




'''simple docstring'''
import warnings
from collections.abc import Callable
from typing import Any

import numpy
from matplotlib import pyplot

A				: Tuple					      =			0.25 + 0.0j
A				: List[str]					      =			-0.4 + 0.6j
A				: Dict					      =			-0.1 + 0.651j
A				: str					      =			-2.0
A				: Any					      =			56
A				: List[Any]					      =			2.0
A				: int					      =			666
def 							snake_case_    (  a__						:      complex       ,a__						:      numpy.ndarray     ):



    """simple docstring"""




    return numpy.exp(a__     ) + c_parameter
def 							snake_case_    (  a__						:      complex       ,a__						:      numpy.ndarray     ):



    """simple docstring"""




    return z_values * z_values + c_parameter
def 							snake_case_    (  a__						:      float       ,a__						:      int     ):



    """simple docstring"""




    __lowercase     							=		numpy.linspace(-window_size       ,a__       ,a__     )
    __lowercase     							=		x.reshape((nb_pixels, 1)     )
    __lowercase     							=		numpy.linspace(-window_size       ,a__       ,a__     )
    __lowercase     							=		y.reshape((1, nb_pixels)     )
    return x + 1.0J * y
def 							snake_case_    (  a__						:      Callable[[Any, numpy.ndarray], numpy.ndarray]       ,a__						:      Any       ,a__						:      int       ,a__						:      numpy.ndarray       ,a__						:      float | None = None       ,):



    """simple docstring"""




    __lowercase     							=		z_a.astype("""complex64"""     )
    for _ in range(a__     ):
        __lowercase     							=		eval_function(a__       ,a__     )
        if infinity is not None:
            numpy.nan_to_num(a__       ,copy=a__       ,nan=a__     )
            __lowercase     							=		infinity
    return z_n
def 							snake_case_    (  a__						:      str       ,a__						:      Any       ,a__						:      float       ,a__						:      numpy.ndarray       ,):



    """simple docstring"""




    __lowercase     							=		(abs(a__     )).transpose()
    __lowercase     							=		abs_z_final[::-1, :]
    pyplot.matshow(abs_z_final < escape_radius     )
    pyplot.title(f'Julia set of ${function_label}$, $c={function_params}$'     )
    pyplot.show()
def 							snake_case_    (  ):



    """simple docstring"""




    warnings.filterwarnings(
        """ignore"""       ,category=a__       ,message="""overflow encountered in multiply"""     )
    warnings.filterwarnings(
        """ignore"""       ,category=a__       ,message="""invalid value encountered in multiply"""       ,)
    warnings.filterwarnings(
        """ignore"""       ,category=a__       ,message="""overflow encountered in absolute"""     )
    warnings.filterwarnings(
        """ignore"""       ,category=a__       ,message="""overflow encountered in exp"""     )


if __name__ == "__main__":
  A				: int					      =			prepare_grid(window_size, nb_pixels)

  ignore_overflow_warnings()  # See file header for explanations

  A				: List[Any]					      =			24
  A				: str					      =			2 * abs(c_cauliflower) + 1
  A				: Optional[Any]					      =			iterate_function(
      eval_quadratic_polynomial,
      c_cauliflower,
      nb_iterations,
      z_a,
      infinity=1.1 * escape_radius,
  )
  show_results("""z^2+c""", c_cauliflower, escape_radius, z_final)

  A				: int					      =			64
  A				: Any					      =			2 * abs(c_polynomial_a) + 1
  A				: Any					      =			iterate_function(
      eval_quadratic_polynomial,
      c_polynomial_a,
      nb_iterations,
      z_a,
      infinity=1.1 * escape_radius,
  )
  show_results("""z^2+c""", c_polynomial_a, escape_radius, z_final)

  A				: str					      =			161
  A				: Dict					      =			2 * abs(c_polynomial_a) + 1
  A				: str					      =			iterate_function(
      eval_quadratic_polynomial,
      c_polynomial_a,
      nb_iterations,
      z_a,
      infinity=1.1 * escape_radius,
  )
  show_results("""z^2+c""", c_polynomial_a, escape_radius, z_final)

  A				: Optional[int]					      =			12
  A				: List[Any]					      =			1_00_00.0
  A				: Union[str, Any]					      =			iterate_function(
      eval_exponential,
      c_exponential,
      nb_iterations,
      z_a + 2,
      infinity=1.0E10,
  )
  show_results("""e^z+c""", c_exponential, escape_radius, z_final)










'''simple docstring'''
import numpy as np
def 							snake_case_    (  a__						:      Tuple     ):



    """simple docstring"""




    __lowercase							,__lowercase     							=		a.shape
    __lowercase     							=		min(a__       ,a__     )
    __lowercase     							=		np.eye(a__     )
    __lowercase     							=		a.copy()

    for k in range(t - 1     ):
        # select a column of modified matrix A':
        __lowercase     							=		r[k:, [k]]
        # construct first basis vector
        __lowercase     							=		np.zeros_like(a__     )
        __lowercase     							=		1.0
        # determine scaling factor
        __lowercase     							=		np.linalg.norm(a__     )
        # construct vector v for Householder reflection
        __lowercase     							=		x + np.sign(x[0]     ) * alpha * ea
        v /= np.linalg.norm(a__     )

        # construct the Householder matrix
        __lowercase     							=		np.eye(m - k     ) - 2.0 * v @ v.T
        # pad with ones and zeros as necessary
        __lowercase     							=		np.block([[np.eye(a__     ), np.zeros((k, m - k)     )], [np.zeros((m - k, k)     ), q_k]]     )

        __lowercase     							=		q @ q_k.T
        __lowercase     							=		q_k @ r

    return q, r


if __name__ == "__main__":
  import doctest

  doctest.testmod()




