simple_continued_fraction: added public functions to access and modify the coefficients

doc/internals/simple_continued_fraction.qbk

@@ -1,5 +1,6 @@
 [/
   Copyright Nick Thompson, 2020
+  Copyright Matt Borland, 2023
   Distributed under 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).
@@ -19,9 +20,17 @@
 
         Real khinchin_harmonic_mean() const;
 
-        template<typename T, typename Z_>
-        friend std::ostream& operator<<(std::ostream& out, simple_continued_fraction<T, Z>& scf);
+        const std::vector<Z>& partial_denominators() const;
+
+        inline std::vector<Z>&& get_data() noexcept;
+
+        template<typename T, typename Z2>
+        friend std::ostream& operator<<(std::ostream& out, simple_continued_fraction<T, Z2>& scf);
     };
+
+    template<typename Real, typename Z = int64_t>
+    inline std::vector<Z> simple_continued_fraction_coefficients(Real x);
+
     }
 
 
@@ -47,6 +56,35 @@ This is because when examining known values like π, it creates a large number o
 It may be the case the a few incorrect partial convergents is harmless, but we compute continued fractions because we would like to do something with them.
 One sensible thing to do it to ask whether the number is in some sense "random"; a question that can be partially answered by computing the Khinchin geometric mean
 
+If you only require the coefficients of the simple continued fraction for example in the calculation of [@https://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations best rational approximations] there is a free function for that.
+
+An example of this calculation follows: 
+
+    using boost::math::tools::simple_continued_fraction_coefficients;
+
+    auto coefs1 = simple_continued_fraction_coefficients(static_cast<Real>(3.14155L)); // [3; 7, 15, 2, 7, 1, 4, 2]
+    auto coefs2 = simple_continued_fraction_coefficients(static_cast<Real>(3.14165L)); // [3; 7, 16, 1, 3, 4, 2, 4]
+
+    const std::size_t max_size = (std::min)(coefs1.size(), coefs2.size());
+    std::vector<std::int64_t> coefs;
+    coefs.reserve(max_size);
+
+    for (std::size_t i = 0; i < max_size; ++i)
+    {
+        const auto c1 = coefs1[i];
+        const auto c2 = coefs2[i];
+        if (c1 == c2) 
+        {
+            coefs.emplace_back(c1);
+            continue;
+        }
+
+        coefs.emplace_back((std::min)(c1, c2) + 1);
+        break;
+    }
+
+    // Result is [3; 7, 16]
+
 [$../equations/khinchin_geometric.svg]
 
 and Khinchin harmonic mean

include/boost/math/tools/simple_continued_fraction.hpp

@@ -1,4 +1,5 @@
 //  (C) Copyright Nick Thompson 2020.
+//  (C) Copyright Matt Borland 2023.
 //  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)
@@ -14,12 +15,15 @@
 #include <limits>
 #include <stdexcept>
 #include <sstream>
+#include <utility>
+#include <cstdint>
+#include <cassert>
 
 #include <boost/math/tools/is_standalone.hpp>
 #ifndef BOOST_MATH_STANDALONE
 #include <boost/config.hpp>
 #ifdef BOOST_NO_CXX17_IF_CONSTEXPR
-#error "The header <boost/math/norms.hpp> can only be used in C++17 and later."
+#error "The header <boost/math/simple_continued_fraction.hpp> can only be used in C++17 and later."
 #endif
 #endif
 
