
'''simple docstring'''
from typing import Any

import numpy as np




def    _UpperCamelCase						(       SCREAMING_SNAKE_CASE__			)  ->							bool:


		'''simple docstring'''


		return np.array_equal(SCREAMING_SNAKE_CASE__ ,       matrix.conjugate().T			)




def    _UpperCamelCase						(       SCREAMING_SNAKE_CASE__ ,       SCREAMING_SNAKE_CASE__			)  ->							Any:


		'''simple docstring'''


		snake_case  :       int     			=						v.conjugate().T
		snake_case  :       Tuple     			=						v_star.dot(SCREAMING_SNAKE_CASE__			)
		assert isinstance(SCREAMING_SNAKE_CASE__ ,       np.ndarray			)
		return (v_star_dot.dot(SCREAMING_SNAKE_CASE__			)) / (v_star.dot(SCREAMING_SNAKE_CASE__			))




def    _UpperCamelCase						(       )  ->							None:


		'''simple docstring'''


		snake_case  :       Optional[int]     			=						np.array([[2, 2 + 1J, 4], [2 - 1J, 3, 1J], [4, -1J, 1]]			)
		snake_case  :       Any     			=						np.array([[1], [2], [3]]			)
		assert is_hermitian(SCREAMING_SNAKE_CASE__			), F'{a} is not hermitian.'
		print(rayleigh_quotient(SCREAMING_SNAKE_CASE__ ,       SCREAMING_SNAKE_CASE__			)			)

		snake_case  :       List[Any]     			=						np.array([[1, 2, 4], [2, 3, -1], [4, -1, 1]]			)
		assert is_hermitian(SCREAMING_SNAKE_CASE__			), F'{a} is not hermitian.'
		assert rayleigh_quotient(SCREAMING_SNAKE_CASE__ ,       SCREAMING_SNAKE_CASE__			) == float(3			)


if __name__ == "__main__":
						import doctest

						doctest.testmod()
						tests()










from __future__ import annotations






class 						lowercase_   :






      '''simple docstring'''




      def __init__( self :		List[str]					,   __UpperCAmelCase :		int    )       ->None:




       """simple docstring"""


       a						    = order

       # a_{0} ... a_{k}
       a						    = [1.0] + [0.0] * order
       # b_{0} ... b_{k}
       a						    = [1.0] + [0.0] * order

       # x[n-1] ... x[n-k]
       a						    = [0.0] * self.order
       # y[n-1] ... y[n-k]
       a						    = [0.0] * self.order







      def  __lowerCAmelCase   ( self :		Union[str, Any]					,   __UpperCAmelCase :		list[float]					,   __UpperCAmelCase :		list[float]    )       ->None:




       """simple docstring"""


       if len(__UpperCAmelCase    ) < self.order:
        a						    = [1.0, *a_coeffs]

       if len(__UpperCAmelCase    ) != self.order + 1:
        a						    = (
            F"""Expected a_coeffs to have {self.order + 1} elements """
            F"""for {self.order}-order filter, got {len(__UpperCAmelCase    )}"""
        )
        raise ValueError(__UpperCAmelCase    )

       if len(__UpperCAmelCase    ) != self.order + 1:
        a						    = (
            F"""Expected b_coeffs to have {self.order + 1} elements """
            F"""for {self.order}-order filter, got {len(__UpperCAmelCase    )}"""
        )
        raise ValueError(__UpperCAmelCase    )

       a						    = a_coeffs
       a						    = b_coeffs







      def  __lowerCAmelCase   ( self :		Any					,   __UpperCAmelCase :		float    )       ->float:




       """simple docstring"""


       a						    = 0.0

       # Start at index 1 and do index 0 at the end.
       for i in range(1					,   self.order + 1    ):
        result += (
            self.b_coeffs[i] * self.input_history[i - 1]
            - self.a_coeffs[i] * self.output_history[i - 1]
        )

       a						    = (result + self.b_coeffs[0] * sample) / self.a_coeffs[0]

       a						    = self.input_history[:-1]
       a						    = self.output_history[:-1]

       a						    = sample
       a						    = result

       return result


