'''simple docstring'''




import numpy as np

def _UpperCamelCase   (       SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_ = 1e-12  ,       SCREAMING_SNAKE_CASE_ = 100  ,       ):
     assert np.shape(SCREAMING_SNAKE_CASE_							)[0] == np.shape(SCREAMING_SNAKE_CASE_							)[1]
     # Ensure proper dimensionality.
     assert np.shape(SCREAMING_SNAKE_CASE_							)[0] == np.shape(SCREAMING_SNAKE_CASE_							)[0]
     # Ensure inputs are either both complex or both real
     assert np.iscomplexobj(SCREAMING_SNAKE_CASE_							) == np.iscomplexobj(SCREAMING_SNAKE_CASE_							)
     lowercase_					:       Union[str, Any]     				=					np.iscomplexobj(SCREAMING_SNAKE_CASE_							)
     if is_complex:
          # Ensure complex input_matrix is Hermitian
          assert np.array_equal(SCREAMING_SNAKE_CASE_  ,       input_matrix.conj().T							)

     # Set convergence to False. Will define convergence when we exceed max_iterations
     # or when we have small changes from one iteration to next.

     lowercase_					:       List[str]     				=					False
     lowercase_					:       Optional[int]     				=					0
     lowercase_					:       List[str]     				=					0
     lowercase_					:       str     				=					1e12

     while not convergence:
          # Multiple matrix by the vector.
          lowercase_					:       int     				=					np.dot(SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							)
          # Normalize the resulting output vector.
          lowercase_					:       Optional[Any]     				=					w / np.linalg.norm(SCREAMING_SNAKE_CASE_							)
          # Find rayleigh quotient
          # (faster than usual b/c we know vector is normalized already)
          lowercase_					:       Optional[Any]     				=					vector.conj().T if is_complex else vector.T
          lowercase_					:       Dict     				=					np.dot(SCREAMING_SNAKE_CASE_  ,       np.dot(SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							)							)

          # Check convergence.
          lowercase_					:       List[str]     				=					np.abs(lambda_ - lambda_previous							) / lambda_
          iterations += 1

          if error <= error_tol or iterations >= max_iterations:
               lowercase_					:       Optional[Any]     				=					True

          lowercase_					:       str     				=					lambda_

     if is_complex:
          lowercase_					:       List[Any]     				=					np.real(lambda_							)

     return lambda_, vector

def _UpperCamelCase   (       ):
     lowercase_					:       Optional[Any]     				=					np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]							)
     lowercase_					:       Dict     				=					np.array([41, 4, 20]							)
     lowercase_					:       Tuple     				=					real_input_matrix.astype(np.complexaaa							)
     lowercase_					:       List[Any]     				=					np.triu(1J * complex_input_matrix  ,       1							)
     complex_input_matrix += imag_matrix
     complex_input_matrix += -1 * imag_matrix.T
     lowercase_					:       Dict     				=					np.array([41, 4, 20]							).astype(np.complexaaa							)

     for problem_type in ["real", "complex"]:
          if problem_type == "real":
               lowercase_					:       Any     				=					real_input_matrix
               lowercase_					:       List[str]     				=					real_vector
          elif problem_type == "complex":
               lowercase_					:       int     				=					complex_input_matrix
               lowercase_					:       Dict     				=					complex_vector

          # Our implementation.
          lowercase_					:       Union[str, Any]     				=					power_iteration(SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							)

          # Numpy implementation.

          # Get eigenvalues and eigenvectors using built-in numpy
          # eigh (eigh used for symmetric or hermetian matrices).
          lowercase_					:       int     				=					np.linalg.eigh(SCREAMING_SNAKE_CASE_							)
          # Last eigenvalue is the maximum one.
          lowercase_					:       int     				=					eigen_values[-1]
          # Last column in this matrix is eigenvector corresponding to largest eigenvalue.
          lowercase_					:       List[str]     				=					eigen_vectors[:, -1]

          # Check our implementation and numpy gives close answers.
          assert np.abs(eigen_value - eigen_value_max							) <= 1e-6
          # Take absolute values element wise of each eigenvector.
          # as they are only unique to a minus sign.
          assert np.linalg.norm(np.abs(SCREAMING_SNAKE_CASE_							) - np.abs(SCREAMING_SNAKE_CASE_							)							) <= 1e-6


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 test_power_iteration()





'''simple docstring'''




from __future__ import annotations

def _UpperCamelCase   (       SCREAMING_SNAKE_CASE_							):

     # preprocessing the first row
     for i in range(1  ,       len(matrix[0]							)							):
          matrix[0][i] += matrix[0][i - 1]

     # preprocessing the first column
     for i in range(1  ,       len(SCREAMING_SNAKE_CASE_							)							):
          matrix[i][0] += matrix[i - 1][0]

     # updating the path cost for current position
     for i in range(1  ,       len(SCREAMING_SNAKE_CASE_							)							):
          for j in range(1  ,       len(matrix[0]							)							):
               matrix[i][j] += min(matrix[i - 1][j]  ,       matrix[i][j - 1]							)

     return matrix[-1][-1]


if __name__ == "__main__":
 import doctest

 doctest.testmod()




