polymerist.maths.fractions.continued
====================================

.. py:module:: polymerist.maths.fractions.continued

.. autoapi-nested-parse::

   Representations and computation methods for continued fractions and ration approximations to real numbers



Attributes
----------

.. autoapisummary::

   polymerist.maths.fractions.continued.Real
   polymerist.maths.fractions.continued.I
   polymerist.maths.fractions.continued.DEFAULT_INT_TYPE
   polymerist.maths.fractions.continued.DEFAULT_EPS
   polymerist.maths.fractions.continued.DEFAULT_TOL


Functions
---------

.. autoapisummary::

   polymerist.maths.fractions.continued.continuant_matrix
   polymerist.maths.fractions.continued.continued_fraction_to_continuants
   polymerist.maths.fractions.continued.real_to_continued_fraction_coeffs
   polymerist.maths.fractions.continued.extended_euclidean_algorithm
   polymerist.maths.fractions.continued.rational_approxes
   polymerist.maths.fractions.continued.best_rational_approx


Module Contents
---------------

.. py:type:: Real
   :canonical: Union[float, int]


.. py:data:: I

.. py:data:: DEFAULT_INT_TYPE
   :type:  Type

.. py:data:: DEFAULT_EPS
   :value: 1e-08


.. py:data:: DEFAULT_TOL
   :value: 1e-06


.. py:function:: continuant_matrix(a: int, int_type: I = DEFAULT_INT_TYPE) -> numpy.ndarray[polymerist.genutils.typetools.numpytypes.Shape[2, 2], int]

   Homographic matrix for a evaluating the next continuant from a continued fraction coefficient


.. py:function:: continued_fraction_to_continuants(coeffs: Iterable[int], int_type: I = DEFAULT_INT_TYPE) -> Generator[tuple[int, int], None, None]

   Fold a sequence of continued fraction coefficients into successive continuants (rational approximations)


.. py:function:: real_to_continued_fraction_coeffs(x: Real, eps: float = DEFAULT_EPS, int_type: I = DEFAULT_INT_TYPE) -> Generator[int, None, None]

   Euclidean algorithm on a real number to produce the integral continued fraction coefficients


.. py:function:: extended_euclidean_algorithm(a: int, b: int, int_type: I = DEFAULT_INT_TYPE) -> tuple[int, int, int]

   Compute the Extended Euclidean Algorithm between two integers "a" and "b"
   Returns the greatest common divisor of a and b, along with a pair of Bezout coefficients"x" and "y" which satisfy a*x + b*y = gcd(a, b)


.. py:function:: rational_approxes(x: Real, tol: float = DEFAULT_TOL, eps: float = DEFAULT_EPS) -> Generator[tuple[int, int], None, None]

   Unfold a real number into its continued fraction representation, then generate successive continuants of it


.. py:function:: best_rational_approx(x: Real, tol: float = DEFAULT_TOL, eps: float = DEFAULT_EPS) -> tuple[int, int]

   Provide a rational approximation to a value with the smallest denominator that is within some tolerance