117 lines
3.1 KiB
C++
117 lines
3.1 KiB
C++
|
// (C) Copyright Nick Thompson 2020.
|
||
|
|
// 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_ENGEL_EXPANSION_HPP
|
||
|
|
#define BOOST_MATH_TOOLS_ENGEL_EXPANSION_HPP
|
||
|
|
|
||
|
|
#include <cmath>
|
||
|
|
#include <cstdint>
|
||
|
|
#include <vector>
|
||
|
|
#include <ostream>
|
||
|
|
#include <iomanip>
|
||
|
|
#include <limits>
|
||
|
|
#include <stdexcept>
|
||
|
|
|
||
|
|
namespace boost::math::tools {
|
||
|
|
|
||
|
|
template<typename Real, typename Z = int64_t>
|
||
|
|
class engel_expansion {
|
||
|
|
public:
|
||
|
|
engel_expansion(Real x) : x_{x}
|
||
|
|
{
|
||
|
|
using std::floor;
|
||
|
|
using std::abs;
|
||
|
|
using std::sqrt;
|
||
|
|
using std::isfinite;
|
||
|
|
if (!isfinite(x))
|
||
|
|
{
|
||
|
|
throw std::domain_error("Cannot convert non-finites into an Engel expansion.");
|
||
|
|
}
|
||
|
|
|
||
|
|
if(x==0)
|
||
|
|
{
|
||
|
|
throw std::domain_error("Zero does not have an Engel expansion.");
|
||
|
|
}
|
||
|
|
a_.reserve(64);
|
||
|
|
// Let the error bound grow by 1 ULP/iteration.
|
||
|
|
// I haven't done the error analysis to show that this is an expected rate of error growth,
|
||
|
|
// but if you don't do this, you can easily get into an infinite loop.
|
||
|
|
Real i = 1;
|
||
|
|
Real computed = 0;
|
||
|
|
Real term = 1;
|
||
|
|
Real scale = std::numeric_limits<Real>::epsilon()*abs(x_)/2;
|
||
|
|
Real u = x;
|
||
|
|
while (abs(x_ - computed) > (i++)*scale)
|
||
|
|
{
|
||
|
|
Real recip = 1/u;
|
||
|
|
Real ak = ceil(recip);
|
||
|
|
a_.push_back(static_cast<Z>(ak));
|
||
|
|
u = u*ak - 1;
|
||
|
|
if (u==0)
|
||
|
|
{
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
term /= ak;
|
||
|
|
computed += term;
|
||
|
|
}
|
||
|
|
|
||
|
|
for (size_t j = 1; j < a_.size(); ++j)
|
||
|
|
{
|
||
|
|
// Sanity check: This should only happen when wraparound occurs:
|
||
|
|
if (a_[j] < a_[j-1])
|
||
|
|
{
|
||
|
|
throw std::domain_error("The digits of an Engel expansion must form a non-decreasing sequence; consider increasing the wide of the integer type.");
|
||
|
|
}
|
||
|
|
// Watch out for saturating behavior:
|
||
|
|
if (a_[j] == (std::numeric_limits<Z>::max)())
|
||
|
|
{
|
||
|
|
throw std::domain_error("The integer type Z does not have enough width to hold the terms of the Engel expansion; please widen the type.");
|
||
|
|
}
|
||
|
|
}
|
||
|
|
a_.shrink_to_fit();
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
const std::vector<Z>& digits() const
|
||
|
|
{
|
||
|
|
return a_;
|
||
|
|
}
|
||
|
|
|
||
|
|
template<typename T, typename Z2>
|
||
|
|
friend std::ostream& operator<<(std::ostream& out, engel_expansion<T, Z2>& eng);
|
||
|
|
|
||
|
|
private:
|
||
|
|
Real x_;
|
||
|
|
std::vector<Z> a_;
|
||
|
|
};
|
||
|
|
|
||
|
|
|
||
|
|
template<typename Real, typename Z2>
|
||
|
|
std::ostream& operator<<(std::ostream& out, engel_expansion<Real, Z2>& engel)
|
||
|
|
{
|
||
|
|
constexpr const int p = std::numeric_limits<Real>::max_digits10;
|
||
|
|
if constexpr (p == 2147483647)
|
||
|
|
{
|
||
|
|
out << std::setprecision(engel.x_.backend().precision());
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
out << std::setprecision(p);
|
||
|
|
}
|
||
|
|
|
||
|
|
out << "{";
|
||
|
|
for (size_t i = 0; i < engel.a_.size() - 1; ++i)
|
||
|
|
{
|
||
|
|
out << engel.a_[i] << ", ";
|
||
|
|
}
|
||
|
|
out << engel.a_.back();
|
||
|
|
out << "}";
|
||
|
|
return out;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
}
|
||
|
|
#endif
|