





'''simple docstring'''

from typing import Any

import numpy as np


def 						__UpperCamelCase						(			_UpperCAmelCase  ):
		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	:							int     =     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 						__UpperCamelCase						(			_UpperCAmelCase  ):
		__UpperCAmelCase	:							str     =     np.random.randn(_UpperCAmelCase,       _UpperCAmelCase  )
		__UpperCAmelCase	:							int     =     np.dot(_UpperCAmelCase,       random_matrix.T  )
		assert _is_matrix_spd(_UpperCAmelCase  )
		return spd_matrix


def 						__UpperCamelCase						(			_UpperCAmelCase,       _UpperCAmelCase,       _UpperCAmelCase = 1000,       _UpperCAmelCase = 1E-8,       ):
		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	:							List[Any]     =     np.zeros((np.shape(_UpperCAmelCase  )[0], 1)  )
		__UpperCAmelCase	:							Dict     =     np.copy(_UpperCAmelCase  )
		__UpperCAmelCase	:							Optional[int]     =     np.copy(_UpperCAmelCase  )

		# Set initial errors in solution guess and residual.
		__UpperCAmelCase	:							int     =     1E9
		__UpperCAmelCase	:							Optional[int]     =     1E9
		__UpperCAmelCase	:							Tuple     =     1E9

		# Set iteration counter to threshold number of iterations.
		__UpperCAmelCase	:							Optional[Any]     =     0

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

				# The main algorithm.

				# Update search direction magnitude.
				__UpperCAmelCase	:							List[Any]     =     np.dot(ra.T,       _UpperCAmelCase  ) / np.dot(pa.T,       _UpperCAmelCase  )
				# Update solution guess.
				__UpperCAmelCase	:							Dict     =     xa + alpha * pa
				# Calculate new residual.
				__UpperCAmelCase	:							Any     =     ra - alpha * w
				# Calculate new Krylov subspace scale.
				__UpperCAmelCase	:							Tuple     =     np.dot(r.T,       _UpperCAmelCase  ) / np.dot(ra.T,       _UpperCAmelCase  )
				# Calculate new A conjuage search direction.
				__UpperCAmelCase	:							Dict     =     r + beta * pa

				# Calculate errors.
				__UpperCAmelCase	:							Union[str, Any]     =     np.linalg.norm(r - ra  )
				__UpperCAmelCase	:							Optional[Any]     =     np.linalg.norm(x - xa  )
				__UpperCAmelCase	:							List[str]     =     np.maximum(_UpperCAmelCase,       _UpperCAmelCase  )

				# Update variables.
				__UpperCAmelCase	:							Dict     =     np.copy(_UpperCAmelCase  )
				__UpperCAmelCase	:							List[Any]     =     np.copy(_UpperCAmelCase  )
				__UpperCAmelCase	:							str     =     np.copy(_UpperCAmelCase  )

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

		return x


def 						__UpperCamelCase						(			):
		__UpperCAmelCase	:							Tuple     =     3
		__UpperCAmelCase	:							Union[str, Any]     =     _create_spd_matrix(_UpperCAmelCase  )
		__UpperCAmelCase	:							Union[str, Any]     =     np.random.randn(_UpperCAmelCase,       1  )
		__UpperCAmelCase	:							List[Any]     =     np.dot(_UpperCAmelCase,       _UpperCAmelCase  )

		# Numpy solution.
		__UpperCAmelCase	:							str     =     np.linalg.solve(_UpperCAmelCase,       _UpperCAmelCase  )

		# Our implementation.
		__UpperCAmelCase	:							List[Any]     =     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'''

def 						__UpperCamelCase						(			_UpperCAmelCase,       _UpperCAmelCase,       _UpperCAmelCase,       _UpperCAmelCase,       _UpperCAmelCase,       _UpperCAmelCase,       ):
		__UpperCAmelCase	:							Optional[int]     =     list(_UpperCAmelCase  )
		__UpperCAmelCase	:							str     =     list(_UpperCAmelCase  )
		__UpperCAmelCase	:							str     =     len(_UpperCAmelCase  )
		__UpperCAmelCase	:							Dict     =     len(_UpperCAmelCase  )

