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Luroth Expansions

#include <boost/math/tools/luroth_expansion.hpp>
namespace boost::math::tools {

template<typename Real, typename Z = int64_t>
class luroth_expansion {
    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.,

The numbers di are called digits or denominators; we use the terminology digits, since technically in our notation d0 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.