






'''simple docstring'''

from math import sqrt



def 				snake_case       (  a_				:			Union[str, Any]						)							->					Dict:





      """simple docstring"""

      assert isinstance(__a     , __a						) and (
          number >= 0
      ), "'number' must been an int and positive"

      UpperCamelCase_					: str            =				True

      # 0 and 1 are none primes.
      if number <= 1:
            UpperCamelCase_					: List[str]            =				False

      for divisor in range(2     , int(round(sqrt(__a						)						)						) + 1						):
            # if 'number' divisible by 'divisor' then sets 'status'
            # of false and break up the loop.
            if number % divisor == 0:
                  UpperCamelCase_					: Any            =				False
                  break

    # precondition
      assert isinstance(__a     , __a						), "'status' must been from type bool"

      return status



def 				snake_case       (  a_				:			Optional[int]						)							->					Dict:





      """simple docstring"""

      assert isinstance(__a     , __a						) and (n > 2), "'N' must been an int and > 2"

      # beginList: contains all natural numbers from 2 up to N
      UpperCamelCase_					: List[str]            =				list(range(2     , n + 1						)						)

      UpperCamelCase_					: Tuple            =				[]  # this list will be returns.

      # actual sieve of erathostenes
      for i in range(len(__a						)						):
            for j in range(i + 1     , len(__a						)						):
                  if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
                        UpperCamelCase_					: Union[str, Any]            =				0

    # filters actual prime numbers.
      UpperCamelCase_					: Tuple            =				[x for x in begin_list if x != 0]

      # precondition
      assert isinstance(__a     , __a						), "'ans' must been from type list"

      return ans



def 				snake_case       (  a_				:			Optional[Any]						)							->					Union[str, Any]:





      """simple docstring"""

      assert isinstance(__a     , __a						) and (n > 2), "'N' must been an int and > 2"

      UpperCamelCase_					: Dict            =				[]

      # iterates over all numbers between 2 up to N+1
      # if a number is prime then appends to list 'ans'
      for number in range(2     , n + 1						):
            if is_prime(__a						):
                  ans.append(__a						)

    # precondition
      assert isinstance(__a     , __a						), "'ans' must been from type list"

      return ans



def 				snake_case       (  a_				:			str						)							->					int:





      """simple docstring"""

      assert isinstance(__a     , __a						) and number >= 0, "'number' must been an int and >= 0"

      UpperCamelCase_					: Any            =				[]  # this list will be returns of the function.

      # potential prime number factors.

      UpperCamelCase_					: int            =				2

      UpperCamelCase_					: Union[str, Any]            =				number

      if number == 0 or number == 1:
            ans.append(__a						)

      # if 'number' not prime then builds the prime factorization of 'number'
      elif not is_prime(__a						):
            while quotient != 1:
                  if is_prime(__a						) and (quotient % factor == 0):
                        ans.append(__a						)
                        quotient /= factor
                  else:
                        factor += 1

      else:
            ans.append(__a						)

      # precondition
      assert isinstance(__a     , __a						), "'ans' must been from type list"

      return ans



def 				snake_case       (  a_				:			Dict						)							->					Optional[Any]:





      """simple docstring"""

      assert isinstance(__a     , __a						) and (
          number >= 0
      ), "'number' bust been an int and >= 0"

      UpperCamelCase_					: Optional[Any]            =				0

      # prime factorization of 'number'
      UpperCamelCase_					: Union[str, Any]            =				prime_factorization(__a						)

      UpperCamelCase_					: List[str]            =				max(__a						)

      # precondition
      assert isinstance(__a     , __a						), "'ans' must been from type int"

      return ans



def 				snake_case       (  a_				:			Union[str, Any]						)							->					str:





      """simple docstring"""

      assert isinstance(__a     , __a						) and (
          number >= 0
      ), "'number' bust been an int and >= 0"

