In Python, There are several iterative methods to calculate pi. A particularly fast method is the iteration:

P(1) = 3.0

P(n + 1) = P(n) + sin(P(n))

This algorithm is a particular case of a method called Fixed Point Iteration

In this particular case, we have f(x) = x + sin(x), and, as sin(n*pi) = 0 for any integer n, any multiple of pi is a solution of that equation.

An equation like [1] can sometimes (often?) be solved by iterating the formula:

x[n+1] = f(x[n]) where x[n] is the nth approximation of the solution.

Newton's method gives the formula: x[n+1] = x[n] - sin(x[n])/cos(x[n] and, as cos(x[n]) is close to -1, you get your formula by replacing cos(x[n]) by -1 in the above formula.

1. Determine if Newton's method or the fixed point method is better for computing pi (up to 15 digits of accuracy) by counting how many iterations each needs to converge.

2. Determine if Newton's method or the fixed point method is better for computing the Dottie number (0.739085...), which is the unique, real number fixed point of the cos() function.

3. Include comments that discuss your analysis of this problem.

Write a higher-order function to compute an iterative fixed point iteration as follows.

def fixed_point_iteration(f, x=1.0):

step = 0

while not approx_fixed_point(f, x):

x = f(x)

step += 1

return x, step

def approx_fixed_point(f, x) should returns True if and only if f(x) is very close (distance < 1e-15) to x.

Sample Run:

>> from math import sin,cos

>>> print(fixed_point_iteration(lambda x: sin(x) + x, 3.0)

(3.141592653589793, 3) # Fixed point pi:

>>> print(newton_find_zero(lambda x: sin(x) , lambda x: cos(x), 3.0)

(3.141592653589793, 3) # Newtons's pi

>>> print(fixed_point_iteration(lambda x: cos(x), 1.0)

(0.7390851332151611, 86) # Fixed point dottie

>>> print(newton_find_zero(lambda x: cos(x) - x , lambda x: -sin(x)-1, 1.0)

(0.7390851332151606, 7) # Newtons's dottie

Any Help Would Be Greatly Appreciated, Thanks.