


'''simple docstring'''
import logging

import numpy as np
import pytest
from scipy.linalg import eigh

logging.basicConfig(level=logging.INFO, format='''%(message)s''')


def       a				(      __a       )  ->     np.ndarray:



					'''simple docstring'''






					return input_array.reshape((input_array.size, 1)       )


def       a				(      __a			, __a			, __a       )  ->     np.ndarray:



					'''simple docstring'''






					UpperCamelCase__       :Union[str, Any]							 =			np.nan
					for i in range(__a       ):
										UpperCamelCase__       :List[str]							 =			features[:, labels == i]
										UpperCamelCase__       :str							 =			data.mean(1       )
										# Centralize the data of class i
										UpperCamelCase__       :List[str]							 =			data - column_reshape(__a       )
										if i > 0:
															# If covariance_sum is not None
															covariance_sum += np.dot(__a			, centered_data.T       )
										else:
															# If covariance_sum is np.nan (i.e. first loop)
															UpperCamelCase__       :List[str]							 =			np.dot(__a			, centered_data.T       )

					return covariance_sum / features.shape[1]


def       a				(      __a			, __a			, __a       )  ->     np.ndarray:



					'''simple docstring'''






					UpperCamelCase__       :Tuple							 =			features.mean(1       )
					UpperCamelCase__       :Optional[Any]							 =			np.nan
					for i in range(__a       ):
										UpperCamelCase__       :str							 =			features[:, labels == i]
										UpperCamelCase__       :Dict							 =			data.shape[1]
										UpperCamelCase__       :Union[str, Any]							 =			data.mean(1       )
										if i > 0:
															# If covariance_sum is not None
															covariance_sum += device_data * np.dot(
															    column_reshape(__a       ) - column_reshape(__a       )			, (column_reshape(__a       ) - column_reshape(__a       )).T			, )
										else:
															# If covariance_sum is np.nan (i.e. first loop)
															UpperCamelCase__       :str							 =			device_data * np.dot(
															    column_reshape(__a       ) - column_reshape(__a       )			, (column_reshape(__a       ) - column_reshape(__a       )).T			, )

					return covariance_sum / features.shape[1]


def       a				(      __a			, __a       )  ->     np.ndarray:



					'''simple docstring'''






					if features.any():
										UpperCamelCase__       :str							 =			features.mean(1       )
										# Center the dataset
										UpperCamelCase__       :Any							 =			features - np.reshape(__a			, (data_mean.size, 1)       )
										UpperCamelCase__       :Optional[int]							 =			np.dot(__a			, centered_data.T       ) / features.shape[1]
										UpperCamelCase__ ,    UpperCamelCase__       :Any							 =			np.linalg.eigh(__a       )
										# Take all the columns in the reverse order (-1), and then takes only the first
										UpperCamelCase__       :int							 =			eigenvectors[:, ::-1][:, 0:dimensions]
										# Project the database on the new space
										UpperCamelCase__       :List[str]							 =			np.dot(filtered_eigenvectors.T			, __a       )
										logging.info('''Principal Component Analysis computed'''       )

										return projected_data
					else:
										logging.basicConfig(level=logging.ERROR			, format='''%(message)s'''			, force=__a       )
										logging.error('''Dataset empty'''       )
										raise AssertionError


def       a				(      __a			, __a			, __a			, __a       )  ->     np.ndarray:



					'''simple docstring'''






					assert classes > dimensions

					# Check if features have been already loaded
					if features.any:
										UpperCamelCase__ ,    UpperCamelCase__       :Any							 =			eigh(
										    covariance_between_classes(__a			, __a			, __a       )			, covariance_within_classes(__a			, __a			, __a       )			, )
										UpperCamelCase__       :str							 =			eigenvectors[:, ::-1][:, :dimensions]
										UpperCamelCase__ ,    UpperCamelCase__ ,    UpperCamelCase__       :Tuple							 =			np.linalg.svd(__a       )
										UpperCamelCase__       :List[str]							 =			svd_matrix[:, 0:dimensions]
										UpperCamelCase__       :List[str]							 =			np.dot(filtered_svd_matrix.T			, __a       )
										logging.info('''Linear Discriminant Analysis computed'''       )

										return projected_data
					else:
										logging.basicConfig(level=logging.ERROR			, format='''%(message)s'''			, force=__a       )
										logging.error('''Dataset empty'''       )
										raise AssertionError


def       a				(      )  ->     None:



