
from __future__ import annotations

from typing import Any

class        _UpperCamelCase				:






			'''simple docstring'''

			def __init__(       self :       Any						,	snake_case_ :       int						,	snake_case_ :       int						,	snake_case_ :       float = 0      ):
							UpperCamelCase_,   UpperCamelCase_:							Dict	=						row, column
							UpperCamelCase_:							Union[str, Any]	=						[[default_value for c in range(__A      )] for r in range(__A      )]

			def __str__(       self :       int      ):
							UpperCamelCase_:							Any	=						f'''Matrix consist of {self.row} rows and {self.column} columns\n'''

							# Make string identifier
							UpperCamelCase_:							Union[str, Any]	=						0
							for row_vector in self.array:
											for obj in row_vector:
															UpperCamelCase_:							Optional[Any]	=						max(__A						,	len(str(__A      )      )      )
							UpperCamelCase_:							Union[str, Any]	=						f'''%{max_element_length}s'''

							# Make string and return
							def single_line(snake_case_ :       list[float]      ) -> str:
											nonlocal string_format_identifier
											UpperCamelCase_:							List[str]	=						"""["""
											line += ", ".join(string_format_identifier % (obj,) for obj in row_vector      )
											line += "]"
											return line

							s += "\n".join(single_line(__A      ) for row_vector in self.array      )
							return s

			def __repr__(       self :       List[str]      ):
							return str(self      )

			def       lowerCAmelCase__	(       self :       str						,	snake_case_ :       tuple[int, int]      ):
							if not (isinstance(__A						,	(list, tuple)      ) and len(__A      ) == 2):
											return False
							elif not (0 <= loc[0] < self.row and 0 <= loc[1] < self.column):
											return False
							else:
											return True

			def __getitem__(       self :       int						,	snake_case_ :       tuple[int, int]      ):
							assert self.validate_indicies(__A      )
							return self.array[loc[0]][loc[1]]

			def __setitem__(       self :       Any						,	snake_case_ :       tuple[int, int]						,	snake_case_ :       float      ):
							assert self.validate_indicies(__A      )
							UpperCamelCase_:							Any	=						value

			def __add__(       self :       int						,	snake_case_ :       Matrix      ):
							assert isinstance(__A						,	__A      )
							assert self.row == another.row and self.column == another.column

							# Add
							UpperCamelCase_:							Union[str, Any]	=						Matrix(self.row						,	self.column      )
							for r in range(self.row      ):
											for c in range(self.column      ):
															UpperCamelCase_:							str	=						self[r, c] + another[r, c]
							return result

			def __neg__(       self :       Dict      ):
							UpperCamelCase_:							str	=						Matrix(self.row						,	self.column      )
							for r in range(self.row      ):
											for c in range(self.column      ):
															UpperCamelCase_:							Optional[int]	=						-self[r, c]
							return result

			def __sub__(       self :       List[Any]						,	snake_case_ :       Matrix      ):
							return self + (-another)

			def __mul__(       self :       List[str]						,	snake_case_ :       int | float | Matrix      ):

							if isinstance(__A						,	(int, float)      ):  # Scalar multiplication
											UpperCamelCase_:							List[Any]	=						Matrix(self.row						,	self.column      )
											for r in range(self.row      ):
															for c in range(self.column      ):
																			UpperCamelCase_:							Optional[Any]	=						self[r, c] * another
											return result
							elif isinstance(__A						,	__A      ):  # Matrix multiplication
											assert self.column == another.row
											UpperCamelCase_:							Any	=						Matrix(self.row						,	another.column      )
											for r in range(self.row      ):
															for c in range(another.column      ):
																			for i in range(self.column      ):
																							result[r, c] += self[r, i] * another[i, c]
											return result
							else:
											UpperCamelCase_:							Dict	=						f'''Unsupported type given for another ({type(__A      )})'''
											raise TypeError(__A      )

			def       lowerCAmelCase__	(       self :       int      ):
							UpperCamelCase_:							Optional[int]	=						Matrix(self.column						,	self.row      )
							for r in range(self.row      ):
											for c in range(self.column      ):
															UpperCamelCase_:							List[Any]	=						self[r, c]
							return result




			def       lowerCAmelCase__	(       self :       List[str]						,	snake_case_ :       Matrix						,	snake_case_ :       Matrix      ):
							assert isinstance(__A						,	__A      ) and isinstance(__A						,	__A      )
							assert self.row == self.column == u.row == v.row  # u, v should be column vector
							assert u.column == v.column == 1  # u, v should be column vector

							# Calculate
							UpperCamelCase_:							Union[str, Any]	=						v.transpose()
							UpperCamelCase_:							Optional[int]	=						(v_t * self * u)[0, 0] + 1
							if numerator_factor == 0:
											return None  # It's not invertable
							return self - ((self * u) * (v_t * self) * (1.0 / numerator_factor))


# Testing
if __name__ == "__main__":







							def        A__					(						)  ->	int:
											UpperCamelCase_:							Optional[int]	=						Matrix(3     ,    3     ,    0						)
											for i in range(3						):
															UpperCamelCase_:							Union[str, Any]	=						1
											print(F'''a^(-1) is {ainv}'''						)
											# u, v
											UpperCamelCase_:							Optional[Any]	=						Matrix(3     ,    1     ,    0						)
											UpperCamelCase_,   UpperCamelCase_,   UpperCamelCase_:							Optional[int]	=						1, 2, -3
											UpperCamelCase_:							Dict	=						Matrix(3     ,    1     ,    0						)
											UpperCamelCase_,   UpperCamelCase_,   UpperCamelCase_:							Union[str, Any]	=						4, -2, 5
											print(F'''u is {u}'''						)
											print(F'''v is {v}'''						)
											print(F'''uv^T is {u * v.transpose()}'''						)
											# Sherman Morrison
											print(F'''(a + uv^T)^(-1) is {ainv.sherman_morrison(_lowercase     ,    _lowercase						)}'''						)







							def        A__					(						)  ->	Union[str, Any]:
											import doctest

											doctest.testmod()





							testa()






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







def        A__					(						lowerCamelCase     ,    lowerCamelCase     ,    lowerCamelCase = "x"     ,    lowerCamelCase = 10**-10     ,    lowerCamelCase = 1     ,    )  ->	complex:
				UpperCamelCase_:							Optional[Any]	=						symbols(lowerCamelCase						)
				UpperCamelCase_:							int	=						lambdify(lowerCamelCase     ,    lowerCamelCase						)
				UpperCamelCase_:							Optional[Any]	=						lambdify(lowerCamelCase     ,    diff(lowerCamelCase     ,    lowerCamelCase						)						)

				UpperCamelCase_:							Tuple	=						starting_point

				while True:
								if diff_function(lowerCamelCase						) != 0:
												UpperCamelCase_:							List[Any]	=						prev_guess - multiplicity * func(lowerCamelCase						) / diff_function(
												    lowerCamelCase						)
								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_:							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', 10, precision=0.005)}""",
							)

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






