

import numpy as np
def      a      (			A__	:   np.ndarray      ,   A__	:   np.ndarray      ,   A__	:   float = 1e-12      ,   A__	:   int = 100      ,   )    ->      tuple[float, np.ndarray]:





      """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__     )
      _lowercase	       =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.

      _lowercase	       =False
      _lowercase	       =0
      _lowercase	       =0
      _lowercase	       =1e12

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

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

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

            _lowercase	       =lambda_

      if is_complex:
            _lowercase	       =np.real(lambda_     )

      return lambda_, vector
def      a      (			)    ->      None:





      """simple docstring"""



      _lowercase	       =np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]     )
      _lowercase	       =np.array([41, 4, 20]     )
      _lowercase	       =real_input_matrix.astype(np.complexaaa     )
      _lowercase	       =np.triu(1j * complex_input_matrix      ,   1     )
      complex_input_matrix += imag_matrix
      complex_input_matrix += -1 * imag_matrix.T
      _lowercase	       =np.array([41, 4, 20]     ).astype(np.complexaaa     )

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

            # Our implementation.
            _lowercase       ,       _lowercase	       =power_iteration(A__      ,   A__     )

            # Numpy implementation.

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





import torch
import torch.nn as nn
from transformers.modeling_utils import ModuleUtilsMixin
from transformers.models.ta.modeling_ta import TaBlock, TaConfig, TaLayerNorm

from ...configuration_utils import ConfigMixin, register_to_config
from ...models import ModelMixin





class  __lowerCAmelCase     (					SCREAMING_SNAKE_CASE			,			SCREAMING_SNAKE_CASE			,			SCREAMING_SNAKE_CASE							):




  @register_to_config
  def __init__(				self							,   lowerCAmelCase							,   lowerCAmelCase							,   lowerCAmelCase							,   lowerCAmelCase							,   lowerCAmelCase							,   lowerCAmelCase							,   lowerCAmelCase							,   lowerCAmelCase							,   lowerCAmelCase							,   lowerCAmelCase = False							,   )  -> Tuple:
        '''simple docstring'''






        super().__init__()

        _lowercase	       =nn.Embedding(lowerCAmelCase							,   lowerCAmelCase					)

        _lowercase	       =nn.Embedding(lowerCAmelCase							,   lowerCAmelCase					)
        _lowercase	       =False

        _lowercase	       =nn.Dropout(p=lowerCAmelCase					)

        _lowercase	       =TaConfig(
            vocab_size=lowerCAmelCase							,   d_model=lowerCAmelCase							,   num_heads=lowerCAmelCase							,   d_kv=lowerCAmelCase							,   d_ff=lowerCAmelCase							,   dropout_rate=lowerCAmelCase							,   feed_forward_proj=lowerCAmelCase							,   is_decoder=lowerCAmelCase							,   is_encoder_decoder=lowerCAmelCase							,   )

        _lowercase	       =nn.ModuleList()
        for lyr_num in range(lowerCAmelCase					):
              _lowercase	       =TaBlock(lowerCAmelCase					)
              self.encoders.append(lowerCAmelCase					)

        _lowercase	       =TaLayerNorm(lowerCAmelCase					)
        _lowercase	       =nn.Dropout(p=lowerCAmelCase					)




  def   A__     (				self							,   lowerCAmelCase							,   lowerCAmelCase					)  -> Dict:
        '''simple docstring'''






        _lowercase	       =self.token_embedder(lowerCAmelCase					)

        _lowercase	       =encoder_input_tokens.shape[1]
        _lowercase	       =torch.arange(lowerCAmelCase							,   device=encoder_input_tokens.device					)
        x += self.position_encoding(lowerCAmelCase					)

        _lowercase	       =self.dropout_pre(lowerCAmelCase					)

        # inverted the attention mask
        _lowercase	       =encoder_input_tokens.size()
        _lowercase	       =self.get_extended_attention_mask(lowerCAmelCase							,   lowerCAmelCase					)

        for lyr in self.encoders:
              _lowercase	       =lyr(lowerCAmelCase							,   lowerCAmelCase					)[0]
        _lowercase	       =self.layer_norm(lowerCAmelCase					)

        return self.dropout_post(lowerCAmelCase					), encoder_inputs_mask


