from typing import Any

import numpy as np


def SCREAMING_SNAKE_CASE					(  lowercase_:						np.ndarray					):
       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.
       _UpperCamelCase    ,   _UpperCamelCase  :							List[str]	    =	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 SCREAMING_SNAKE_CASE					(  lowercase_:						int					):
       _UpperCamelCase  :							Dict	    =	np.random.randn(lowercase_		,    lowercase_					)
       _UpperCamelCase  :							Any	    =	np.dot(lowercase_		,    random_matrix.T					)
       assert _is_matrix_spd(lowercase_					)
       return spd_matrix


def SCREAMING_SNAKE_CASE					(  lowercase_:						np.ndarray		,    lowercase_:						np.ndarray		,    lowercase_:						int = 1000		,    lowercase_:						float = 1e-8		,    ):
       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.
       _UpperCamelCase  :							Dict	    =	np.zeros((np.shape(lowercase_					)[0], 1)					)
       _UpperCamelCase  :							Optional[Any]	    =	np.copy(lowercase_					)
       _UpperCamelCase  :							Optional[Any]	    =	np.copy(lowercase_					)

       # Set initial errors in solution guess and residual.
       _UpperCamelCase  :							Tuple	    =	1e9
       _UpperCamelCase  :							Optional[int]	    =	1e9
       _UpperCamelCase  :							List[str]	    =	1e9

       # Set iteration counter to threshold number of iterations.
       _UpperCamelCase  :							List[str]	    =	0

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

              # The main algorithm.

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

              # Calculate errors.
              _UpperCamelCase  :							int	    =	np.linalg.norm(r - ra					)
              _UpperCamelCase  :							Union[str, Any]	    =	np.linalg.norm(x - xa					)
              _UpperCamelCase  :							Dict	    =	np.maximum(lowercase_		,    lowercase_					)

              # Update variables.
              _UpperCamelCase  :							Union[str, Any]	    =	np.copy(lowercase_					)
              _UpperCamelCase  :							Any	    =	np.copy(lowercase_					)
              _UpperCamelCase  :							Optional[Any]	    =	np.copy(lowercase_					)

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

       return x


def SCREAMING_SNAKE_CASE					(  ):
       _UpperCamelCase  :							Optional[Any]	    =	3
       _UpperCamelCase  :							List[str]	    =	_create_spd_matrix(lowercase_					)
       _UpperCamelCase  :							Tuple	    =	np.random.randn(lowercase_		,    1					)
       _UpperCamelCase  :							List[Any]	    =	np.dot(lowercase_		,    lowercase_					)

       # Numpy solution.
       _UpperCamelCase  :							int	    =	np.linalg.solve(lowercase_		,    lowercase_					)

       # Our implementation.
       _UpperCamelCase  :							List[Any]	    =	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()






"""simple docstring"""
#
# Sort large text files in a minimum amount of memory
#
import argparse
import os

class        _UpperCamelCase	:


  """simple docstring"""





  snake_case_        =    'block_{0}.dat'



  def __init__( self					: int       ,       snake_case					: int      )	->			List[str]:


         '''simple docstring'''





         __magic_name__			:		Optional[int]        =     filename
         __magic_name__			:		Any        =     []



  def 							_UpperCAmelCase      ( self					: Optional[Any]       ,       snake_case					: int       ,       snake_case					: Any      )	->			List[Any]:


         '''simple docstring'''





         __magic_name__			:		Optional[Any]        =     self.BLOCK_FILENAME_FORMAT.format(snake_case      )
         with open(snake_case       ,       '''w'''      ) as file:
                file.write(snake_case      )
         self.block_filenames.append(snake_case      )



  def 							_UpperCAmelCase      ( self					: str      )	->			List[str]:


         '''simple docstring'''





         return self.block_filenames



  def 							_UpperCAmelCase      ( self					: Union[str, Any]       ,       snake_case					: Dict       ,       snake_case					: int=None      )	->			Union[str, Any]:


         '''simple docstring'''





         __magic_name__			:		List[str]        =     0
         with open(self.filename      ) as file:
                while True:
                       __magic_name__			:		Tuple        =     file.readlines(snake_case      )

                       if lines == []:
                              break

                       if sort_key is None:
                              lines.sort()
                       else:
                              lines.sort(key=snake_case      )

                       self.write_block(''''''.join(snake_case      )       ,       snake_case      )
                       i += 1




  def 							_UpperCAmelCase      ( self					: Tuple      )	->			Optional[int]:


         '''simple docstring'''





         map(os.remove       ,       self.block_filenames      )

class        _UpperCamelCase	:


  """simple docstring"""





  def 							_UpperCAmelCase      ( self					: List[Any]       ,       snake_case					: Dict      )	->			int:


         '''simple docstring'''





         __magic_name__			:		Any        =     -1
         __magic_name__			:		Union[str, Any]        =     None

         for i in range(len(snake_case      )      ):
                if min_str is None or choices[i] < min_str:
                       __magic_name__			:		Dict        =     i

         return min_index

class        _UpperCamelCase	:


