






import warnings
from collections.abc import Callable
from typing import Any

import numpy
from matplotlib import pyplot

lowerCamelCase			:							Any       =0.25 + 0.0j
lowerCamelCase			:							Tuple       =-0.4 + 0.6j
lowerCamelCase			:							Tuple       =-0.1 + 0.651j
lowerCamelCase			:							Union[str, Any]       =-2.0
lowerCamelCase			:							int       =56
lowerCamelCase			:							Optional[Any]       =2.0
lowerCamelCase			:							Tuple       =666







def 	SCREAMING_SNAKE_CASE							(      __lowerCAmelCase      ,					__lowerCAmelCase    )				->			numpy.ndarray:
     return numpy.exp(__lowerCAmelCase    ) + c_parameter







def 	SCREAMING_SNAKE_CASE							(      __lowerCAmelCase      ,					__lowerCAmelCase    )				->			numpy.ndarray:
     return z_values * z_values + c_parameter







def 	SCREAMING_SNAKE_CASE							(      __lowerCAmelCase      ,					__lowerCAmelCase    )				->			numpy.ndarray:
     UpperCamelCase__    :     Dict        =						numpy.linspace(-window_size      ,					__lowerCAmelCase      ,					__lowerCAmelCase    )
     UpperCamelCase__    :     Union[str, Any]        =						x.reshape((nb_pixels, 1)    )
     UpperCamelCase__    :     Optional[int]        =						numpy.linspace(-window_size      ,					__lowerCAmelCase      ,					__lowerCAmelCase    )
     UpperCamelCase__    :     Union[str, Any]        =						y.reshape((1, nb_pixels)    )
     return x + 1.0J * y







def 	SCREAMING_SNAKE_CASE							(      __lowerCAmelCase      ,					__lowerCAmelCase      ,					__lowerCAmelCase      ,					__lowerCAmelCase      ,					__lowerCAmelCase = None      ,					)				->			numpy.ndarray:
     UpperCamelCase__    :     Any        =						z_a.astype("complex64"    )
     for _ in range(__lowerCAmelCase    ):
          UpperCamelCase__    :     Dict        =						eval_function(__lowerCAmelCase      ,					__lowerCAmelCase    )
          if infinity is not None:
               numpy.nan_to_num(__lowerCAmelCase      ,					copy=__lowerCAmelCase      ,					nan=__lowerCAmelCase    )
               UpperCamelCase__    :     List[Any]        =						infinity
     return z_n







def 	SCREAMING_SNAKE_CASE							(      __lowerCAmelCase      ,					__lowerCAmelCase      ,					__lowerCAmelCase      ,					__lowerCAmelCase      ,					)				->			None:
     UpperCamelCase__    :     Union[str, Any]        =						(abs(__lowerCAmelCase    )).transpose()
     UpperCamelCase__    :     Optional[int]        =						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 	SCREAMING_SNAKE_CASE							(      )				->			None:
     warnings.filterwarnings(
         "ignore"      ,					category=__lowerCAmelCase      ,					message="overflow encountered in multiply"    )
     warnings.filterwarnings(
         "ignore"      ,					category=__lowerCAmelCase      ,					message="invalid value encountered in multiply"      ,					)
     warnings.filterwarnings(
         "ignore"      ,					category=__lowerCAmelCase      ,					message="overflow encountered in absolute"    )
     warnings.filterwarnings(
         "ignore"      ,					category=__lowerCAmelCase      ,					message="overflow encountered in exp"    )


if __name__ == "__main__":
 lowerCamelCase			:							Optional[Any]       =prepare_grid(window_size, nb_pixels)

 ignore_overflow_warnings()  # See file header for explanations

 lowerCamelCase			:							List[Any]       =24
 lowerCamelCase			:							Union[str, Any]       =2 * abs(c_cauliflower) + 1
 lowerCamelCase			:							Optional[int]       =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)

