63 lines
2.1 KiB
Plaintext
63 lines
2.1 KiB
Plaintext
[/
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Copyright Nick Thompson, 2020
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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[section:luroth_expansion Luroth Expansions]
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#include <boost/math/tools/luroth_expansion.hpp>
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namespace boost::math::tools {
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template<typename Real, typename Z = int64_t>
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class luroth_expansion {
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public:
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luroth_expansion(Real x);
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std::vector<Z> const & digits() const;
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Real digit_geometric_mean() const;
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template<typename T, typename Z_>
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friend std::ostream& operator<<(std::ostream& out, luroth_expansion<T, Z_>& luroth);
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};
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}
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The `luroth_expansion` class provided by Boost expands a floating point number into a Lüroth representation, i.e.,
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[$../equations/luroth_expansion.svg]
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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.
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Here's a minimal working example:
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using boost::math::constants::pi;
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using boost::math::tools::luroth_expansion;
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auto luroth = luroth_expansion(pi<long double>());
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std::cout << "π ≈ " << luroth << "\n";
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// Prints:
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// π ≈ ((3; 7, 1, 1, 1, 2, 1, 4, 23, 4, 1, 1, 1, 1, 80, 1, 1, 5))
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The class computes denominators while simultaneously computing convergents.
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Once a convergent is within a few ulps of the input value, the computation stops.
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/Nota bene:/ There is an alternative definition of the Lüroth representation where every digit is shifted by 1.
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We follow the definition given in Kalpazidou; with the modification that we do not constrain the input to be in the interval [0,1]
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and let the first digit be the floor of the input.
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For almost all real numbers, the geometric mean of the digits converges to a constant which is approximately 2.2001610580.
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This is "Khinchin's constant" for the Lüroth representation.
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[heading References]
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* Kalpazidou, Sofia. "Khintchine's constant for Lüroth representation." Journal of Number Theory 29.2 (1988): 196-205.
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* Finch, Steven R. Mathematical constants. Cambridge university press, 2003.
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[endsect]
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