Question

Problem 4. PYTHON: (Root Finding) Write a program root.py (a variant of the sqrt.py program we discussed in class) that accepts k (int), c (float), and epsilon (float) as command-line arguments, and writes to standard output the kth root of c, up to epsilon decimal places.

Example: $ python3 root . py 3 2 1e -15

1.2599210498948732

Hints:

Set t (our guess) to c

Use the condition |1 − c/tk| > to continue the loop

At each iteration, replace the estimate t by t − f(t)/f0 (t), where f(t) = t k − c and f 0 (t) = ktk−1

Answer 1

The method described in the question is called Newton Raphson method. Below is the python code for the question. Copy it to a file called "root.py" and run it through command line as "python3 root.py 3 2 1e-15".

import sys

# Read command line arguments
args = sys.argv
k = int(args[1])
c = float(args[2])
e = float(args[3])

# At any iteration, 't' will hold the current iteration's guess and 'prev_t' will hold the previous iteration's guess
t = c
while True:
   prev_t = t
   ft = t**k - c
   f0t = k*(t**(k-1))
   t = t - ft/f0t
  
   # If absolute differnt of previous guess and current guess is less than equal to epsilon, then we have found our answer
   if abs(prev_t - t) <= e:
       break

print("The answer is: {}".format(t))