@@ -32,34 +36,59 @@ namespace boost::math::tools {
 template<typename Real, typename Z = int64_t>
 class simple_continued_fraction {
 public:
-    simple_continued_fraction(Real x) : x_{x} {
+    typedef Z int_type;
+
+    simple_continued_fraction(std::vector<Z> data) : b_{std::move(data)} {
+        const size_t size_ = b_.size();
+        if (size_ == 0) {
+            throw std::length_error("Array of coefficients is empty.");
+        }
+
+        for (size_t i = 1; i < size_; ++i) {
+            if (b_[i] <= 0) {
+                std::ostringstream oss;
+                oss << "Found a negative partial denominator: b[" << i << "] = " << b_[i] << ".";
+                throw std::domain_error(oss.str());
+            }
+        }
+
+        canonicalize();
+    }
+
+    simple_continued_fraction(Real x) : b_{} {
         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.");  
+            throw std::domain_error("Cannot convert non-finites into continued fractions.");
+        }
+
+        if constexpr (std_precision == 2147483647) {
+           precision_ = x.backend().precision();
         }
+
         b_.reserve(50);
         Real bj = floor(x);
         b_.push_back(static_cast<Z>(bj));
         if (bj == x) {
            b_.shrink_to_fit();
            return;
         }
+
+        const Real orig_x = x;
         x = 1/(x-bj);
         Real f = bj;
         if (bj == 0) {
            f = 16*(std::numeric_limits<Real>::min)();
         }
         Real C = f;
         Real D = 0;
-        int i = 0;
-        // the "1 + i++" lets the error bound grow slowly with the number of convergents.
+        // 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<Real>::epsilon()*abs(x_))
-        {
+        const Real eps_abs_orig_x = std::numeric_limits<Real>::epsilon()*abs(orig_x);
+        for (int i = 0; abs(f - orig_x) >= (1 + i)*eps_abs_orig_x; ++i) {
           bj = floor(x);
           b_.push_back(static_cast<Z>(bj));
           x = 1/(x-bj);
@@ -74,16 +103,10 @@ class simple_continued_fraction {
           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) {
+       canonicalize();
+
+       const size_t size_ = b_.size();
+       for (size_t i = 1; i < size_; ++i) {
          if (b_[i] <= 0) {
             std::ostringstream oss;
             oss << "Found a negative partial denominator: b[" << i << "] = " << b_[i] << "."
@@ -98,19 +121,20 @@ class simple_continued_fraction {
          }
        }
     }
-    
+
     Real khinchin_geometric_mean() const {
-        if (b_.size() == 1) { 
+        const size_t size_ = b_.size();
+        if (size_ == 1) {
          return std::numeric_limits<Real>::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<Real, 7> logs{std::numeric_limits<Real>::quiet_NaN(), Real(0), log(static_cast<Real>(2)), log(static_cast<Real>(3)), log(static_cast<Real>(4)), log(static_cast<Real>(5)), log(static_cast<Real>(6))};
+         const std::array<Real, 7> logs{std::numeric_limits<Real>::quiet_NaN(), static_cast<Real>(0), log(static_cast<Real>(2)), log(static_cast<Real>(3)), log(static_cast<Real>(4)), log(static_cast<Real>(5)), log(static_cast<Real>(6))};
          Real log_prod = 0;
-         for (size_t i = 1; i < b_.size(); ++i) {
+         for (size_t i = 1; i < size_; ++i) {
             if (b_[i] < static_cast<Z>(logs.size())) {
                log_prod += logs[b_[i]];
             }
@@ -119,44 +143,57 @@ class simple_continued_fraction {
                log_prod += log(static_cast<Real>(b_[i]));
             }
          }
-         log_prod /= (b_.size()-1);
+         log_prod /= (size_-1);
          return exp(log_prod);
     }
-    
+
     Real khinchin_harmonic_mean() const {
-        if (b_.size() == 1) {
+        const size_t size_ = b_.size();
+        if (size_ == 1) {
           return std::numeric_limits<Real>::quiet_NaN();
         }
-        Real n = b_.size() - 1;
+        Real n = size_ - 1;
         Real denom = 0;
-        for (size_t i = 1; i < b_.size(); ++i) {
+        for (size_t i = 1; i < size_; ++i) {
             denom += 1/static_cast<Real>(b_[i]);
         }
         return n/denom;
     }
-
+
+    // Note that this also includes the integer part (i.e. all the coefficients)
     const std::vector<Z>& partial_denominators() const {
       return b_;
     }
-
+
+    inline std::vector<Z>&& get_data() noexcept {
+        return std::move(b_);
+    }
+
     template<typename T, typename Z2>
     friend std::ostream& operator<<(std::ostream& out, simple_continued_fraction<T, Z2>& scf);
-
 private:
-    const Real x_;
+    static constexpr int std_precision = std::numeric_limits<Real>::max_digits10;
+
+    void canonicalize() {
+        // 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_.pop_back();
+            b_.back() += 1;
+        }
+        b_.shrink_to_fit();
+    }
+
     std::vector<Z> b_;
+
+    int precision_{std_precision};
 };
 
