720 lines
18 KiB
Plaintext
720 lines
18 KiB
Plaintext
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// (C) Copyright John Maddock 2006.
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// (C) Copyright Jeremy William Murphy 2015.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_TOOLS_POLYNOMIAL_HPP
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#define BOOST_MATH_TOOLS_POLYNOMIAL_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <boost/assert.hpp>
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#include <boost/config.hpp>
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#include <boost/config/suffix.hpp>
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#include <boost/function.hpp>
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#include <boost/lambda/lambda.hpp>
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#include <boost/math/tools/rational.hpp>
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#include <boost/math/tools/real_cast.hpp>
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#include <boost/math/policies/error_handling.hpp>
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#include <boost/math/special_functions/binomial.hpp>
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#include <boost/operators.hpp>
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#include <vector>
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#include <ostream>
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#include <algorithm>
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#ifndef BOOST_NO_CXX11_HDR_INITIALIZER_LIST
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#include <initializer_list>
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#endif
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namespace boost{ namespace math{ namespace tools{
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template <class T>
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T chebyshev_coefficient(unsigned n, unsigned m)
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{
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BOOST_MATH_STD_USING
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if(m > n)
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return 0;
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if((n & 1) != (m & 1))
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return 0;
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if(n == 0)
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return 1;
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T result = T(n) / 2;
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unsigned r = n - m;
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r /= 2;
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BOOST_ASSERT(n - 2 * r == m);
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if(r & 1)
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result = -result;
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result /= n - r;
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result *= boost::math::binomial_coefficient<T>(n - r, r);
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result *= ldexp(1.0f, m);
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return result;
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}
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template <class Seq>
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Seq polynomial_to_chebyshev(const Seq& s)
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{
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// Converts a Polynomial into Chebyshev form:
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typedef typename Seq::value_type value_type;
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typedef typename Seq::difference_type difference_type;
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Seq result(s);
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difference_type order = s.size() - 1;
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difference_type even_order = order & 1 ? order - 1 : order;
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difference_type odd_order = order & 1 ? order : order - 1;
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for(difference_type i = even_order; i >= 0; i -= 2)
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{
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value_type val = s[i];
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for(difference_type k = even_order; k > i; k -= 2)
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{
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val -= result[k] * chebyshev_coefficient<value_type>(static_cast<unsigned>(k), static_cast<unsigned>(i));
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}
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val /= chebyshev_coefficient<value_type>(static_cast<unsigned>(i), static_cast<unsigned>(i));
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result[i] = val;
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}
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result[0] *= 2;
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for(difference_type i = odd_order; i >= 0; i -= 2)
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{
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value_type val = s[i];
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for(difference_type k = odd_order; k > i; k -= 2)
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{
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val -= result[k] * chebyshev_coefficient<value_type>(static_cast<unsigned>(k), static_cast<unsigned>(i));
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}
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val /= chebyshev_coefficient<value_type>(static_cast<unsigned>(i), static_cast<unsigned>(i));
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result[i] = val;
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}
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return result;
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}
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template <class Seq, class T>
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T evaluate_chebyshev(const Seq& a, const T& x)
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{
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// Clenshaw's formula:
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typedef typename Seq::difference_type difference_type;
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T yk2 = 0;
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T yk1 = 0;
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T yk = 0;
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for(difference_type i = a.size() - 1; i >= 1; --i)
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{
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yk2 = yk1;
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yk1 = yk;
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yk = 2 * x * yk1 - yk2 + a[i];
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}
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return a[0] / 2 + yk * x - yk1;
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}
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template <typename T>
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class polynomial;
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namespace detail {
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/**
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* Knuth, The Art of Computer Programming: Volume 2, Third edition, 1998
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* Chapter 4.6.1, Algorithm D: Division of polynomials over a field.
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*
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* @tparam T Coefficient type, must be not be an integer.
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*
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* Template-parameter T actually must be a field but we don't currently have that
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* subtlety of distinction.
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*/
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template <typename T, typename N>
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BOOST_DEDUCED_TYPENAME disable_if_c<std::numeric_limits<T>::is_integer, void >::type
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division_impl(polynomial<T> &q, polynomial<T> &u, const polynomial<T>& v, N n, N k)
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{
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q[k] = u[n + k] / v[n];
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for (N j = n + k; j > k;)
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{
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j--;
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u[j] -= q[k] * v[j - k];
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}
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}
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template <class T, class N>
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T integer_power(T t, N n)
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{
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switch(n)
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{
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case 0:
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return static_cast<T>(1u);
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case 1:
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return t;
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case 2:
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return t * t;
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case 3:
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return t * t * t;
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}
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T result = integer_power(t, n / 2);
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result *= result;
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if(n & 1)
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result *= t;
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return result;
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}
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/**
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* Knuth, The Art of Computer Programming: Volume 2, Third edition, 1998
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* Chapter 4.6.1, Algorithm R: Pseudo-division of polynomials.
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*
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* @tparam T Coefficient type, must be an integer.
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*
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* Template-parameter T actually must be a unique factorization domain but we
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* don't currently have that subtlety of distinction.