      UpperCamelCase_					: Optional[Any]            =				0

      # prime factorization of 'number'
      UpperCamelCase_					: Optional[int]            =				prime_factorization(__a						)

      UpperCamelCase_					: List[str]            =				min(__a						)

      # precondition
      assert isinstance(__a     , __a						), "'ans' must been from type int"

      return ans



def 				snake_case       (  a_				:			Tuple						)							->					Tuple:





      """simple docstring"""

      assert isinstance(__a     , __a						), "'number' must been an int"
      assert isinstance(number % 2 == 0     , __a						), "compare bust been from type bool"

      return number % 2 == 0



def 				snake_case       (  a_				:			List[Any]						)							->					Union[str, Any]:





      """simple docstring"""

      assert isinstance(__a     , __a						), "'number' must been an int"
      assert isinstance(number % 2 != 0     , __a						), "compare bust been from type bool"

      return number % 2 != 0



def 				snake_case       (  a_				:			Optional[Any]						)							->					Optional[Any]:





      """simple docstring"""

      assert (
          isinstance(__a     , __a						) and (number > 2) and is_even(__a						)
      ), "'number' must been an int, even and > 2"

      UpperCamelCase_					: Tuple            =				[]  # this list will returned

      # creates a list of prime numbers between 2 up to 'number'
      UpperCamelCase_					: str            =				get_prime_numbers(__a						)
      UpperCamelCase_					: Any            =				len(__a						)

      # run variable for while-loops.
      UpperCamelCase_					: int            =				0
      UpperCamelCase_					: Dict            =				None

      # exit variable. for break up the loops
      UpperCamelCase_					: List[Any]            =				True

      while i < len_pn and loop:
            UpperCamelCase_					: int            =				i + 1

            while j < len_pn and loop:
                  if prime_numbers[i] + prime_numbers[j] == number:
                        UpperCamelCase_					: Dict            =				False
                        ans.append(prime_numbers[i]						)
                        ans.append(prime_numbers[j]						)

                  j += 1

            i += 1

      # precondition
      assert (
          isinstance(__a     , __a						)
          and (len(__a						) == 2)
          and (ans[0] + ans[1] == number)
          and is_prime(ans[0]						)
          and is_prime(ans[1]						)
      ), "'ans' must contains two primes. And sum of elements must been eq 'number'"

      return ans



def 				snake_case       (  a_				:			int     , a_				:			Any						)							->					int:





      """simple docstring"""

      assert (
          isinstance(__a     , __a						)
          and isinstance(__a     , __a						)
          and (numbera >= 0)
          and (numbera >= 0)
      ), "'number1' and 'number2' must been positive integer."

      UpperCamelCase_					: Dict            =				0

      while numbera != 0:
            UpperCamelCase_					: str            =				numbera % numbera
            UpperCamelCase_					: List[str]            =				numbera
            UpperCamelCase_					: Optional[int]            =				rest

      # precondition
      assert isinstance(__a     , __a						) and (
          numbera >= 0
      ), "'number' must been from type int and positive"

      return numbera



def 				snake_case       (  a_				:			Dict     , a_				:			int						)							->					Optional[Any]:





      """simple docstring"""

      assert (
          isinstance(__a     , __a						)
          and isinstance(__a     , __a						)
          and (numbera >= 1)
          and (numbera >= 1)
      ), "'number1' and 'number2' must been positive integer."

      UpperCamelCase_					: int            =				1  # actual answer that will be return.

