Luroth expansions (#401)

doc/internals/luroth_expansion.qbk

@@ -0,0 +1,62 @@
+[/
+  Copyright Nick Thompson, 2020
+  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).
+]
+
+[section:luroth_expansion Luroth Expansions]
+
+    #include <boost/math/tools/luroth_expansion.hpp>
+    namespace boost::math::tools {
+
+    template<typename Real, typename Z = int64_t>
+    class luroth_expansion {
+    public:
+        luroth_expansion(Real x);
+
+        std::vector<Z> const & digits() const;
+
+        Real digit_geometric_mean() const;
+
+        template<typename T, typename Z_>
+        friend std::ostream& operator<<(std::ostream& out, luroth_expansion<T, Z_>& luroth);
+    };
+    }
+
+
+The `luroth_expansion` class provided by Boost expands a floating point number into a Lüroth representation, i.e.,
+
+[$../equations/luroth_expansion.svg]
+
+The numbers /d/[sub i] are called digits or denominators; we use the terminology digits, since technically in our notation /d/[sub 0] is not a denominator.
+
+Here's a minimal working example:
+
+    using boost::math::constants::pi;
+    using boost::math::tools::luroth_expansion;
+    auto luroth = luroth_expansion(pi<long double>());
+    std::cout << "π ≈ " << luroth << "\n";
+    // Prints:
+    // π ≈ ((3; 7, 1, 1, 1, 2, 1, 4, 23, 4, 1, 1, 1, 1, 80, 1, 1, 5))
+
+The class computes denominators while simultaneously computing convergents.
+Once a convergent is within a few ulps of the input value, the computation stops.
+
+/Nota bene:/ There is an alternative definition of the Lüroth representation where every digit is shifted by 1.
+We follow the definition given in Kalpazidou; with the modification that we do not constrain the input to be in the interval [0,1]
+and let the first digit be the floor of the input.
+
+For almost all real numbers, the geometric mean of the digits converges to a constant which is approximately 2.2001610580.
+This is "Khinchin's constant" for the Lüroth representation.
+
+
+
+[heading References]
+
+* Kalpazidou, Sofia. "Khintchine's constant for Lüroth representation." Journal of Number Theory 29.2 (1988): 196-205.
+
+* Finch, Steven R. Mathematical constants. Cambridge university press, 2003.
+
+
+[endsect]

doc/math.qbk

@@ -753,6 +753,7 @@ and as a CD ISBN 0-9504833-2-X  978-0-9504833-2-0, Classification 519.2-dc22.
 [include internals/fraction.qbk]
 [include internals/simple_continued_fraction.qbk]
 [include internals/centered_continued_fraction.qbk]
+[include internals/luroth_expansion.qbk]
 [include internals/recurrence.qbk]
 [/include internals/rational.qbk] [/moved to tools]
 [include internals/tuple.qbk]