		__UpperCAmelCase	:							int     =     [
		    [0 for _ in range(len_destination_seq + 1  )] for _ in range(len_source_seq + 1  )
		]
		__UpperCAmelCase	:							Any     =     [
		    ["0" for _ in range(len_destination_seq + 1  )] for _ in range(len_source_seq + 1  )
		]

		for i in range(1,       len_source_seq + 1  ):
				__UpperCAmelCase	:							Optional[Any]     =     i * delete_cost
				__UpperCAmelCase	:							Optional[int]     =     F"D{source_seq[i - 1]:c}"

		for i in range(1,       len_destination_seq + 1  ):
				__UpperCAmelCase	:							Optional[int]     =     i * insert_cost
				__UpperCAmelCase	:							List[Any]     =     F"I{destination_seq[i - 1]:c}"

		for i in range(1,       len_source_seq + 1  ):
				for j in range(1,       len_destination_seq + 1  ):
						if source_seq[i - 1] == destination_seq[j - 1]:
								__UpperCAmelCase	:							List[str]     =     costs[i - 1][j - 1] + copy_cost
								__UpperCAmelCase	:							Tuple     =     F"C{source_seq[i - 1]:c}"
						else:
								__UpperCAmelCase	:							Optional[int]     =     costs[i - 1][j - 1] + replace_cost
								__UpperCAmelCase	:							Dict     =     F"R{source_seq[i - 1]:c}" + str(destination_seq[j - 1]  )

						if costs[i - 1][j] + delete_cost < costs[i][j]:
								__UpperCAmelCase	:							Union[str, Any]     =     costs[i - 1][j] + delete_cost
								__UpperCAmelCase	:							int     =     F"D{source_seq[i - 1]:c}"

						if costs[i][j - 1] + insert_cost < costs[i][j]:
								__UpperCAmelCase	:							List[str]     =     costs[i][j - 1] + insert_cost
								__UpperCAmelCase	:							Dict     =     F"I{destination_seq[j - 1]:c}"

		return costs, ops


def 						__UpperCamelCase						(			_UpperCAmelCase,       _UpperCAmelCase,       _UpperCAmelCase  ):
		if i == 0 and j == 0:
				return []
		else:
				if ops[i][j][0] == "C" or ops[i][j][0] == "R":
						__UpperCAmelCase	:							int     =     assemble_transformation(_UpperCAmelCase,       i - 1,       j - 1  )
						seq.append(ops[i][j]  )
						return seq
				elif ops[i][j][0] == "D":
						__UpperCAmelCase	:							List[Any]     =     assemble_transformation(_UpperCAmelCase,       i - 1,       _UpperCAmelCase  )
						seq.append(ops[i][j]  )
						return seq
				else:
						__UpperCAmelCase	:							Optional[Any]     =     assemble_transformation(_UpperCAmelCase,       _UpperCAmelCase,       j - 1  )
						seq.append(ops[i][j]  )
						return seq


if __name__ == "__main__":
			lowerCAmelCase__	,			lowerCAmelCase__      :   Optional[int]      	=			compute_transform_tables("Python", "Algorithms", -1, 1, 2, 2)

			lowerCAmelCase__      :   Optional[Any]      	=			len(operations)
			lowerCAmelCase__      :   int      	=			len(operations[0])
			lowerCAmelCase__      :   Dict      	=			assemble_transformation(operations, m - 1, n - 1)

			lowerCAmelCase__      :   str      	=			list("Python")
			lowerCAmelCase__      :   Any      	=			0
			lowerCAmelCase__      :   Dict      	=			0

			with open("min_cost.txt", "w") as file:
						for op in sequence:
									print("".join(string))

									if op[0] == "C":
												file.write("%-16s" % "Copy %c" % op[1])
												file.write("\t\t\t" + "".join(string))
												file.write("\r\n")

												cost -= 1
									elif op[0] == "R":
												lowerCAmelCase__      :   Optional[int]      	=			op[2]

												file.write("%-16s" % ("Replace %c" % op[1] + " with " + str(op[2])))
												file.write("\t\t" + "".join(string))
												file.write("\r\n")

												cost += 1
									elif op[0] == "D":
												string.pop(i)

												file.write("%-16s" % "Delete %c" % op[1])
												file.write("\t\t\t" + "".join(string))
												file.write("\r\n")

												cost += 2
									else:
												string.insert(i, op[1])

												file.write("%-16s" % "Insert %c" % op[1])
												file.write("\t\t\t" + "".join(string))
												file.write("\r\n")

												cost += 2

									i += 1

						print("".join(string))
						print("Cost: ", cost)

						file.write("\r\nMinimum cost: " + str(cost))




