[/
  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]