
def 		_lowerCamelCase       (   _a			,   _a			,   _a			,   _a			,   _a			,   _a			,   ):
  """simple docstring"""




  _lowerCamelCase      =						list(lowercase__  )
  _lowerCamelCase      =						list(lowercase__  )
  _lowerCamelCase      =						len(lowercase__  )
  _lowerCamelCase      =						len(lowercase__  )

  _lowerCamelCase      =						[
      [0 for _ in range(len_destination_seq + 1  )] for _ in range(len_source_seq + 1  )
  ]
  _lowerCamelCase      =						[
      ['''0''' for _ in range(len_destination_seq + 1  )] for _ in range(len_source_seq + 1  )
  ]

  for i in range(1			,   len_source_seq + 1  ):
    _lowerCamelCase      =						i * delete_cost
    _lowerCamelCase      =						F'''D{source_seq[i - 1]:c}'''

  for i in range(1			,   len_destination_seq + 1  ):
    _lowerCamelCase      =						i * insert_cost
    _lowerCamelCase      =						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]:
        _lowerCamelCase      =						costs[i - 1][j - 1] + copy_cost
        _lowerCamelCase      =						F'''C{source_seq[i - 1]:c}'''
      else:
        _lowerCamelCase      =						costs[i - 1][j - 1] + replace_cost
        _lowerCamelCase      =						F'''R{source_seq[i - 1]:c}''' + str(destination_seq[j - 1]  )

      if costs[i - 1][j] + delete_cost < costs[i][j]:
        _lowerCamelCase      =						costs[i - 1][j] + delete_cost
        _lowerCamelCase      =						F'''D{source_seq[i - 1]:c}'''

      if costs[i][j - 1] + insert_cost < costs[i][j]:
        _lowerCamelCase      =						costs[i][j - 1] + insert_cost
        _lowerCamelCase      =						F'''I{destination_seq[j - 1]:c}'''

  return costs, ops


def 		_lowerCamelCase       (   _a			,   _a			,   _a  ):
  """simple docstring"""




  if i == 0 and j == 0:
    return []
  else:
    if ops[i][j][0] == "C" or ops[i][j][0] == "R":
      _lowerCamelCase      =						assemble_transformation(lowercase__			,   i - 1			,   j - 1  )
      seq.append(ops[i][j]  )
      return seq
    elif ops[i][j][0] == "D":
      _lowerCamelCase      =						assemble_transformation(lowercase__			,   i - 1			,   lowercase__  )
      seq.append(ops[i][j]  )
      return seq
    else:
      _lowerCamelCase      =						assemble_transformation(lowercase__			,   lowercase__			,   j - 1  )
      seq.append(ops[i][j]  )
      return seq


if __name__ == "__main__":
  _UpperCAmelCase    			=					compute_transform_tables("Python", "Algorithms", -1, 1, 2, 2)

  _UpperCAmelCase    			=					len(operations)
  _UpperCAmelCase    			=					len(operations[0])
  _UpperCAmelCase    			=					assemble_transformation(operations, m - 1, n - 1)

  _UpperCAmelCase    			=					list("Python")
  _UpperCAmelCase    			=					0
  _UpperCAmelCase    			=					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":
        _UpperCAmelCase    			=					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))









from typing import Any

import numpy as np




def    UpperCAmelCase					(  lowercase__						:					np.ndarray	):



			'''simple docstring'''





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

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

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

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




def    UpperCAmelCase					(  lowercase__						:					int	):



			'''simple docstring'''





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




def    UpperCAmelCase					(  lowercase__						:					np.ndarray				,					lowercase__						:					np.ndarray				,					lowercase__						:					int = 1000				,					lowercase__						:					float = 1e-8				,					):



			'''simple docstring'''





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

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

			# Set initial errors in solution guess and residual.
			a__					=  1e9
			a__					=  1e9
			a__					=  1e9

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

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

						# The main algorithm.

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

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

						# Update variables.
						a__					=  np.copy(lowercase__	)
						a__					=  np.copy(lowercase__	)
						a__					=  np.copy(lowercase__	)

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

			return x




def    UpperCAmelCase					(  ):



			'''simple docstring'''





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

			# Numpy solution.
			a__					=  np.linalg.solve(lowercase__				,					lowercase__	)

			# Our implementation.
			a__					=  conjugate_gradient(lowercase__				,					lowercase__	)

			# 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()

