


'''simple docstring'''

from sympy import diff, lambdify, symbols
from sympy.functions import *  # noqa: F403

def _SCREAMING_SNAKE_CASE (A    ,					A    ,					A = "x"    ,					A = 10**-10    ,					A = 1    ,					)			->			complex:






       """simple docstring"""







       lowercase__      =					symbols(A			)
       lowercase__      =					lambdify(A    ,					A			)
       lowercase__      =					lambdify(A    ,					diff(A    ,					A			)			)

       lowercase__      =					starting_point

       while True:
              if diff_function(A			) != 0:
                     lowercase__      =					prev_guess - multiplicity * func(A			) / diff_function(
                         A			)
              else:
                     raise ZeroDivisionError('''Could not find root'''			) from None

              # Precision is checked by comparing the difference of consecutive guesses
              if abs(next_guess - prev_guess			) < precision:
                     return next_guess

              lowercase__      =					next_guess


# Let's Execute
if __name__ == "__main__":
 # Find root of trigonometric function
 # Find value of pi
 print(f"""The root of sin(x) = 0 is {newton_raphson('sin(x)', 2)}""")

 # Find root of polynomial
 # Find fourth Root of 5
 print(f"""The root of x**4 - 5 = 0 is {newton_raphson('x**4 -5', 0.4 +5j)}""")

 # Find value of e
 print(
     'The root of log(y) - 1 = 0 is ',
     f"""{newton_raphson('log(y) - 1', 2, variable='y')}""",
 )

 # Exponential Roots
 print(
     'The root of exp(x) - 1 = 0 is',
     f"""{newton_raphson('exp(x) - 1', 10, precision=0.0_0_5)}""",
 )

 # Find root of cos(x)
 print(f"""The root of cos(x) = 0 is {newton_raphson('cos(x)', 0)}""")








'''simple docstring'''

import unittest

import numpy as np

def _SCREAMING_SNAKE_CASE (A    ,					A    ,					A    ,					A = None    ,					)			->			np.ndarray:






       """simple docstring"""







       lowercase__      =					np.shape(A			)
       lowercase__      =					np.shape(A			)
       lowercase__      =					np.shape(A			)

       if shape_a[0] != shape_b[0]:
              lowercase__      =					(
                  '''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(A			)

       if shape_b[1] != shape_c[1]:
              lowercase__      =					(
                  '''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(A			)

       lowercase__      =					pseudo_inv
       if a_inv is None:
              try:
                     lowercase__      =					np.linalg.inv(A			)
              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     __lowerCAmelCase		(unittest.TestCase	):





      '''simple docstring'''






      def  UpperCamelCase__						(self       :       Dict					):

             '''simple docstring'''







             lowercase__      =					np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]					)
             lowercase__      =					np.array([[0, 3], [3, 0], [2, 3]]					)
             lowercase__      =					np.array([[2, 1], [6, 3]]					)

             lowercase__      =					schur_complement(UpperCamelCase  ,		UpperCamelCase  ,		UpperCamelCase					)

             lowercase__      =					np.block([[a, b], [b.T, c]]					)

             lowercase__      =					np.linalg.det(UpperCamelCase					)
             lowercase__      =					np.linalg.det(UpperCamelCase					)
             lowercase__      =					np.linalg.det(UpperCamelCase					)

             self.assertAlmostEqual(UpperCamelCase  ,		det_a * det_s					)



      def  UpperCamelCase__						(self       :       Optional[int]					):

             '''simple docstring'''







             lowercase__      =					np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]					)
             lowercase__      =					np.array([[0, 3], [3, 0], [2, 3]]					)
             lowercase__      =					np.array([[2, 1], [6, 3]]					)

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






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

             '''simple docstring'''







             lowercase__      =					np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]					)
             lowercase__      =					np.array([[0, 3], [3, 0], [2, 3]]					)
             lowercase__      =					np.array([[2, 1, 3], [6, 3, 5]]					)

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


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 unittest.main()




