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// (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_LUROTH_EXPANSION_HPP #define BOOST_MATH_TOOLS_LUROTH_EXPANSION_HPP #include #include #include #include #include #include #include #include #ifndef BOOST_MATH_STANDALONE #include #ifdef BOOST_MATH_NO_CXX17_IF_CONSTEXPR #error "The header can only be used in C++17 and later." #endif #endif namespace boost::math::tools { template class luroth_expansion { public: luroth_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 a Luroth representation."); } d_.reserve(50); Real dn1 = floor(x); d_.push_back(static_cast(dn1)); if (dn1 == x) { d_.shrink_to_fit(); return; } // This attempts to follow the notation of: // "Khinchine's constant for Luroth Representation", by Sophia Kalpazidou. x = x - dn1; Real computed = dn1; Real prod = 1; // 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 scale = std::numeric_limits::epsilon()*abs(x_)/2; while (abs(x_ - computed) > (i++)*scale) { Real recip = 1/x; Real dn = floor(recip); // x = n + 1/k => lur(x) = ((n; k - 1)) // Note that this is a bit different than Kalpazidou (examine the half-open interval of definition carefully). // One way to examine this definition is better for rationals (it never happens for irrationals) // is to consider i + 1/3. If you follow Kalpazidou, then you get ((i, 3, 0)); a zero digit! // That's bad since it destroys uniqueness and also breaks the computation of the geometric mean. if (recip == dn) { d_.push_back(static_cast(dn - 1)); break; } d_.push_back(static_cast(dn)); Real tmp = 1/(dn+1); computed += prod*tmp; prod *= tmp/dn; x = dn*(dn+1)*(x - tmp); } for (size_t i = 1; i < d_.size(); ++i) { // Sanity check: if (d_[i] <= 0) { throw std::domain_error("Found a digit <= 0; this is an error."); } } d_.shrink_to_fit(); } const std::vector& digits() const { return d_; } // Under the assumption of 'randomness', this mean converges to 2.2001610580. // See Finch, Mathematical Constants, section 1.8.1. Real digit_geometric_mean() const { if (d_.size() == 1) { return std::numeric_limits::quiet_NaN(); } using std::log; using std::exp; Real g = 0; for (size_t i = 1; i < d_.size(); ++i) { g += log(static_cast(d_[i])); } return exp(g/(d_.size() - 1)); } template friend std::ostream& operator<<(std::ostream& out, luroth_expansion& scf); private: const Real x_; std::vector d_; }; template std::ostream& operator<<(std::ostream& out, luroth_expansion& luroth) { constexpr const int p = std::numeric_limits::max_digits10; if constexpr (p == 2147483647) { out << std::setprecision(luroth.x_.backend().precision()); } else { out << std::setprecision(p); } out << "((" << luroth.d_.front(); if (luroth.d_.size() > 1) { out << "; "; for (size_t i = 1; i < luroth.d_.size() -1; ++i) { out << luroth.d_[i] << ", "; } out << luroth.d_.back(); } out << "))"; return out; } } #endif