

import numpy as np




def     __UpperCamelCase			(	_A :				str	,   _A :				List[Any]	,   _A :				Optional[Any] = 1E-12	,   _A :				List[Any] = 100	,   )							->Dict:



							"""simple docstring"""
							assert np.shape(__A						)[0] == np.shape(__A						)[1]
							# Ensure proper dimensionality.
							assert np.shape(__A						)[0] == np.shape(__A						)[0]
							# Ensure inputs are either both complex or both real
							assert np.iscomplexobj(__A						) == np.iscomplexobj(__A						)
							lowerCamelCase_				     =np.iscomplexobj(__A						)
							if is_complex:
														# Ensure complex input_matrix is Hermitian
														assert np.array_equal(__A	,   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.

							lowerCamelCase_				     =False
							lowerCamelCase_				     =0
							lowerCamelCase_				     =0
							lowerCamelCase_				     =1E12

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

														# Check convergence.
														lowerCamelCase_				     =np.abs(lambda_ - lambda_previous						) / lambda_
														iterations += 1

														if error <= error_tol or iterations >= max_iterations:
																					lowerCamelCase_				     =True

														lowerCamelCase_				     =lambda_

							if is_complex:
														lowerCamelCase_				     =np.real(lambda_						)

							return lambda_, vector




def     __UpperCamelCase			(	)							->Optional[int]:



							"""simple docstring"""
							lowerCamelCase_				     =np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]						)
							lowerCamelCase_				     =np.array([41, 4, 20]						)
							lowerCamelCase_				     =real_input_matrix.astype(np.complexaaa						)
							lowerCamelCase_				     =np.triu(1J * complex_input_matrix	,   1						)
							complex_input_matrix += imag_matrix
							complex_input_matrix += -1 * imag_matrix.T
							lowerCamelCase_				     =np.array([41, 4, 20]						).astype(np.complexaaa						)

							for problem_type in ["real", "complex"]:
														if problem_type == "real":
																					lowerCamelCase_				     =real_input_matrix
																					lowerCamelCase_				     =real_vector
														elif problem_type == "complex":
																					lowerCamelCase_				     =complex_input_matrix
																					lowerCamelCase_				     =complex_vector

														# Our implementation.
														lowerCamelCase_     ,					lowerCamelCase_				     =power_iteration(__A	,   __A						)

														# Numpy implementation.

														# Get eigenvalues and eigenvectors using built-in numpy
														# eigh (eigh used for symmetric or hermetian matrices).
														lowerCamelCase_     ,					lowerCamelCase_				     =np.linalg.eigh(__A						)
														# Last eigenvalue is the maximum one.
														lowerCamelCase_				     =eigen_values[-1]
														# Last column in this matrix is eigenvector corresponding to largest eigenvalue.
														lowerCamelCase_				     =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(__A						) - np.abs(__A						)						) <= 1E-6


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	test_power_iteration()





'''simple docstring'''





import random




def lowerCAmelCase  (__A):



     """simple docstring"""



     _a      =  num - 1
     _a      =  0

     while s % 2 == 0:
          _a      =  s // 2
          t += 1

     for _ in range(5):
          _a      =  random.randrange(2  ,	num - 1)
          _a      =  pow(__A  ,	__A  ,	__A)
          if v != 1:
               _a      =  0
               while v != (num - 1):
                    if i == t - 1:
                         return False
                    else:
                         _a      =  i + 1
                         _a      =  (v**2) % num
     return True




def lowerCAmelCase  (__A):



     """simple docstring"""



     if num < 2:
          return False

     _a      =  [
         2,
         3,
         5,
         7,
         11,
         13,
         17,
         19,
         23,
         29,
         31,
         37,
         41,
         43,
         47,
         53,
         59,
         61,
         67,
         71,
         73,
         79,
         83,
         89,
         97,
         101,
         103,
         107,
         109,
         113,
         127,
         131,
         137,
         139,
         149,
         151,
         157,
         163,
         167,
         173,
         179,
         181,
         191,
         193,
         197,
         199,
         211,
         223,
         227,
         229,
         233,
         239,
         241,
         251,
         257,
         263,
         269,
         271,
         277,
         281,
         283,
         293,
         307,
         311,
         313,
         317,
         331,
         337,
         347,
         349,
         353,
         359,
         367,
         373,
         379,
         383,
         389,
         397,
         401,
         409,
         419,
         421,
         431,
         433,
         439,
         443,
         449,
         457,
         461,
         463,
         467,
         479,
         487,
         491,
         499,
         503,
         509,
         521,
         523,
         541,
         547,
         557,
         563,
         569,
         571,
         577,
         587,
         593,
         599,
         601,
         607,
         613,
         617,
         619,
         631,
         641,
         643,
         647,
         653,
         659,
         661,
         673,
         677,
         683,
         691,
         701,
         709,
         719,
         727,
         733,
         739,
         743,
         751,
         757,
         761,
         769,
         773,
         787,
         797,
         809,
         811,
         821,
         823,
         827,
         829,
         839,
         853,
         857,
         859,
         863,
         877,
         881,
         883,
         887,
         907,
         911,
         919,
         929,
         937,
         941,
         947,
         953,
         967,
         971,
         977,
         983,
         991,
         997,
     ]

     if num in low_primes:
          return True

     for prime in low_primes:
          if (num % prime) == 0:
               return False

     return rabin_miller(__A)




def lowerCAmelCase  (__A = 1_024):



     """simple docstring"""



     while True:
          _a      =  random.randrange(2 ** (keysize - 1)  ,	2 ** (keysize))
          if is_prime_low_num(__A):
               return num


if __name__ == "__main__":
  lowercase_		       =  generate_large_prime()
  print(("Prime number:", num))
  print(("is_prime_low_num:", is_prime_low_num(num)))