      # for kgV (x,1)
      if numbera > 1 and numbera > 1:
            # builds the prime factorization of 'number1' and 'number2'
            UpperCamelCase_					: List[Any]            =				prime_factorization(__a						)
            UpperCamelCase_					: str            =				prime_factorization(__a						)

      elif numbera == 1 or numbera == 1:
            UpperCamelCase_					: Union[str, Any]            =				[]
            UpperCamelCase_					: Optional[int]            =				[]
            UpperCamelCase_					: Union[str, Any]            =				max(__a     , __a						)

      UpperCamelCase_					: Dict            =				0
      UpperCamelCase_					: str            =				0

      UpperCamelCase_					: Any            =				[]  # captured numbers int both 'primeFac1' and 'primeFac2'

      # iterates through primeFac1
      for n in prime_fac_a:
            if n not in done:
                  if n in prime_fac_a:
                        UpperCamelCase_					: List[str]            =				prime_fac_a.count(__a						)
                        UpperCamelCase_					: Tuple            =				prime_fac_a.count(__a						)

                        for _ in range(max(__a     , __a						)						):
                              ans *= n

                  else:
                        UpperCamelCase_					: Tuple            =				prime_fac_a.count(__a						)

                        for _ in range(__a						):
                              ans *= n

                  done.append(__a						)

    # iterates through primeFac2
      for n in prime_fac_a:
            if n not in done:
                  UpperCamelCase_					: str            =				prime_fac_a.count(__a						)

                  for _ in range(__a						):
                        ans *= n

                  done.append(__a						)

    # precondition
      assert isinstance(__a     , __a						) and (
          ans >= 0
      ), "'ans' must been from type int and positive"

      return ans



def 				snake_case       (  a_				:			Tuple						)							->					Union[str, Any]:





      """simple docstring"""

      assert isinstance(__a     , __a						) and (n >= 0), "'number' must been a positive int"

      UpperCamelCase_					: Optional[Any]            =				0
      UpperCamelCase_					: List[Any]            =				2  # this variable holds the answer

      while index < n:
            index += 1

            ans += 1  # counts to the next number

            # if ans not prime then
            # runs to the next prime number.
            while not is_prime(__a						):
                  ans += 1

    # precondition
      assert isinstance(__a     , __a						) and is_prime(
          __a						), "'ans' must been a prime number and from type int"

      return ans



def 				snake_case       (  a_				:			Union[str, Any]     , a_				:			List[Any]						)							->					List[Any]:





      """simple docstring"""

      assert (
          is_prime(__a						) and is_prime(__a						) and (p_number_a < p_number_a)
      ), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

      UpperCamelCase_					: List[Any]            =				p_number_a + 1  # jump to the next number

      UpperCamelCase_					: List[Any]            =				[]  # this list will be returns.

      # if number is not prime then
      # fetch the next prime number.
      while not is_prime(__a						):
            number += 1

      while number < p_number_a:
            ans.append(__a						)

            number += 1

            # fetch the next prime number.
            while not is_prime(__a						):
                  number += 1

    # precondition
      assert (
          isinstance(__a     , __a						)
          and ans[0] != p_number_a
          and ans[len(__a						) - 1] != p_number_a
      ), "'ans' must been a list without the arguments"

      # 'ans' contains not 'pNumber1' and 'pNumber2' !
      return ans



def 				snake_case       (  a_				:			Optional[int]						)							->					str:





      """simple docstring"""

      assert isinstance(__a     , __a						) and (n >= 1), "'n' must been int and >= 1"

      UpperCamelCase_					: List[str]            =				[]  # will be returned.

      for divisor in range(1     , n + 1						):
            if n % divisor == 0:
                  ans.append(__a						)

    # precondition
      assert ans[0] == 1 and ans[len(__a						) - 1] == n, "Error in function getDivisiors(...)"

      return ans



def 				snake_case       (  a_				:			Union[str, Any]						)							->					List[str]:





      """simple docstring"""

      assert isinstance(__a     , __a						) and (
          number > 1
      ), "'number' must been an int and >= 1"

      UpperCamelCase_					: List[Any]            =				get_divisors(__a						)

      # precondition
      assert (
          isinstance(__a     , __a						)
          and (divisors[0] == 1)
          and (divisors[len(__a						) - 1] == number)
      ), "Error in help-function getDivisiors(...)"

