




'''simple docstring'''


import unittest

import numpy as np



def  _SCREAMING_SNAKE_CASE     (      UpperCamelCase__    :					np.ndarray ,    UpperCamelCase__    :					np.ndarray ,    UpperCamelCase__    :					np.ndarray ,    UpperCamelCase__    :					np.ndarray | None = None ,    ):




     """simple docstring"""


     a_     :							Tuple     						=  np.shape(UpperCamelCase__				)
     a_     :							str     						=  np.shape(UpperCamelCase__				)
     a_     :							List[str]     						=  np.shape(UpperCamelCase__				)

     if shape_a[0] != shape_b[0]:
          a_     :							List[str]     						=  (
              """Expected the same number of rows for A and B. """
              F"Instead found A of size {shape_a} and B of size {shape_b}"
          )
          raise ValueError(UpperCamelCase__				)

     if shape_b[1] != shape_c[1]:
          a_     :							Any     						=  (
              """Expected the same number of columns for B and C. """
              F"Instead found B of size {shape_b} and C of size {shape_c}"
          )
          raise ValueError(UpperCamelCase__				)

     a_     :							Any     						=  pseudo_inv
     if a_inv is None:
          try:
               a_     :							Tuple     						=  np.linalg.inv(UpperCamelCase__				)
          except np.linalg.LinAlgError:
               raise ValueError(
                   """Input matrix A is not invertible. Cannot compute Schur complement."""				)

     return mat_c - mat_b.T @ a_inv @ mat_b







class      SCREAMING_SNAKE_CASE    (						unittest.TestCase				):






      def 							lowercase_						( self   :  Union[str, Any]       ):


           '''simple docstring'''

           a_     :							List[str]     						=  np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]       )
           a_     :							Union[str, Any]     						=  np.array([[0, 3], [3, 0], [2, 3]]       )
           a_     :							int     						=  np.array([[2, 1], [6, 3]]       )

           a_     :							List[str]     						=  schur_complement(lowercase__						,						lowercase__						,						lowercase__       )

           a_     :							Optional[int]     						=  np.block([[a, b], [b.T, c]]       )

           a_     :							str     						=  np.linalg.det(lowercase__       )
           a_     :							Dict     						=  np.linalg.det(lowercase__       )
           a_     :							List[Any]     						=  np.linalg.det(lowercase__       )

           self.assertAlmostEqual(lowercase__						,						det_a * det_s       )






      def 							lowercase_						( self   :  int       ):


           '''simple docstring'''

           a_     :							str     						=  np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]       )
           a_     :							Optional[int]     						=  np.array([[0, 3], [3, 0], [2, 3]]       )
           a_     :							str     						=  np.array([[2, 1], [6, 3]]       )

           with self.assertRaises(lowercase__       ):
                schur_complement(lowercase__						,						lowercase__						,						lowercase__       )






      def 							lowercase_						( self   :  Optional[Any]       ):


           '''simple docstring'''

           a_     :							Optional[Any]     						=  np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]       )
           a_     :							List[str]     						=  np.array([[0, 3], [3, 0], [2, 3]]       )
           a_     :							List[Any]     						=  np.array([[2, 1, 3], [6, 3, 5]]       )

           with self.assertRaises(lowercase__       ):
                schur_complement(lowercase__						,						lowercase__						,						lowercase__       )


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 unittest.main()






'''simple docstring'''


from math import sqrt

import numpy as np
from sympy import symbols

# Coefficient
# Speed of light (m/s)
lowerCAmelCase_      :  List[Any]				     =  2_9979_2458

# Symbols
lowerCAmelCase_     ,   lowerCAmelCase_     ,   lowerCAmelCase_     ,   lowerCAmelCase_      :  Tuple				     =  symbols('ct x y z')



def  _SCREAMING_SNAKE_CASE     (      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  _SCREAMING_SNAKE_CASE     (      UpperCamelCase__    :					float				):




     """simple docstring"""


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



def  _SCREAMING_SNAKE_CASE     (      UpperCamelCase__    :					float				):




     """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  _SCREAMING_SNAKE_CASE     (      UpperCamelCase__    :					float ,    UpperCamelCase__    :					np.ndarray | None = None				):




     """simple docstring"""


     if event is None:
          a_     :							str     						=  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:
 lowerCAmelCase_      :  Union[str, Any]				     =  transform(2997_9245)
 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
 lowerCAmelCase_      :  Tuple				     =  {ct: c, x: 1, y: 1, z: 1}
 lowerCAmelCase_      :  Optional[int]				     =  [four_vector[i].subs(sub_dict) for i in range(4)]

 print(f"\n{numerical_vector}")
