
'''simple docstring'''

import math as m







def 					A       ( UpperCamelCase_   :       Union[str, Any]  ,		UpperCamelCase_   :       Tuple  ,		UpperCamelCase_   :       str=0.001 )   ->    Optional[int]:







			'''simple docstring'''



			return (f(a + h ) - f(a - h )) / (2 * h)







def 					A       ( UpperCamelCase_   :       Dict  ,		UpperCamelCase_   :       str=0  ,		UpperCamelCase_   :       Dict=1_00  ,		UpperCamelCase_   :       str=0.0_001  ,		UpperCamelCase_   :       Union[str, Any]=1E-6  ,		UpperCamelCase_   :       str=False )   ->    List[Any]:







			'''simple docstring'''



			lowerCAmelCase__									=							xa  # set the initial guess
			lowerCAmelCase__									=							[a]
			lowerCAmelCase__									=							abs(f(UpperCamelCase_ ) )
			lowerCAmelCase__									=							lambda UpperCamelCase_ : calc_derivative(UpperCamelCase_  ,		UpperCamelCase_  ,		h=UpperCamelCase_ )  # noqa: E731  Derivative of f(x)
			for _ in range(UpperCamelCase_ ):
						if fa(UpperCamelCase_ ) == 0:
									raise ValueError("No converging solution found" )
						lowerCAmelCase__									=							a - f(UpperCamelCase_ ) / fa(UpperCamelCase_ )  # Calculate the next estimate
						if logsteps:
									steps.append(UpperCamelCase_ )
						if error < maxerror:
									break
			else:
						raise ValueError("Iteration limit reached, no converging solution found" )
			if logsteps:
						# If logstep is true, then log intermediate steps
						return a, error, steps
			return a, error


if __name__ == "__main__":
		from matplotlib import pyplot as plt

		UpperCAmelCase__				:	Any					=						lambda x: m.tanh(x) ** 2 - m.exp(3 * x)  # noqa: E731
		UpperCAmelCase__				:	Tuple					=						newton_raphson(
		    f, xa=10, maxiter=10_00, step=1e-6, logsteps=True
		)
		plt.plot([abs(f(x)) for x in steps])
		plt.xlabel("step")
		plt.ylabel("error")
		plt.show()
		print(F"solution = {{{solution:f}}}, error = {{{error:f}}}")








'''simple docstring'''

import math







def 					A       ( UpperCamelCase_   :       int )   ->    list:







			'''simple docstring'''



			if n <= 0:
						raise ValueError("Only positive integers have prime factors" )
			lowerCAmelCase__									=							[]
			while n % 2 == 0:
						pf.append(2 )
						lowerCAmelCase__									=							int(n / 2 )
			for i in range(3  ,		int(math.sqrt(UpperCamelCase_ ) ) + 1  ,		2 ):
						while n % i == 0:
									pf.append(UpperCamelCase_ )
									lowerCAmelCase__									=							int(n / i )
			if n > 2:
						pf.append(UpperCamelCase_ )
			return pf







def 					A       ( UpperCamelCase_   :       int )   ->    int:







			'''simple docstring'''



			if n <= 0:
						raise ValueError("Only positive numbers are accepted" )
			lowerCAmelCase__									=							1
			lowerCAmelCase__									=							1
			while n % 2 == 0:
						temp += 1
						lowerCAmelCase__									=							int(n / 2 )
			div *= temp
			for i in range(3  ,		int(math.sqrt(UpperCamelCase_ ) ) + 1  ,		2 ):
						lowerCAmelCase__									=							1
						while n % i == 0:
									temp += 1
									lowerCAmelCase__									=							int(n / i )
						div *= temp
			if n > 1:
						div *= 2
			return div







def 					A       ( UpperCamelCase_   :       int )   ->    int:







			'''simple docstring'''



			if n <= 0:
						raise ValueError("Only positive numbers are accepted" )
			lowerCAmelCase__									=							1
			lowerCAmelCase__									=							1
			while n % 2 == 0:
						temp += 1
						lowerCAmelCase__									=							int(n / 2 )
			if temp > 1:
						s *= (2**temp - 1) / (2 - 1)
			for i in range(3  ,		int(math.sqrt(UpperCamelCase_ ) ) + 1  ,		2 ):
						lowerCAmelCase__									=							1
						while n % i == 0:
									temp += 1
									lowerCAmelCase__									=							int(n / i )
						if temp > 1:
									s *= (i**temp - 1) / (i - 1)
			return int(UpperCamelCase_ )







def 					A       ( UpperCamelCase_   :       int )   ->    int:







			'''simple docstring'''



			lowerCAmelCase__									=							n
			for x in set(prime_factors(UpperCamelCase_ ) ):
						s *= (x - 1) / x
			return int(UpperCamelCase_ )


if __name__ == "__main__":
		import doctest

		doctest.testmod()