      # summed all divisors up to 'number' (exclusive), hence [:-1]
      return sum(divisors[:-1]						) == number



def 				snake_case       (  a_				:			Tuple     , a_				:			Optional[Any]						)							->					Optional[Any]:





      """simple docstring"""

      assert (
          isinstance(__a     , __a						)
          and isinstance(__a     , __a						)
          and (denominator != 0)
      ), "The arguments must been from type int and 'denominator' != 0"

      # build the greatest common divisor of numerator and denominator.
      UpperCamelCase_					: Optional[int]            =				gcd(abs(__a						)     , abs(__a						)						)

      # precondition
      assert (
          isinstance(__a     , __a						)
          and (numerator % gcd_of_fraction == 0)
          and (denominator % gcd_of_fraction == 0)
      ), "Error in function gcd(...,...)"

      return (numerator // gcd_of_fraction, denominator // gcd_of_fraction)



def 				snake_case       (  a_				:			Union[str, Any]						)							->					Union[str, Any]:





      """simple docstring"""

      assert isinstance(__a     , __a						) and (n >= 0), "'n' must been a int and >= 0"

      UpperCamelCase_					: str            =				1  # this will be return.

      for factor in range(1     , n + 1						):
            ans *= factor

      return ans



def 				snake_case       (  a_				:			Union[str, Any]						)							->					int:





      """simple docstring"""

      assert isinstance(__a     , __a						) and (n >= 0), "'n' must been an int and >= 0"

      UpperCamelCase_					: int            =				0
      UpperCamelCase_					: Any            =				1
      UpperCamelCase_					: Any            =				1  # this will be return

      for _ in range(n - 1						):
            UpperCamelCase_					: int            =				ans
            ans += fiba
            UpperCamelCase_					: List[Any]            =				tmp

      return ans








"""simple docstring"""




import logging

import numpy as np
import pytest
from scipy.linalg import eigh

logging.basicConfig(level=logging.INFO, format='%(message)s')



def 							_lowerCamelCase			(			__a     ):
		return input_array.reshape((input_array.size, 1)     )



def 							_lowerCamelCase			(			__a,				__a,				__a     ):
		SCREAMING_SNAKE_CASE_    			=			np.nan
		for i in range(__a     ):
				SCREAMING_SNAKE_CASE_    			=			features[:, labels == i]
				SCREAMING_SNAKE_CASE_    			=			data.mean(1     )
				# Centralize the data of class i
				SCREAMING_SNAKE_CASE_    			=			data - column_reshape(__a     )
				if i > 0:
						# If covariance_sum is not None
						covariance_sum += np.dot(__a,				centered_data.T     )
				else:
						# If covariance_sum is np.nan (i.e. first loop)
						SCREAMING_SNAKE_CASE_    			=			np.dot(__a,				centered_data.T     )

		return covariance_sum / features.shape[1]



def 							_lowerCamelCase			(			__a,				__a,				__a     ):
		SCREAMING_SNAKE_CASE_    			=			features.mean(1     )
		SCREAMING_SNAKE_CASE_    			=			np.nan
		for i in range(__a     ):
				SCREAMING_SNAKE_CASE_    			=			features[:, labels == i]
				SCREAMING_SNAKE_CASE_    			=			data.shape[1]
				SCREAMING_SNAKE_CASE_    			=			data.mean(1     )
				if i > 0:
						# If covariance_sum is not None
						covariance_sum += device_data * np.dot(
						    column_reshape(__a     ) - column_reshape(__a     ),				(column_reshape(__a     ) - column_reshape(__a     )).T,				)
				else:
						# If covariance_sum is np.nan (i.e. first loop)
						SCREAMING_SNAKE_CASE_    			=			device_data * np.dot(
						    column_reshape(__a     ) - column_reshape(__a     ),				(column_reshape(__a     ) - column_reshape(__a     )).T,				)

		return covariance_sum / features.shape[1]



def 							_lowerCamelCase			(			__a,				__a     ):

