





from __future__ import annotations

from typing import Any






class        _a:







		def __init__(    self ,						__snake_case ,						__snake_case ,						__snake_case = 0				)      ->      None:


							'''simple docstring'''

							_snake_case       :					Optional[int]				       =   row, column
							_snake_case       :					Union[str, Any]				       =   [[default_value for c in range(__snake_case				)] for r in range(__snake_case				)]







		def __str__(    self				)      ->      str:


							'''simple docstring'''

							_snake_case       :					List[Any]				       =   f"""Matrix consist of {self.row} rows and {self.column} columns\n"""

							# Make string identifier
							_snake_case       :					Union[str, Any]				       =   0
							for row_vector in self.array:
												for obj in row_vector:
																	_snake_case       :					int				       =   max(__snake_case ,						len(str(__snake_case				)				)				)
							_snake_case       :					int				       =   f"""%{max_element_length}s"""

							# Make string and return
							def single_line(__snake_case				) -> str:
												nonlocal string_format_identifier
												_snake_case       :					Any				       =   "["
												line += ", ".join(string_format_identifier % (obj,) for obj in row_vector				)
												line += "]"
												return line

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







		def __repr__(    self				)      ->      str:


							'''simple docstring'''

							return str(self				)







		def        lowercase       (    self ,						__snake_case				)      ->      bool:


							'''simple docstring'''

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







		def __getitem__(    self ,						__snake_case				)      ->      Any:


							'''simple docstring'''

							assert self.validate_indicies(__snake_case				)
							return self.array[loc[0]][loc[1]]







		def __setitem__(    self ,						__snake_case ,						__snake_case				)      ->      None:


							'''simple docstring'''

							assert self.validate_indicies(__snake_case				)
							_snake_case       :					List[str]				       =   value







		def __add__(    self ,						__snake_case				)      ->      Matrix:


							'''simple docstring'''

							assert isinstance(__snake_case ,						__snake_case				)
							assert self.row == another.row and self.column == another.column

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







		def __neg__(    self				)      ->      Matrix:


							'''simple docstring'''

							_snake_case       :					int				       =   Matrix(self.row ,						self.column				)
							for r in range(self.row				):
												for c in range(self.column				):
																	_snake_case       :					Optional[Any]				       =   -self[r, c]
							return result







		def __sub__(    self ,						__snake_case				)      ->      Matrix:


							'''simple docstring'''

							return self + (-another)







		def __mul__(    self ,						__snake_case				)      ->      Matrix:


							'''simple docstring'''

							if isinstance(__snake_case ,						(int, float)				):  # Scalar multiplication
												_snake_case       :					Optional[int]				       =   Matrix(self.row ,						self.column				)
												for r in range(self.row				):
																	for c in range(self.column				):
																						_snake_case       :					List[Any]				       =   self[r, c] * another
												return result
							elif isinstance(__snake_case ,						__snake_case				):  # Matrix multiplication
												assert self.column == another.row
												_snake_case       :					Optional[int]				       =   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:
												_snake_case       :					List[str]				       =   f"""Unsupported type given for another ({type(__snake_case				)})"""
												raise TypeError(__snake_case				)







		def        lowercase       (    self				)      ->      Matrix:


							'''simple docstring'''

							_snake_case       :					Optional[Any]				       =   Matrix(self.column ,						self.row				)
							for r in range(self.row				):
												for c in range(self.column				):
																	_snake_case       :					Tuple				       =   self[r, c]
							return result







		def        lowercase       (    self ,						__snake_case ,						__snake_case				)      ->      Any:


							'''simple docstring'''

							assert isinstance(__snake_case ,						__snake_case				) and isinstance(__snake_case ,						__snake_case				)
							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
							_snake_case       :					Tuple				       =   v.transpose()
							_snake_case       :					Tuple				       =   (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				(							):
									# a^(-1)
									_snake_case       :					int				       =   Matrix(3  ,  3  ,  0				)
									for i in range(3				):
														_snake_case       :					Optional[Any]				       =   1
									print(F"""a^(-1) is {ainv}"""				)
									# u, v
									_snake_case       :					List[Any]				       =   Matrix(3  ,  1  ,  0				)
									_snake_case       :					List[str]				       =   1, 2, -3
									_snake_case       :					Optional[int]				       =   Matrix(3  ,  1  ,  0				)
									_snake_case       :					Optional[int]				       =   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(UpperCAmelCase  ,  UpperCAmelCase				)}"""				)







				def    A				(							):
									import doctest

									doctest.testmod()

				testa()






import math

from numpy import inf
from scipy.integrate import quad







def    A				(							UpperCAmelCase				):
					if num <= 0:
										raise ValueError("math domain error"				)

					return quad(UpperCAmelCase  ,  0  ,  UpperCAmelCase  ,  args=(UpperCAmelCase)				)[0]







def    A				(							UpperCAmelCase  ,  UpperCAmelCase				):
					return math.pow(UpperCAmelCase  ,  z - 1				) * math.exp(-x				)


if __name__ == "__main__":
							from doctest import testmod

							testmod()