






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


def 	lowerCamelCase__      (  lowercase    ,  lowercase    ,  lowercase = "x"    ,  lowercase = 10**-10    ,  lowercase = 1    ,  ):

  """simple docstring"""






  SCREAMING_SNAKE_CASE							:   List[Any]						    =     symbols(lowercase					)
  SCREAMING_SNAKE_CASE							:   Tuple						    =     lambdify(lowercase    ,  lowercase					)
  SCREAMING_SNAKE_CASE							:   str						    =     lambdify(lowercase    ,  diff(lowercase    ,  lowercase					)					)

  SCREAMING_SNAKE_CASE							:   Union[str, Any]						    =     starting_point

  while True:
    if diff_function(lowercase					) != 0:
      SCREAMING_SNAKE_CASE							:   str						    =     prev_guess - multiplicity * func(lowercase					) / diff_function(
          lowercase					)
    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

    SCREAMING_SNAKE_CASE							:   str						    =     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=0.005)}""",
   )

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








from __future__ import annotations




def 							a		(				snake_case__:   list[list[int]]   ):


				'''simple docstring'''

				# preprocessing the first row
				for i in range(1		,  len(matrix[0]   )   ):
								matrix[0][i] += matrix[0][i - 1]

				# preprocessing the first column
				for i in range(1		,  len(snake_case__   )   ):
								matrix[i][0] += matrix[i - 1][0]

				# updating the path cost for current position
				for i in range(1		,  len(snake_case__   )   ):
								for j in range(1		,  len(matrix[0]   )   ):
												matrix[i][j] += min(matrix[i - 1][j]		,  matrix[i][j - 1]   )

				return matrix[-1][-1]


if __name__ == "__main__":
						import doctest

						doctest.testmod()




