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




def    lowerCAmelCase_  (		lowercase:				Any	,       lowercase:				int	,       lowercase:				Any = "x"	,       lowercase:				int = 10**-10	,       lowercase:				Dict = 1	,       )     ->					complex:







 '''simple docstring'''
 _UpperCamelCase:  List[str]    						=			symbols(lowercase		)
 _UpperCamelCase:  Union[str, Any]    						=			lambdify(lowercase	,       lowercase		)
 _UpperCamelCase:  Tuple    						=			lambdify(lowercase	,       diff(lowercase	,       lowercase		)		)

 _UpperCamelCase:  List[Any]    						=			starting_point

 while True:
  if diff_function(lowercase		) != 0:
   _UpperCamelCase:  Union[str, Any]    						=			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

  _UpperCamelCase:  Optional[Any]    						=			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', 1_0, precision=0.0_0_5)}""",
  )

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






from __future__ import annotations

import numpy as np
from numpy import floataa
from numpy.typing import NDArray





def  __magic_name__  (				lowercase				,       lowercase				,       lowercase				,       lowercase				,       )				-> list[float]:

							"""simple docstring"""



							lowercase_				,		lowercase_	:   List[str]  				=     coefficient_matrix.shape
							lowercase_				,		lowercase_	:   Any  				=     constant_matrix.shape

							if rowsa != colsa:
														lowercase_	:   List[Any]  				=     f"""Coefficient matrix dimensions must be nxn but received {rowsa}x{colsa}"""
														raise ValueError(lowercase				)

							if colsa != 1:
														lowercase_	:   Optional[int]  				=     f"""Constant matrix must be nx1 but received {rowsa}x{colsa}"""
														raise ValueError(lowercase				)

							if rowsa != rowsa:
														lowercase_	:   Tuple  				=     (
														    """Coefficient and constant matrices dimensions must be nxn and nx1 but """
														    f"""received {rowsa}x{colsa} and {rowsa}x{colsa}"""
														)
														raise ValueError(lowercase				)

							if len(lowercase				) != rowsa:
														lowercase_	:   int  				=     (
														    """Number of initial values must be equal to number of rows in coefficient """
														    f"""matrix but received {len(lowercase				)} and {rowsa}"""
														)
														raise ValueError(lowercase				)

							if iterations <= 0:
														raise ValueError("""Iterations must be at least 1"""				)

							lowercase_	:   NDArray[floataa]  				=     np.concatenate(
							    (coefficient_matrix, constant_matrix)				,       axis=1				)

							lowercase_				,		lowercase_	:   Dict  				=     table.shape

							strictly_diagonally_dominant(lowercase				)

							# Iterates the whole matrix for given number of times
							for _ in range(lowercase				):
														lowercase_	:   str  				=     []
														for row in range(lowercase				):
																					lowercase_	:   Dict  				=     0
																					for col in range(lowercase				):
																												if col == row:
																																			lowercase_	:   Optional[Any]  				=     table[row][col]
																												elif col == cols - 1:
																																			lowercase_	:   Tuple  				=     table[row][col]
																												else:
																																			temp += (-1) * table[row][col] * init_val[col]
																					lowercase_	:   List[Any]  				=     (temp + val) / denom
																					new_val.append(lowercase				)
														lowercase_	:   Optional[int]  				=     new_val

							return [float(lowercase				) for i in new_val]





def  __magic_name__  (				lowercase				)				-> bool:

							"""simple docstring"""



							lowercase_				,		lowercase_	:   str  				=     table.shape

							lowercase_	:   str  				=     True

							for i in range(0				,       lowercase				):
														lowercase_	:   Any  				=     0
														for j in range(0				,       cols - 1				):
																					if i == j:
																												continue
																					else:
																												total += table[i][j]

														if table[i][i] <= total:
																					raise ValueError("""Coefficient matrix is not strictly diagonally dominant"""				)

							return is_diagonally_dominant


# Test Cases
if __name__ == "__main__":
							import doctest

							doctest.testmod()