		# Check if the features have been loaded
		if features.any():
				SCREAMING_SNAKE_CASE_    			=			features.mean(1     )
				# Center the dataset
				SCREAMING_SNAKE_CASE_    			=			features - np.reshape(__a,				(data_mean.size, 1)     )
				SCREAMING_SNAKE_CASE_    			=			np.dot(__a,				centered_data.T     ) / features.shape[1]
				SCREAMING_SNAKE_CASE_			,SCREAMING_SNAKE_CASE_    			=			np.linalg.eigh(__a     )
				# Take all the columns in the reverse order (-1), and then takes only the first
				SCREAMING_SNAKE_CASE_    			=			eigenvectors[:, ::-1][:, 0:dimensions]
				# Project the database on the new space
				SCREAMING_SNAKE_CASE_    			=			np.dot(filtered_eigenvectors.T,				__a     )
				logging.info('''Principal Component Analysis computed'''     )

				return projected_data
		else:
				logging.basicConfig(level=logging.ERROR,				format='''%(message)s''',				force=__a     )
				logging.error('''Dataset empty'''     )
				raise AssertionError



def 							_lowerCamelCase			(			__a,				__a,				__a,				__a     ):
		assert classes > dimensions

		# Check if features have been already loaded
		if features.any:
				SCREAMING_SNAKE_CASE_			,SCREAMING_SNAKE_CASE_    			=			eigh(
				    covariance_between_classes(__a,				__a,				__a     ),				covariance_within_classes(__a,				__a,				__a     ),				)
				SCREAMING_SNAKE_CASE_    			=			eigenvectors[:, ::-1][:, :dimensions]
				SCREAMING_SNAKE_CASE_			,SCREAMING_SNAKE_CASE_			,SCREAMING_SNAKE_CASE_    			=			np.linalg.svd(__a     )
				SCREAMING_SNAKE_CASE_    			=			svd_matrix[:, 0:dimensions]
				SCREAMING_SNAKE_CASE_    			=			np.dot(filtered_svd_matrix.T,				__a     )
				logging.info('''Linear Discriminant Analysis computed'''     )

				return projected_data
		else:
				logging.basicConfig(level=logging.ERROR,				format='''%(message)s''',				force=__a     )
				logging.error('''Dataset empty'''     )
				raise AssertionError



def 							_lowerCamelCase			(			):
		# Create dummy dataset with 2 classes and 3 features
		SCREAMING_SNAKE_CASE_    			=			np.array([[1, 2, 3, 4, 5], [2, 3, 4, 5, 6], [3, 4, 5, 6, 7]]     )
		SCREAMING_SNAKE_CASE_    			=			np.array([0, 0, 0, 1, 1]     )
		SCREAMING_SNAKE_CASE_    			=			2
		SCREAMING_SNAKE_CASE_    			=			2

		# Assert that the function raises an AssertionError if dimensions > classes
		with pytest.raises(__a     ) as error_info:
				SCREAMING_SNAKE_CASE_    			=			linear_discriminant_analysis(
				    __a,				__a,				__a,				__a     )
				if isinstance(__a,				np.ndarray     ):
						raise AssertionError(
						    '''Did not raise AssertionError for dimensions > classes'''     )
				assert error_info.type is AssertionError



def 							_lowerCamelCase			(			):
		SCREAMING_SNAKE_CASE_    			=			np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]     )
		SCREAMING_SNAKE_CASE_    			=			2
		SCREAMING_SNAKE_CASE_    			=			np.array([[6.9_2_8_2_0_3_2_3, 8.6_6_0_2_5_4_0_4, 1_0.3_9_2_3_0_4_8_5], [3.0, 3.0, 3.0]]     )

		with pytest.raises(__a     ) as error_info:
				SCREAMING_SNAKE_CASE_    			=			principal_component_analysis(__a,				__a     )
				if not np.allclose(__a,				__a     ):
						raise AssertionError
				assert error_info.type is AssertionError


if __name__ == "__main__":
						import doctest

						doctest.testmod()