& value)
|
||
|
|
{
|
||
|
|
addition(value);
|
||
|
|
normalize();
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class U>
|
||
|
|
polynomial& operator -=(const U& value)
|
||
|
|
{
|
||
|
|
subtraction(value);
|
||
|
|
normalize();
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class U>
|
||
|
|
polynomial& operator *=(const U& value)
|
||
|
|
{
|
||
|
|
multiplication(value);
|
||
|
|
normalize();
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class U>
|
||
|
|
polynomial& operator /=(const U& value)
|
||
|
|
{
|
||
|
|
division(value);
|
||
|
|
normalize();
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class U>
|
||
|
|
polynomial& operator %=(const U& /*value*/)
|
||
|
|
{
|
||
|
|
// We can always divide by a scalar, so there is no remainder:
|
||
|
|
this->set_zero();
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class U>
|
||
|
|
polynomial& operator +=(const polynomial<U>& value)
|
||
|
|
{
|
||
|
|
addition(value);
|
||
|
|
normalize();
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class U>
|
||
|
|
polynomial& operator -=(const polynomial<U>& value)
|
||
|
|
{
|
||
|
|
subtraction(value);
|
||
|
|
normalize();
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
template <class U>
|
||
|
|
polynomial& operator *=(const polynomial<U>& value)
|
||
|
|
{
|
||
|
|
// TODO: FIXME: use O(N log(N)) algorithm!!!
|
||
|
|
if (!value)
|
||
|
|
{
|
||
|
|
this->set_zero();
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
std::vector<T> prod(size() + value.size() - 1, T(0));
|
||
|
|
for (size_type i = 0; i < value.size(); ++i)
|
||
|
|
for (size_type j = 0; j < size(); ++j)
|
||
|
|
prod[i+j] += m_data[j] * value[i];
|
||
|
|
m_data.swap(prod);
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <typename U>
|
||
|
|
polynomial& operator /=(const polynomial<U>& value)
|
||
|
|
{
|
||
|
|
*this = quotient_remainder(*this, value).first;
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <typename U>
|
||
|
|
polynomial& operator %=(const polynomial<U>& value)
|
||
|
|
{
|
||
|
|
*this = quotient_remainder(*this, value).second;
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <typename U>
|
||
|
|
polynomial& operator >>=(U const &n)
|
||
|
|
{
|
||
|
|
BOOST_ASSERT(n <= m_data.size());
|
||
|
|
m_data.erase(m_data.begin(), m_data.begin() + n);
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <typename U>
|
||
|
|
polynomial& operator <<=(U const &n)
|
||
|
|
{
|
||
|
|
m_data.insert(m_data.begin(), n, static_cast<T>(0));
|
||
|
|
normalize();
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
// Convenient and efficient query for zero.
|
||
|
|
bool is_zero() const
|
||
|
|
{
|
||
|
|
return m_data.empty();
|
||
|
|
}
|
||
|
|
|
||
|
|
// Conversion to bool.
|
||
|
|
#ifdef BOOST_NO_CXX11_EXPLICIT_CONVERSION_OPERATORS
|
||
|
|
typedef bool (polynomial::*unmentionable_type)() const;
|
||
|
|
|
||
|
|
BOOST_FORCEINLINE operator unmentionable_type() const
|
||
|
|
{
|
||
|
|
return is_zero() ? false : &polynomial::is_zero;
|
||
|
|
}
|
||
|
|
#else
|
||
|
|
BOOST_FORCEINLINE explicit operator bool() const
|
||
|
|
{
|
||
|
|
return !m_data.empty();
|
||
|
|
}
|
||
|
|
#endif
|
||
|
|
|
||
|
|
// Fast way to set a polynomial to zero.
