from math import sqrt

import numpy as np
from sympy import symbols

# Coefficient
# Speed of light (m/s)
A_      =      299792458

# Symbols
A_    ,    A_    ,    A_    ,    A_      =      symbols("ct x y z")






def        __UpperCAmelCase    (       UpperCAmelCase       )->       float:



			"""simple docstring"""







			if velocity > c:
						raise ValueError('''Speed must not exceed light speed 299,792,458 [m/s]!'''       )
			elif velocity < 1:
						# Usually the speed should be much higher than 1 (c order of magnitude)
						raise ValueError('''Speed must be greater than or equal to 1!'''       )

			return velocity / c






def        __UpperCAmelCase    (       UpperCAmelCase       )->       float:



			"""simple docstring"""







			return 1 / sqrt(1 - beta(UpperCAmelCase       ) ** 2       )






def        __UpperCAmelCase    (       UpperCAmelCase       )->       np.ndarray:



			"""simple docstring"""







			return np.array(
			    [
			        [gamma(UpperCAmelCase       ), -gamma(UpperCAmelCase       ) * beta(UpperCAmelCase       ), 0, 0],
			        [-gamma(UpperCAmelCase       ) * beta(UpperCAmelCase       ), gamma(UpperCAmelCase       ), 0, 0],
			        [0, 0, 1, 0],
			        [0, 0, 0, 1],
			    ]       )






def        __UpperCAmelCase    (       UpperCAmelCase,		UpperCAmelCase = None       )->       np.ndarray:



			"""simple docstring"""







			if event is None:
						lowercase							=		np.array([ct, x, y, z]       )  # Symbolic four vector
			else:
						event[0] *= c  # x0 is ct (speed of light * time)

			return transformation_matrix(UpperCAmelCase       ) @ event


if __name__ == "__main__":
						import doctest

						doctest.testmod()

						# Example of symbolic vector:
						A_      =      transform(29979245)
						print("Example of four vector: ")
						print(F"ct' = {four_vector[0]}")
						print(F"x' = {four_vector[1]}")
						print(F"y' = {four_vector[2]}")
						print(F"z' = {four_vector[3]}")

						# Substitute symbols with numerical values
						A_      =      {ct: c, x: 1, y: 1, z: 1}
						A_      =      [four_vector[i].subs(sub_dict) for i in range(4)]

						print(F"\n{numerical_vector}")






from __future__ import annotations

from collections.abc import MutableSequence






class      __lowercase					:

						def __init__(	self				:      Optional[Any]							,							__lowerCamelCase				:      int							,							__lowerCamelCase				:      MutableSequence[float]   )							->		None:


									'''simple docstring'''

									if len(__lowerCamelCase   ) != degree + 1:
												raise ValueError(
												    '''The number of coefficients should be equal to the degree + 1.'''   )

									lowercase							=		list(__lowerCamelCase   )
									lowercase							=		degree

						def __add__(	self				:      Any							,							__lowerCamelCase				:      Polynomial   )							->		Polynomial:


									'''simple docstring'''

									if self.degree > polynomial_a.degree:
												lowercase							=		self.coefficients[:]
												for i in range(polynomial_a.degree + 1   ):
															coefficients[i] += polynomial_a.coefficients[i]
												return Polynomial(self.degree							,							__lowerCamelCase   )
									else:
												lowercase							=		polynomial_a.coefficients[:]
												for i in range(self.degree + 1   ):
															coefficients[i] += self.coefficients[i]
												return Polynomial(polynomial_a.degree							,							__lowerCamelCase   )

						def __sub__(	self				:      str							,							__lowerCamelCase				:      Polynomial   )							->		Polynomial:


									'''simple docstring'''

									return self + polynomial_a * Polynomial(0							,							[-1]   )

						def __neg__(	self				:      str   )							->		Polynomial:


									'''simple docstring'''

									return Polynomial(self.degree							,							[-c for c in self.coefficients]   )

						def __mul__(	self				:      List[str]							,							__lowerCamelCase				:      Polynomial   )							->		Polynomial:


									'''simple docstring'''

									lowercase							=		[0] * (self.degree + polynomial_a.degree + 1)
									for i in range(self.degree + 1   ):
												for j in range(polynomial_a.degree + 1   ):
															coefficients[i + j] += (
															    self.coefficients[i] * polynomial_a.coefficients[j]
															)

									return Polynomial(self.degree + polynomial_a.degree							,							__lowerCamelCase   )

						def        __a		(	self				:      List[str]							,							__lowerCamelCase				:      int | float   )							->		int | float:


									'''simple docstring'''

									lowercase							=		0
									for i in range(self.degree + 1   ):
												result += self.coefficients[i] * (substitution**i)
									return result

						def __str__(	self				:      str   )							->		str:


									'''simple docstring'''

									lowercase							=		''''''
									for i in range(self.degree							,							-1							,							-1   ):
												if self.coefficients[i] == 0:
															continue
												elif self.coefficients[i] > 0:
															if polynomial:
																		polynomial += " + "
												else:
															polynomial += " - "

												if i == 0:
															polynomial += str(abs(self.coefficients[i]   )   )
												elif i == 1:
															polynomial += str(abs(self.coefficients[i]   )   ) + "x"
												else:
															polynomial += str(abs(self.coefficients[i]   )   ) + "x^" + str(__lowerCamelCase   )

									return polynomial

						def __repr__(	self				:      Tuple   )							->		str:


									'''simple docstring'''

									return self.__str__()

						def        __a		(	self				:      Union[str, Any]   )							->		Polynomial:


									'''simple docstring'''

									lowercase							=		[0] * self.degree
									for i in range(self.degree   ):
												lowercase							=		self.coefficients[i + 1] * (i + 1)
									return Polynomial(self.degree - 1							,							__lowerCamelCase   )

						def        __a		(	self				:      Union[str, Any]							,							__lowerCamelCase				:      int | float = 0   )							->		Polynomial:


									'''simple docstring'''

									lowercase							=		[0] * (self.degree + 2)
									lowercase							=		constant
									for i in range(self.degree + 1   ):
												lowercase							=		self.coefficients[i] / (i + 1)
									return Polynomial(self.degree + 1							,							__lowerCamelCase   )

						def __eq__(	self				:      Tuple							,							__lowerCamelCase				:      object   )							->		bool:


									'''simple docstring'''

									if not isinstance(__lowerCamelCase							,							__lowerCamelCase   ):
												return False

									if self.degree != polynomial_a.degree:
												return False

									for i in range(self.degree + 1   ):
												if self.coefficients[i] != polynomial_a.coefficients[i]:
															return False

									return True



						def __ne__(	self				:      Tuple							,							__lowerCamelCase				:      object   )							->		bool:


									'''simple docstring'''

									return not self.__eq__(__lowerCamelCase   )