					'''simple docstring'''






					UpperCamelCase__       :Any							 =			np.array([[1, 2, 3, 4, 5], [2, 3, 4, 5, 6], [3, 4, 5, 6, 7]]       )
					UpperCamelCase__       :Optional[Any]							 =			np.array([0, 0, 0, 1, 1]       )
					UpperCamelCase__       :str							 =			2
					UpperCamelCase__       :Optional[int]							 =			2

					# Assert that the function raises an AssertionError if dimensions > classes
					with pytest.raises(__a       ) as error_info:
										UpperCamelCase__       :Any							 =			linear_discriminant_analysis(
										    __a			, __a			, __a			, __a       )
										if isinstance(__a			, np.ndarray       ):
															raise AssertionError(
															    '''Did not raise AssertionError for dimensions > classes'''       )
										assert error_info.type is AssertionError


def       a				(      )  ->     None:



					'''simple docstring'''






					UpperCamelCase__       :int							 =			np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]       )
					UpperCamelCase__       :Union[str, Any]							 =			2
					UpperCamelCase__       :List[Any]							 =			np.array([[6.9_2_8_2_0_3_2_3, 8.6_6_0_2_5_4_0_4, 1_0.3_9_2_3_0_4_8_5], [3.0, 3.0, 3.0]]       )

					with pytest.raises(__a       ) as error_info:
										UpperCamelCase__       :Dict							 =			principal_component_analysis(__a			, __a       )
										if not np.allclose(__a			, __a       ):
															raise AssertionError
										assert error_info.type is AssertionError


if __name__ == "__main__":
		import doctest

		doctest.testmod()



'''simple docstring'''
from __future__ import annotations

__snake_case											=					list[list[int]]

# assigning initial values to the grid
__snake_case											=					[
    [3, 0, 6, 5, 0, 8, 4, 0, 0],
    [5, 2, 0, 0, 0, 0, 0, 0, 0],
    [0, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]

# a grid with no solution
__snake_case											=					[
    [5, 0, 6, 5, 0, 8, 4, 0, 3],
    [5, 2, 0, 0, 0, 0, 0, 0, 2],
    [1, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]


def       a				(      __a			, __a			, __a			, __a       )  ->     bool:



					'''simple docstring'''






					for i in range(9       ):
										if grid[row][i] == n or grid[i][column] == n:
															return False

					for i in range(3       ):
										for j in range(3       ):
															if grid[(row - row % 3) + i][(column - column % 3) + j] == n:
																				return False

					return True


def       a				(      __a       )  ->     tuple[int, int] | None:



					'''simple docstring'''






					for i in range(9       ):
										for j in range(9       ):
															if grid[i][j] == 0:
																				return i, j
					return None


def       a				(      __a       )  ->     Matrix | None:



					'''simple docstring'''






					if location := find_empty_location(__a       ):
										UpperCamelCase__ ,    UpperCamelCase__       :Optional[int]							 =			location
					else:
										# If the location is ``None``, then the grid is solved.
										return grid

					for digit in range(1			, 10       ):
										if is_safe(__a			, __a			, __a			, __a       ):
															UpperCamelCase__       :Tuple							 =			digit

															if sudoku(__a       ) is not None:
																				return grid

															UpperCamelCase__       :Union[str, Any]							 =			0

					return None


def       a				(      __a       )  ->     None:



					'''simple docstring'''






					for row in grid:
										for cell in row:
															print(__a			, end=''' '''       )
										print()


if __name__ == "__main__":
		# make a copy of grid so that you can compare with the unmodified grid
		for example_grid in (initial_grid, no_solution):
				print('''\nExample grid:\n''' + '''=''' * 20)
				print_solution(example_grid)
				print('''\nExample grid solution:''')
				__snake_case											=					sudoku(example_grid)
				if solution is not None:
						print_solution(solution)
				else:
						print('''Cannot find a solution.''')