 
 template<typename Real, typename Z2>
 std::ostream& operator<<(std::ostream& out, simple_continued_fraction<Real, Z2>& scf) {
-   constexpr const int p = std::numeric_limits<Real>::max_digits10;
-   if constexpr (p == 2147483647) {
-      out << std::setprecision(scf.x_.backend().precision());
-   } else {
-      out << std::setprecision(p);
-   }
-
+   out << std::setprecision(scf.precision_);
    out << "[" << scf.b_.front();
    if (scf.b_.size() > 1)
    {
@@ -171,6 +208,41 @@ std::ostream& operator<<(std::ostream& out, simple_continued_fraction<Real, Z2>&
    return out;
 }
 
+template<typename Real, typename Z = std::int64_t>
+inline auto simple_continued_fraction_coefficients(Real x)
+{
+    auto temp = simple_continued_fraction<Real, Z>(x);
+    return temp.get_data();
+}
+
+// Can be used with `boost::rational` from <boost/rational.hpp>
+template <typename Rational, typename Real, typename Z = std::int64_t>
+inline Rational to_rational(const simple_continued_fraction<Real, Z>& scf)
+{
+    using int_t = typename Rational::int_type;
+
+    auto& coefs = scf.partial_denominators();
+    const size_t size_ = coefs.size();
+    assert(size_ >= 1);
+    if (size_ == 1) return static_cast<int_t>(coefs[0]);
+
+    // p0 = a0, p1 = a1.a0 + 1, pn = an.pn-1 + pn-2 for 2 <= n
+    // q0 = 1,  q1 = a1,        qn = an.qn-1 + qn-2 for 2 <= n
+
+    int_t p0 = coefs[0];
+    int_t p1 = p0*coefs[1] + 1;
+    int_t q0 = 1;
+    int_t q1 = coefs[1];
+    for (size_t i = 2; i < size_; ++i) {
+        const Z cn = coefs[i];
+        const int_t pn = cn*p1 + p0;
+        const int_t qn = cn*q1 + q0;
+        p0 = std::exchange(p1, pn);
+        q0 = std::exchange(q1, qn);
+    }
+
+    return {p1, q1};
+}
 
 }
 #endif

test/simple_continued_fraction_test.cpp

@@ -1,5 +1,6 @@
 /*
  * Copyright Nick Thompson, 2020
+ * Copyright Matt Borland, 2023
  * 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)
@@ -131,6 +132,37 @@ void test_khinchin()
     CHECK_ULP_CLOSE(Real(2), Km1, 10);
 }
 
+template <typename Real>
+void test_git_issue_970()
+{
+    using boost::math::tools::simple_continued_fraction_coefficients;
+
+    auto coefs1 = simple_continued_fraction_coefficients(static_cast<Real>(3.14155L)); // [3; 7, 15, 2, 7, 1, 4, 2]
+    auto coefs2 = simple_continued_fraction_coefficients(static_cast<Real>(3.14165L)); // [3; 7, 16, 1, 3, 4, 2, 4]
+
+    const std::size_t max_size = (std::min)(coefs1.size(), coefs2.size());
+    std::vector<std::int64_t> coefs;
+    coefs.reserve(max_size);
+
+    for (std::size_t i = 0; i < max_size; ++i)
+    {
+        const auto c1 = coefs1[i];
+        const auto c2 = coefs2[i];
+        if (c1 == c2) 
+        {
+            coefs.emplace_back(c1);
+            continue;
+        }
+
+        coefs.emplace_back((std::min)(c1, c2) + 1);
+        break;
+    }
+
+    // Result is [3; 7, 16]
+    CHECK_EQUAL(coefs[0], static_cast<std::int64_t>(3));
+    CHECK_EQUAL(coefs[1], static_cast<std::int64_t>(7));
+    CHECK_EQUAL(coefs[2], static_cast<std::int64_t>(16));
+}
 
 int main()
 {
@@ -157,5 +189,10 @@ int main()
     test_simple<float128>();
     test_khinchin<float128>();
     #endif
+
+    test_git_issue_970<float>();
+    test_git_issue_970<double>();
+    test_git_issue_970<long double>();
+
     return boost::math::test::report_errors();
 }