





'''simple docstring'''



from typing import Any

import numpy as np



def  lowercase__(		_UpperCamelCase  : np.ndarray				)->   bool:






					"""simple docstring"""

					assert np.shape(_UpperCamelCase				)[0] == np.shape(_UpperCamelCase				)[1]

					# If matrix not symmetric, exit right away.
					if np.allclose(_UpperCamelCase ,   matrix.T				) is False:
										return False

					# Get eigenvalues and eignevectors for a symmetric matrix.
					_UpperCamelCase				,	_UpperCamelCase											=   np.linalg.eigh(_UpperCamelCase				)

					# Check sign of all eigenvalues.
					# np.all returns a value of type np.bool_
					return bool(np.all(eigen_values > 0				)				)



def  lowercase__(		_UpperCamelCase  : int				)->   Any:






					"""simple docstring"""

					_UpperCamelCase											=   np.random.randn(_UpperCamelCase ,   _UpperCamelCase				)
					_UpperCamelCase											=   np.dot(_UpperCamelCase ,   random_matrix.T				)
					assert _is_matrix_spd(_UpperCamelCase				)
					return spd_matrix



def  lowercase__(		_UpperCamelCase  : np.ndarray ,   _UpperCamelCase  : np.ndarray ,   _UpperCamelCase  : int = 1000 ,   _UpperCamelCase  : float = 1E-8 ,   )->   Any:






					"""simple docstring"""

					assert np.shape(_UpperCamelCase				)[0] == np.shape(_UpperCamelCase				)[1]
					assert np.shape(_UpperCamelCase				)[0] == np.shape(_UpperCamelCase				)[0]
					assert _is_matrix_spd(_UpperCamelCase				)

					# Initialize solution guess, residual, search direction.
					_UpperCamelCase											=   np.zeros((np.shape(_UpperCamelCase				)[0], 1)				)
					_UpperCamelCase											=   np.copy(_UpperCamelCase				)
					_UpperCamelCase											=   np.copy(_UpperCamelCase				)

					# Set initial errors in solution guess and residual.
					_UpperCamelCase											=   1E9
					_UpperCamelCase											=   1E9
					_UpperCamelCase											=   1E9

					# Set iteration counter to threshold number of iterations.
					_UpperCamelCase											=   0

					while error > tol:
										# Save this value so we only calculate the matrix-vector product once.
										_UpperCamelCase											=   np.dot(_UpperCamelCase ,   _UpperCamelCase				)

										# The main algorithm.

										# Update search direction magnitude.
										_UpperCamelCase											=   np.dot(ra.T ,   _UpperCamelCase				) / np.dot(pa.T ,   _UpperCamelCase				)
										# Update solution guess.
										_UpperCamelCase											=   xa + alpha * pa
										# Calculate new residual.
										_UpperCamelCase											=   ra - alpha * w
										# Calculate new Krylov subspace scale.
										_UpperCamelCase											=   np.dot(r.T ,   _UpperCamelCase				) / np.dot(ra.T ,   _UpperCamelCase				)
										# Calculate new A conjuage search direction.
										_UpperCamelCase											=   r + beta * pa

										# Calculate errors.
										_UpperCamelCase											=   np.linalg.norm(r - ra				)
										_UpperCamelCase											=   np.linalg.norm(x - xa				)
										_UpperCamelCase											=   np.maximum(_UpperCamelCase ,   _UpperCamelCase				)

										# Update variables.
										_UpperCamelCase											=   np.copy(_UpperCamelCase				)
										_UpperCamelCase											=   np.copy(_UpperCamelCase				)
										_UpperCamelCase											=   np.copy(_UpperCamelCase				)

										# Update number of iterations.
										iterations += 1
										if iterations > max_iterations:
															break

					return x



def  lowercase__(		)->   None:






					"""simple docstring"""

					_UpperCamelCase											=   3
					_UpperCamelCase											=   _create_spd_matrix(_UpperCamelCase				)
					_UpperCamelCase											=   np.random.randn(_UpperCamelCase ,   1				)
					_UpperCamelCase											=   np.dot(_UpperCamelCase ,   _UpperCamelCase				)

					# Numpy solution.
					_UpperCamelCase											=   np.linalg.solve(_UpperCamelCase ,   _UpperCamelCase				)

					# Our implementation.
					_UpperCamelCase											=   conjugate_gradient(_UpperCamelCase ,   _UpperCamelCase				)

					# Ensure both solutions are close to x_true (and therefore one another).
					assert np.linalg.norm(x_numpy - x_true				) <= 1E-6
					assert np.linalg.norm(x_conjugate_gradient - x_true				) <= 1E-6


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			test_conjugate_gradient()










'''simple docstring'''



from __future__ import annotations



def  lowercase__(		_UpperCamelCase  : list[list[int]] ,   _UpperCamelCase  : int ,   _UpperCamelCase  : int ,   _UpperCamelCase  : int | float				)->   None:






					"""simple docstring"""

					_UpperCamelCase				,	_UpperCamelCase											=   m - 1, 0
					while i >= 0 and j < n:
										if key == mat[i][j]:
															print(f"Key {key} found at row- {i + 1} column- {j + 1}"				)
															return
										if key < mat[i][j]:
															i -= 1
										else:
															j += 1
					print(f"Key {key} not found"				)



def  lowercase__(		)->   None:






					"""simple docstring"""

					_UpperCamelCase											=   [[2, 5, 7], [4, 8, 13], [9, 11, 15], [12, 17, 20]]
					_UpperCamelCase											=   int(input("Enter the element to be searched:"				)				)
					print(_UpperCamelCase				)
					search_in_a_sorted_matrix(_UpperCamelCase ,   len(_UpperCamelCase				) ,   len(mat[0]				) ,   _UpperCamelCase				)


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			main()