|
||
|
|
void set_zero()
|
||
|
|
{
|
||
|
|
m_data.clear();
|
||
|
|
}
|
||
|
|
|
||
|
|
/** Remove zero coefficients 'from the top', that is for which there are no
|
||
|
|
* non-zero coefficients of higher degree. */
|
||
|
|
void normalize()
|
||
|
|
{
|
||
|
|
using namespace boost::lambda;
|
||
|
|
m_data.erase(std::find_if(m_data.rbegin(), m_data.rend(), _1 != T(0)).base(), m_data.end());
|
||
|
|
}
|
||
|
|
|
||
|
|
private:
|
||
|
|
template <class U, class R1, class R2>
|
||
|
|
polynomial& addition(const U& value, R1 sign, R2 op)
|
||
|
|
{
|
||
|
|
if(m_data.size() == 0)
|
||
|
|
m_data.push_back(sign(value));
|
||
|
|
else
|
||
|
|
m_data[0] = op(m_data[0], value);
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class U>
|
||
|
|
polynomial& addition(const U& value)
|
||
|
|
{
|
||
|
|
return addition(value, detail::identity<U>(), std::plus<U>());
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class U>
|
||
|
|
polynomial& subtraction(const U& value)
|
||
|
|
{
|
||
|
|
return addition(value, std::negate<U>(), std::minus<U>());
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class U, class R1, class R2>
|
||
|
|
polynomial& addition(const polynomial<U>& value, R1 sign, R2 op)
|
||
|
|
{
|
||
|
|
size_type s1 = (std::min)(m_data.size(), value.size());
|
||
|
|
for(size_type i = 0; i < s1; ++i)
|
||
|
|
m_data[i] = op(m_data[i], value[i]);
|
||
|
|
for(size_type i = s1; i < value.size(); ++i)
|
||
|
|
m_data.push_back(sign(value[i]));
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class U>
|
||
|
|
polynomial& addition(const polynomial<U>& value)
|
||
|
|
{
|
||
|
|
return addition(value, detail::identity<U>(), std::plus<U>());
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class U>
|
||
|
|
polynomial& subtraction(const polynomial<U>& value)
|
||
|
|
{
|
||
|
|
return addition(value, std::negate<U>(), std::minus<U>());
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class U>
|
||
|
|
polynomial& multiplication(const U& value)
|
||
|
|
{
|
||
|
|
using namespace boost::lambda;
|
||
|
|
std::transform(m_data.begin(), m_data.end(), m_data.begin(), ret<T>(_1 * value));
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class U>
|
||
|
|
polynomial& division(const U& value)
|
||
|
|
{
|
||
|
|
using namespace boost::lambda;
|
||
|
|
std::transform(m_data.begin(), m_data.end(), m_data.begin(), ret<T>(_1 / value));
|
||
|
|
return *this;
|
||
|
|
}
|
||
|
|
|
||
|
|
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 T>
|
||
|
|
bool operator == (const polynomial<T> &a, const polynomial<T> &b)
|
||
|
|
{
|
||
|
|
return a.data() == b.data();
|
||
|
|
}
|
||
|
|
|
||
|
|
template <typename T, typename U>
|
||
|
|
polynomial<T> operator >> (const polynomial<T>& a, const U& b)
|
||
|
|
{
|
||
|
|
polynomial<T> result(a);
|
||
|
|
result >>= b;
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <typename T, typename U>
|
||
|
|
polynomial<T> operator << (const polynomial<T>& a, const U& b)
|
||
|
|
{
|
||
|
|
polynomial<T> result(a);
|
||
|
|
result <<= b;
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
|
||
|
|
// Unary minus (negate).
|
||
|
|
template <class T>
|
||
|
|
polynomial<T> operator - (polynomial<T> a)
|
||
|
|
{
|
||
|
|
std::transform(a.data().begin(), a.data().end(), a.data().begin(), std::negate<T>());
|
||
|
|
return a;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
bool odd(polynomial<T> const &a)
|
||
|
|
{
|
||
|
|
return a.size() > 0 && a[0] != static_cast<T>(0);
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
bool even(polynomial<T> const &a)
|
||
|
|
{
|
||
|
|
return !odd(a);
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
polynomial<T> pow(polynomial<T> base, int exp)
|
||
|
|
{
|
||
|
|
if (exp < 0)
|
||
|
|
return policies::raise_domain_error(
|
||
|
|
"boost::math::tools::pow<%1%>",
|
||
|
|
"Negative powers are not supported for polynomials.",
|
||
|
|
base, policies::policy<>());
|
||
|
|
// if the policy is ignore_error or errno_on_error, raise_domain_error
|
||
|
|
// will return std::numeric_limits<polynomial<T>>::quiet_NaN(), which
|
||
|
|
// defaults to polynomial<T>(), which is the zero polynomial
|
||
|
|
polynomial<T> result(T(1));
|
||
|
|
if (exp & 1)
|
||
|
|
result = base;
|
||
|
|
/* "Exponentiation by squaring" */
|
||
|
|
while (exp >>= 1)
|
||
|
|
{
|
||
|
|
base *= base;
|
||
|
|
if (exp & 1)
|
||
|
|
result *= base;
|
||
|
|
}
|
||
|
|
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
|
||
|
|
|
||
|
|
|
||
|
|
|