





import numpy as np



def 		A_				(     _lowerCAmelCase							,   _lowerCAmelCase							,   _lowerCAmelCase = 1e-12							,   _lowerCAmelCase = 100							,   )						-> tuple[float, np.ndarray]:
 assert np.shape(_lowerCAmelCase     )[0] == np.shape(_lowerCAmelCase     )[1]
 # Ensure proper dimensionality.
 assert np.shape(_lowerCAmelCase     )[0] == np.shape(_lowerCAmelCase     )[0]
 # Ensure inputs are either both complex or both real
 assert np.iscomplexobj(_lowerCAmelCase     ) == np.iscomplexobj(_lowerCAmelCase     )
 UpperCamelCase	:					Optional[int]       	=   np.iscomplexobj(_lowerCAmelCase     )
 if is_complex:
  # Ensure complex input_matrix is Hermitian
  assert np.array_equal(_lowerCAmelCase							,   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.

 UpperCamelCase	:					str       	=   False
 UpperCamelCase	:					int       	=   0
 UpperCamelCase	:					Optional[int]       	=   0
 UpperCamelCase	:					Tuple       	=   1e12

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

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

  if error <= error_tol or iterations >= max_iterations:
   UpperCamelCase	:					str       	=   True

  UpperCamelCase	:					Union[str, Any]       	=   lambda_

 if is_complex:
  UpperCamelCase	:					Optional[Any]       	=   np.real(lambda_     )

 return lambda_, vector



def 		A_				(     )						-> None:
 UpperCamelCase	:					Any       	=   np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]     )
 UpperCamelCase	:					str       	=   np.array([41, 4, 20]     )
 UpperCamelCase	:					Optional[Any]       	=   real_input_matrix.astype(np.complexaaa     )
 UpperCamelCase	:					Dict       	=   np.triu(1j * complex_input_matrix							,   1     )
 complex_input_matrix += imag_matrix
 complex_input_matrix += -1 * imag_matrix.T
 UpperCamelCase	:					Optional[int]       	=   np.array([41, 4, 20]     ).astype(np.complexaaa     )

 for problem_type in ["real", "complex"]:
  if problem_type == "real":
   UpperCamelCase	:					int       	=   real_input_matrix
   UpperCamelCase	:					Any       	=   real_vector
  elif problem_type == "complex":
   UpperCamelCase	:					Union[str, Any]       	=   complex_input_matrix
   UpperCamelCase	:					Tuple       	=   complex_vector

  # Our implementation.
  UpperCamelCase	,    UpperCamelCase	:					List[Any]       	=   power_iteration(_lowerCAmelCase							,   _lowerCAmelCase     )

  # Numpy implementation.

  # Get eigenvalues and eigenvectors using built-in numpy
  # eigh (eigh used for symmetric or hermetian matrices).
  UpperCamelCase	,    UpperCamelCase	:					Optional[int]       	=   np.linalg.eigh(_lowerCAmelCase     )
  # Last eigenvalue is the maximum one.
  UpperCamelCase	:					Tuple       	=   eigen_values[-1]
  # Last column in this matrix is eigenvector corresponding to largest eigenvalue.
  UpperCamelCase	:					List[Any]       	=   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(_lowerCAmelCase     ) - np.abs(_lowerCAmelCase     )     ) <= 1e-6


if __name__ == "__main__":
    import doctest

    doctest.testmod()
    test_power_iteration()


from math import cos, sin, sqrt, tau

from audio_filters.iir_filter import IIRFilter


def     __lowerCAmelCase			(			a__						,    a__						,    a__ = 1 / sqrt(2  )  )       ->  IIRFilter:
				__a										=    tau * frequency / samplerate
				__a										=    sin(a__  )
				__a										=    cos(a__  )
				__a										=    _sin / (2 * q_factor)

				__a										=    (1 - _cos) / 2
				__a										=    1 - _cos

				__a										=    1 + alpha
				__a										=    -2 * _cos
				__a										=    1 - alpha

				__a										=    IIRFilter(2  )
				filt.set_coefficients([aa, aa, aa]						,    [ba, ba, ba]  )
				return filt


def     __lowerCAmelCase			(			a__						,    a__						,    a__ = 1 / sqrt(2  )  )       ->  IIRFilter:
				__a										=    tau * frequency / samplerate
				__a										=    sin(a__  )
				__a										=    cos(a__  )
				__a										=    _sin / (2 * q_factor)

