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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_SIMPLE_CONTINUED_FRACTION_HPP #define BOOST_MATH_TOOLS_SIMPLE_CONTINUED_FRACTION_HPP #include #include #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 #ifndef BOOST_MATH_STANDALONE #include #endif namespace boost::math::tools { template class simple_continued_fraction { public: simple_continued_fraction(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 continued fractions."); } b_.reserve(50); Real bj = floor(x); b_.push_back(static_cast(bj)); if (bj == x) { b_.shrink_to_fit(); return; } x = 1/(x-bj); Real f = bj; if (bj == 0) { f = 16*(std::numeric_limits::min)(); } Real C = f; Real D = 0; int i = 0; // the "1 + i++" lets the error bound grow slowly with the number of convergents. // I have not worked out the error propagation of the Modified Lentz's method to see if it does indeed grow at this rate. // Numerical Recipes claims that no one has worked out the error analysis of the modified Lentz's method. while (abs(f - x_) >= (1 + i++)*std::numeric_limits::epsilon()*abs(x_)) { bj = floor(x); b_.push_back(static_cast(bj)); x = 1/(x-bj); D += bj; if (D == 0) { D = 16*(std::numeric_limits::min)(); } C = bj + 1/C; if (C==0) { C = 16*(std::numeric_limits::min)(); } D = 1/D; f *= (C*D); } // Deal with non-uniqueness of continued fractions: [a0; a1, ..., an, 1] = a0; a1, ..., an + 1]. // The shorter representation is considered the canonical representation, // so if we compute a non-canonical representation, change it to canonical: if (b_.size() > 2 && b_.back() == 1) { b_[b_.size() - 2] += 1; b_.resize(b_.size() - 1); } b_.shrink_to_fit(); for (size_t i = 1; i < b_.size(); ++i) { if (b_[i] <= 0) { std::ostringstream oss; oss << "Found a negative partial denominator: b[" << i << "] = " << b_[i] << "." #ifndef BOOST_MATH_STANDALONE << " This means the integer type '" << boost::core::demangle(typeid(Z).name()) #else << " This means the integer type '" << typeid(Z).name() #endif << "' has overflowed and you need to use a wider type," << " or there is a bug."; throw std::overflow_error(oss.str()); } } } Real khinchin_geometric_mean() const { if (b_.size() == 1) { return std::numeric_limits::quiet_NaN(); } using std::log; using std::exp; // Precompute the most probable logarithms. See the Gauss-Kuzmin distribution for details. // Example: b_i = 1 has probability -log_2(3/4) ~ .415: // A random partial denominator has ~80% chance of being in this table: const std::array logs{std::numeric_limits::quiet_NaN(), Real(0), log(static_cast(2)), log(static_cast(3)), log(static_cast(4)), log(static_cast(5)), log(static_cast(6))}; Real log_prod = 0; for (size_t i = 1; i < b_.size(); ++i) { if (b_[i] < static_cast(logs.size())) { log_prod += logs[b_[i]]; } else { log_prod += log(static_cast(b_[i])); } } log_prod /= (b_.size()-1); return exp(log_prod); } Real khinchin_harmonic_mean() const { if (b_.size() == 1) { return std::numeric_limits::quiet_NaN(); } Real n = b_.size() - 1; Real denom = 0; for (size_t i = 1; i < b_.size(); ++i) { denom += 1/static_cast(b_[i]); } return n/denom; } const std::vector& partial_denominators() const { return b_; } template friend std::ostream& operator<<(std::ostream& out, simple_continued_fraction& scf); private: const Real x_; std::vector b_; }; template std::ostream& operator<<(std::ostream& out, simple_continued_fraction& scf) { constexpr const int p = std::numeric_limits::max_digits10; if constexpr (p == 2147483647) { out << std::setprecision(scf.x_.backend().precision()); } else { out << std::setprecision(p); } out << "[" << scf.b_.front(); if (scf.b_.size() > 1) { out << "; "; for (size_t i = 1; i < scf.b_.size() -1; ++i) { out << scf.b_[i] << ", "; } out << scf.b_.back(); } out << "]"; return out; } } #endif