// (C) Copyright Nick Thompson 2020.
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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_LUROTH_EXPANSION_HPP
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#define BOOST_MATH_TOOLS_LUROTH_EXPANSION_HPP
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#include <vector>
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#include <ostream>
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#include <iomanip>
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#include <cmath>
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#include <limits>
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#include <stdexcept>
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namespace boost::math::tools {
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template<typename Real, typename Z = int64_t>
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class luroth_expansion {
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public:
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luroth_expansion(Real x) : x_{x}
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{
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using std::floor;
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using std::abs;
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using std::sqrt;
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using std::isfinite;
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if (!isfinite(x))
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{
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throw std::domain_error("Cannot convert non-finites into a Lüroth representation.");
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}
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d_.reserve(50);
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Real dn = floor(x);
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d_.push_back(static_cast<Z>(dn));
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if (dn == x)
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{
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d_.shrink_to_fit();
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return;
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}
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// This attempts to follow the notation of:
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// "Khinchine's constant for Luroth Representation", by Sophia Kalpazidou.
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x = x - dn;
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Real computed = dn;
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Real prod = 1;
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// Let the error bound grow by 1 ULP/iteration.
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// I haven't done the error analysis to show that this is an expected rate of error growth,
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// but if you don't do this, you can easily get into an infinite loop.
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Real i = 1;
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Real scale = std::numeric_limits<Real>::epsilon()*abs(x_)/2;
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while (abs(x_ - computed) > (i++)*scale)
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{
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Real recip = 1/x;
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Real dn = floor(recip);
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// x = n + 1/k => lur(x) = ((n; k - 1))
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// Note that this is a bit different than Kalpazidou (examine the half-open interval of definition carefully).
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// One way to examine this definition is better for rationals (it never happens for irrationals)
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// is to consider i + 1/3. If you follow Kalpazidou, then you get ((i, 3, 0)); a zero digit!
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// That's bad since it destroys uniqueness and also breaks the computation of the geometric mean.
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if (recip == dn) {
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d_.push_back(static_cast<Z>(dn - 1));
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break;
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}
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d_.push_back(static_cast<Z>(dn));
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Real tmp = 1/(dn+1);
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computed += prod*tmp;
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prod *= tmp/dn;
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x = dn*(dn+1)*(x - tmp);
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}
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for (size_t i = 1; i < d_.size(); ++i)
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{
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// Sanity check:
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if (d_[i] <= 0)
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{
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throw std::domain_error("Found a digit <= 0; this is an error.");
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}
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}
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d_.shrink_to_fit();
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}
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const std::vector<Z>& digits() const {
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return d_;
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}
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// Under the assumption of 'randomness', this mean converges to 2.2001610580.
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// See Finch, Mathematical Constants, section 1.8.1.
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Real digit_geometric_mean() const {
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if (d_.size() == 1) {
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return std::numeric_limits<Real>::quiet_NaN();
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}
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using std::log;
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using std::exp;
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Real g = 0;
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for (size_t i = 1; i < d_.size(); ++i) {
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g += log(static_cast<Real>(d_[i]));
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}
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return exp(g/(d_.size() - 1));
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}
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template<typename T, typename Z2>
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friend std::ostream& operator<<(std::ostream& out, luroth_expansion<T, Z2>& scf);
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private:
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const Real x_;
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std::vector<Z> d_;
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};
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template<typename Real, typename Z2>
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std::ostream& operator<<(std::ostream& out, luroth_expansion<Real, Z2>& luroth)
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{
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constexpr const int p = std::numeric_limits<Real>::max_digits10;
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if constexpr (p == 2147483647)
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{
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out << std::setprecision(luroth.x_.backend().precision());
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}
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else
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{
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out << std::setprecision(p);
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}
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out << "((" << luroth.d_.front();
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if (luroth.d_.size() > 1)
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{
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out << "; ";
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for (size_t i = 1; i < luroth.d_.size() -1; ++i)
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{
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out << luroth.d_[i] << ", ";
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}
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out << luroth.d_.back();
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}
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out << "))";
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return out;
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}
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}
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#endif
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