 lowerCamelCase			:							Union[str, Any]       =64
 lowerCamelCase			:							Optional[Any]       =2 * abs(c_polynomial_a) + 1
 lowerCamelCase			:							Optional[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)

 lowerCamelCase			:							Optional[int]       =161
 lowerCamelCase			:							Union[str, Any]       =2 * abs(c_polynomial_a) + 1
 lowerCamelCase			:							Optional[int]       =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)

 lowerCamelCase			:							Dict       =12
 lowerCamelCase			:							Union[str, Any]       =1_0000.0
 lowerCamelCase			:							int       =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)







def 	SCREAMING_SNAKE_CASE							(      __lowerCAmelCase    )				->			str:
     if len(__lowerCAmelCase    ) == 0 or not all(len(__lowerCAmelCase    ) == 2 for pair in coordinates    ):
          raise ValueError("The program cannot work out a fitting polynomial."    )

     if len({tuple(__lowerCAmelCase    ) for pair in coordinates}    ) != len(__lowerCAmelCase    ):
          raise ValueError("The program cannot work out a fitting polynomial."    )

     UpperCamelCase__    :     List[str]        =						{x for x, _ in coordinates}
     if len(__lowerCAmelCase    ) == 1:
          return f'x={coordinates[0][0]}'

     if len(__lowerCAmelCase    ) != len(__lowerCAmelCase    ):
          raise ValueError("The program cannot work out a fitting polynomial."    )

     UpperCamelCase__    :     Any        =						len(__lowerCAmelCase    )

     UpperCamelCase__    :     str        =						0
     UpperCamelCase__    :     list[list[float]]        =						[]
     # put the x and x to the power values in a matrix
     while count_of_line < x:
          UpperCamelCase__    :     int        =						0
          UpperCamelCase__    :     List[Any]        =						coordinates[count_of_line][0]
          UpperCamelCase__    :     list[float]        =						[]
          while count_in_line < x:
               count_line.append(a ** (x - (count_in_line + 1))    )
               count_in_line += 1
          matrix.append(__lowerCAmelCase    )
          count_of_line += 1

     UpperCamelCase__    :     List[Any]        =						0
     # put the y values into a vector
     UpperCamelCase__    :     list[float]        =						[]
     while count_of_line < x:
          vector.append(coordinates[count_of_line][1]    )
          count_of_line += 1

     UpperCamelCase__    :     Tuple        =						0

     while count < x:
          UpperCamelCase__    :     Dict        =						0
          while zahlen < x:
               if count == zahlen:
                    zahlen += 1
               if zahlen == x:
                    break
               UpperCamelCase__    :     Optional[Any]        =						matrix[zahlen][count] / matrix[count][count]
               for counting_columns, item in enumerate(matrix[count]    ):
                    # manipulating all the values in the matrix
                    matrix[zahlen][counting_columns] -= item * bruch
               # manipulating the values in the vector
               vector[zahlen] -= vector[count] * bruch
               zahlen += 1
          count += 1

     UpperCamelCase__    :     Tuple        =						0
     # make solutions
     UpperCamelCase__    :     list[str]        =						[]
     while count < x:
          solution.append(str(vector[count] / matrix[count][count]    )    )
          count += 1

     UpperCamelCase__    :     int        =						0
     UpperCamelCase__    :     List[str]        =						"f(x)="

     while count < x:
          UpperCamelCase__    :     list[str]        =						solution[count].split("E"    )
          if len(__lowerCAmelCase    ) > 1:
               UpperCamelCase__    :     Union[str, Any]        =						f'{remove_e[0]}*10^{remove_e[1]}'
          solved += f'x^{x - (count + 1)}*{solution[count]}'
          if count + 1 != x:
               solved += "+"
          count += 1

     return solved


if __name__ == "__main__":
 print(points_to_polynomial([]))
 print(points_to_polynomial([[]]))
 print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
 print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
 print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
 print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
 print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
 print(points_to_polynomial([[1, 3], [2, 6], [3, 11]]))
 print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
 print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))