A fixed-point function is a mathematical function that returns the same value when applied repeatedly. Here's an example of a fixed-point function in Python:
def fixed_point(func, initial_guess, tolerance=1e-6, max_iterations=100):
"""
Computes the fixed point of a function using the fixed-point iteration method.
Args:
func (function): The function for which to find the fixed point.
initial_guess (float): The initial guess for the fixed point.
tolerance (float): The tolerance for convergence (default: 1e-6).
max_iterations (int): The maximum number of iterations (default: 100).
Returns:
float: The approximate fixed point of the function.
"""
x = initial_guess
for _ in range(max_iterations):
x_next = func(x)
if abs(x_next - x) < tolerance:
return x_next
x = x_next
return None
In this implementation, the fixed_point function takes three parameters: func, which represents the function for which we want to find the fixed point, initial_guess, which is the initial guess for the fixed point, and optional parameters tolerance and max_iterations for controlling the convergence criteria and maximum number of iterations, respectively.
The function uses a simple fixed-point iteration method: it starts with the initial guess, repeatedly applies the function func to the current value, and checks if the difference between the current value and the next value is below the specified tolerance. If the convergence condition is met, the approximate fixed point is returned. If the maximum number of iterations is reached without convergence, the function returns None.
You can use this fixed_point function by defining your specific function and providing the appropriate initial guess, tolerance, and maximum iterations.
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