  """simple docstring"""





  def __init__( self					: str       ,       snake_case					: Optional[int]      )	->			Any:


         '''simple docstring'''





         __magic_name__			:		Dict        =     files
         __magic_name__			:		List[str]        =     set()
         __magic_name__			:		Any        =     len(snake_case      )
         __magic_name__			:		int        =     {i: None for i in range(self.num_buffers      )}



  def 							_UpperCAmelCase      ( self					: List[str]      )	->			Optional[Any]:


         '''simple docstring'''





         return {
             i: self.buffers[i] for i in range(self.num_buffers      ) if i not in self.empty
         }



  def 							_UpperCAmelCase      ( self					: int      )	->			int:


         '''simple docstring'''





         for i in range(self.num_buffers      ):
                if self.buffers[i] is None and i not in self.empty:
                       __magic_name__			:		int        =     self.files[i].readline()

                       if self.buffers[i] == "":
                              self.empty.add(snake_case      )
                              self.files[i].close()

         if len(self.empty      ) == self.num_buffers:
                return False

         return True




  def 							_UpperCAmelCase      ( self					: Dict       ,       snake_case					: Optional[Any]      )	->			Optional[Any]:


         '''simple docstring'''





         __magic_name__			:		List[str]        =     self.buffers[index]
         __magic_name__			:		Dict        =     None

         return value

class        _UpperCamelCase	:


  """simple docstring"""





  def __init__( self					: Any       ,       snake_case					: str      )	->			Dict:


         '''simple docstring'''





         __magic_name__			:		Optional[Any]        =     merge_strategy



  def 							_UpperCAmelCase      ( self					: Optional[int]       ,       snake_case					: int       ,       snake_case					: Dict       ,       snake_case					: Dict      )	->			Union[str, Any]:


         '''simple docstring'''





         __magic_name__			:		Union[str, Any]        =     FilesArray(self.get_file_handles(snake_case       ,       snake_case      )      )
         with open(snake_case       ,       '''w'''       ,       snake_case      ) as outfile:
                while buffers.refresh():
                       __magic_name__			:		List[str]        =     self.merge_strategy.select(buffers.get_dict()      )
                       outfile.write(buffers.unshift(snake_case      )      )




  def 							_UpperCAmelCase      ( self					: Union[str, Any]       ,       snake_case					: List[str]       ,       snake_case					: Optional[int]      )	->			Union[str, Any]:


         '''simple docstring'''





         __magic_name__			:		str        =     {}

         for i in range(len(snake_case      )      ):
                __magic_name__			:		Any        =     open(filenames[i]       ,       '''r'''       ,       snake_case      )  # noqa: UP015

         return files

class        _UpperCamelCase	:


  """simple docstring"""





  def __init__( self					: Optional[Any]       ,       snake_case					: Union[str, Any]      )	->			Optional[int]:


         '''simple docstring'''





         __magic_name__			:		int        =     block_size



  def 							_UpperCAmelCase      ( self					: str       ,       snake_case					: Optional[int]       ,       snake_case					: Dict=None      )	->			Optional[Any]:


         '''simple docstring'''





         __magic_name__			:		Tuple        =     self.get_number_blocks(snake_case       ,       self.block_size      )
         __magic_name__			:		str        =     FileSplitter(snake_case      )
         splitter.split(self.block_size       ,       snake_case      )

         __magic_name__			:		int        =     FileMerger(NWayMerge()      )
         __magic_name__			:		Any        =     self.block_size / (num_blocks + 1)
         merger.merge(splitter.get_block_filenames()       ,       filename + '''.out'''       ,       snake_case      )

         splitter.cleanup()




  def 							_UpperCAmelCase      ( self					: Tuple       ,       snake_case					: List[str]       ,       snake_case					: Union[str, Any]      )	->			Tuple:


         '''simple docstring'''





         return (os.stat(snake_case      ).st_size / block_size) + 1







def       UpperCamelCase_					(     lowerCamelCase							:					Union[str, Any]   )						->     Union[str, Any]:



       """simple docstring"""







       if string[-1].lower() == "k":
              return int(string[:-1]   ) * 1024
       elif string[-1].lower() == "m":
              return int(string[:-1]   ) * 1024 * 1024
       elif string[-1].lower() == "g":
              return int(string[:-1]   ) * 1024 * 1024 * 1024
       else:
              return int(lowerCamelCase   )







def       UpperCamelCase_					(     )						->     Optional[Any]:



       """simple docstring"""







       __magic_name__			:		Tuple        =     argparse.ArgumentParser()
       parser.add_argument(
           '''-m'''     , '''--mem'''     , help='''amount of memory to use for sorting'''     , default='''100M'''   )
       parser.add_argument(
           '''filename'''     , metavar='''<filename>'''     , nargs=1     , help='''name of file to sort'''   )
       __magic_name__			:		List[str]        =     parser.parse_args()

       __magic_name__			:		Any        =     ExternalSort(parse_memory(args.mem   )   )
       sorter.sort(args.filename[0]   )


if __name__ == "__main__":
 main()