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*/
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template <typename T, typename N>
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BOOST_DEDUCED_TYPENAME enable_if_c<std::numeric_limits<T>::is_integer, void >::type
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division_impl(polynomial<T> &q, polynomial<T> &u, const polynomial<T>& v, N n, N k)
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{
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q[k] = u[n + k] * integer_power(v[n], k);
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for (N j = n + k; j > 0;)
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{
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j--;
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u[j] = v[n] * u[j] - (j < k ? T(0) : u[n + k] * v[j - k]);
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}
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}
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/**
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* Knuth, The Art of Computer Programming: Volume 2, Third edition, 1998
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* Chapter 4.6.1, Algorithm D and R: Main loop.
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*
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* @param u Dividend.
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* @param v Divisor.
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*/
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template <typename T>
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std::pair< polynomial<T>, polynomial<T> >
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division(polynomial<T> u, const polynomial<T>& v)
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{
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BOOST_ASSERT(v.size() <= u.size());
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BOOST_ASSERT(v);
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BOOST_ASSERT(u);
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typedef typename polynomial<T>::size_type N;
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N const m = u.size() - 1, n = v.size() - 1;
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N k = m - n;
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polynomial<T> q;
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q.data().resize(m - n + 1);
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do
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{
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division_impl(q, u, v, n, k);
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}
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while (k-- != 0);
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u.data().resize(n);
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u.normalize(); // Occasionally, the remainder is zeroes.
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return std::make_pair(q, u);
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}
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template <class T>
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struct identity
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{
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T operator()(T const &x) const
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{
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return x;
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}
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};
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} // namespace detail
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/**
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* Returns the zero element for multiplication of polynomials.
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*/
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template <class T>
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polynomial<T> zero_element(std::multiplies< polynomial<T> >)
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{
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return polynomial<T>();
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}
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template <class T>
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polynomial<T> identity_element(std::multiplies< polynomial<T> >)
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{
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return polynomial<T>(T(1));
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}
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/* Calculates a / b and a % b, returning the pair (quotient, remainder) together
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* because the same amount of computation yields both.
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* This function is not defined for division by zero: user beware.
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*/
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template <typename T>
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std::pair< polynomial<T>, polynomial<T> >
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quotient_remainder(const polynomial<T>& dividend, const polynomial<T>& divisor)
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{
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BOOST_ASSERT(divisor);
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if (dividend.size() < divisor.size())
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return std::make_pair(polynomial<T>(), dividend);
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return detail::division(dividend, divisor);
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}
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template <class T>
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class polynomial :
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equality_comparable< polynomial<T>,
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dividable< polynomial<T>,
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dividable2< polynomial<T>, T,
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modable< polynomial<T>,
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modable2< polynomial<T>, T > > > > >
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{
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public:
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// typedefs:
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typedef typename std::vector<T>::value_type value_type;
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typedef typename std::vector<T>::size_type size_type;
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// construct:
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polynomial(){}
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template <class U>
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polynomial(const U* data, unsigned order)
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: m_data(data, data + order + 1)
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{
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normalize();
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}
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template <class I>
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polynomial(I first, I last)
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: m_data(first, last)
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{
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normalize();
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}
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template <class U>
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explicit polynomial(const U& point)
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{
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if (point != U(0))
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m_data.push_back(point);
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}
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// copy:
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polynomial(const polynomial& p)
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: m_data(p.m_data) { }
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template <class U>
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polynomial(const polynomial<U>& p)
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{
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for(unsigned i = 0; i < p.size(); ++i)
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{
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m_data.push_back(boost::math::tools::real_cast<T>(p[i]));
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}
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}
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#if !defined(BOOST_NO_CXX11_HDR_INITIALIZER_LIST) && !BOOST_WORKAROUND(BOOST_GCC_VERSION, < 40500)
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polynomial(std::initializer_list<T> l) : polynomial(std::begin(l), std::end(l))
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{
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}
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polynomial&
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operator=(std::initializer_list<T> l)
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{
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m_data.assign(std::begin(l), std::end(l));
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normalize();
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return *this;
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}
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#endif
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// access:
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size_type size()const { return m_data.size(); }
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size_type degree()const
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{
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if (size() == 0)
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throw std::logic_error("degree() is undefined for the zero polynomial.");
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return m_data.size() - 1;
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}
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value_type& operator[](size_type i)
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{
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return m_data[i];
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}
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const value_type& operator[](size_type i)const
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{
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return m_data[i];
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}
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T evaluate(T z)const
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{
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return m_data.size() > 0 ? boost::math::tools::evaluate_polynomial(&m_data[0], z, m_data.size()) : 0;
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}
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std::vector<T> chebyshev()const
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{
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return polynomial_to_chebyshev(m_data);
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}
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std::vector<T> const& data() const
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{
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return m_data;
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}
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std::vector<T> & data()
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{
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return m_data;
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}
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// operators:
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template <class U>
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polynomial& operator +=(const U& value)
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{
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addition(value);
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normalize();
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return *this;
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}
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template <class U>
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polynomial& operator -=(const U& value)
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{
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subtraction(value);
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normalize();
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return *this;
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}
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template <class U>
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polynomial& operator *=(const U& value)
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{
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multiplication(value);
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normalize();
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return *this;
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}
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template <class U>
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polynomial& operator /=(const U& value)
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{
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division(value);
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normalize();
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return *this;
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}
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template <class U>
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polynomial& operator %=(const U& /*value*/)
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{
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// We can always divide by a scalar, so there is no remainder:
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this->set_zero();
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|
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
|
||
|
|
|
||
|
|
|
||
|
|
|