				__a										=    (1 + _cos) / 2
				__a										=    -1 - _cos

				__a										=    1 + alpha
				__a										=    -2 * _cos
				__a										=    1 - alpha

				__a										=    IIRFilter(2  )
				filt.set_coefficients([aa, aa, aa]						,    [ba, ba, ba]  )
				return filt


def     __lowerCAmelCase			(			a__						,    a__						,    a__ = 1 / sqrt(2  )  )       ->  IIRFilter:
				__a										=    tau * frequency / samplerate
				__a										=    sin(a__  )
				__a										=    cos(a__  )
				__a										=    _sin / (2 * q_factor)

				__a										=    _sin / 2
				__a										=    0
				__a										=    -ba

				__a										=    1 + alpha
				__a										=    -2 * _cos
				__a										=    1 - alpha

				__a										=    IIRFilter(2  )
				filt.set_coefficients([aa, aa, aa]						,    [ba, ba, ba]  )
				return filt


def     __lowerCAmelCase			(			a__						,    a__						,    a__ = 1 / sqrt(2  )  )       ->  IIRFilter:
				__a										=    tau * frequency / samplerate
				__a										=    sin(a__  )
				__a										=    cos(a__  )
				__a										=    _sin / (2 * q_factor)

				__a										=    1 - alpha
				__a										=    -2 * _cos
				__a										=    1 + alpha

				__a										=    IIRFilter(2  )
				filt.set_coefficients([ba, ba, ba]						,    [ba, ba, ba]  )
				return filt


def     __lowerCAmelCase			(			a__						,    a__						,    a__						,    a__ = 1 / sqrt(2  )						,    )       ->  IIRFilter:
				__a										=    tau * frequency / samplerate
				__a										=    sin(a__  )
				__a										=    cos(a__  )
				__a										=    _sin / (2 * q_factor)
				__a										=    10 ** (gain_db / 40)

				__a										=    1 + alpha * big_a
				__a										=    -2 * _cos
				__a										=    1 - alpha * big_a
				__a										=    1 + alpha / big_a
				__a										=    -2 * _cos
				__a										=    1 - alpha / big_a

				__a										=    IIRFilter(2  )
				filt.set_coefficients([aa, aa, aa]						,    [ba, ba, ba]  )
				return filt


def     __lowerCAmelCase			(			a__						,    a__						,    a__						,    a__ = 1 / sqrt(2  )						,    )       ->  IIRFilter:
				__a										=    tau * frequency / samplerate
				__a										=    sin(a__  )
				__a										=    cos(a__  )
				__a										=    _sin / (2 * q_factor)
				__a										=    10 ** (gain_db / 40)
				__a										=    (big_a + 1) - (big_a - 1) * _cos
				__a										=    (big_a + 1) + (big_a - 1) * _cos
				__a										=    (big_a - 1) - (big_a + 1) * _cos
				__a										=    (big_a - 1) + (big_a + 1) * _cos
				__a										=    2 * sqrt(a__  ) * alpha

				__a										=    big_a * (pmc + aaa)
				__a										=    2 * big_a * mpc
				__a										=    big_a * (pmc - aaa)
				__a										=    ppmc + aaa
				__a										=    -2 * pmpc
				__a										=    ppmc - aaa

				__a										=    IIRFilter(2  )
				filt.set_coefficients([aa, aa, aa]						,    [ba, ba, ba]  )
				return filt






def     __lowerCAmelCase			(			a__						,    a__						,    a__						,    a__ = 1 / sqrt(2  )						,    )       ->  IIRFilter:
				__a										=    tau * frequency / samplerate
				__a										=    sin(a__  )
				__a										=    cos(a__  )
				__a										=    _sin / (2 * q_factor)
				__a										=    10 ** (gain_db / 40)
				__a										=    (big_a + 1) - (big_a - 1) * _cos
				__a										=    (big_a + 1) + (big_a - 1) * _cos
				__a										=    (big_a - 1) - (big_a + 1) * _cos
				__a										=    (big_a - 1) + (big_a + 1) * _cos
				__a										=    2 * sqrt(a__  ) * alpha

				__a										=    big_a * (ppmc + aaa)
				__a										=    -2 * big_a * pmpc
				__a										=    big_a * (ppmc - aaa)
				__a										=    pmc + aaa
				__a										=    2 * mpc
				__a										=    pmc - aaa

				__a										=    IIRFilter(2  )
				filt.set_coefficients([aa, aa, aa]						,    [ba, ba, ba]  )
				return filt