How to loop over a condition until the values have converged? (Newton Raphson Method)

Question

For the Newton-Raphson Method, I'm asked to write a loop to call the function until it has converged so that the values of x(n) and x(n+1) differ by less than a small number e=10^-8, however, I am not quite sure how to create this loop.

Here's my code so far

def newton_step(f, fp, x0):
     return x0-(f(x0)/fp(x0))

import numpy as np
def f(x):
    return x**3 + x**2 +2 

def fp(x): 
    return 3*x**2 + 2*x

x = newton_step(f, fp, (x0))
count = 0 #to count the number of iterations required to satisfy the condition
while (x0 - x) <= 10**-8:
    x0 = newton_step(f, fp, (x0))
    count += 1
    print(x0, x)

You're using a function to step 2 values closer to one another, right? You must provide those 2 starting values (x, x0) — Adam Smooch, Oct 20 '21 at 18:06

score 1 · Answer 1 · answered Oct 20 '21 at 17:58

1

Looks like your comparison operator is backwards, try:

while (x0 - x) > 10**-8:

so that it will loop, continuing while the diff is greater than 10^-8 (breaking out of the loop once it is less than that value)

Also, you'll want to make sure that your code sets an initial value for x0 before the loop (so there is something to compare)

answered Oct 20 '21 at 17:58

Adam Smooch

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I was confused at what that x0 is, so it's just any random value or the solution of this expression? – Amnesia Smith Oct 20 '21 at 18:01
Thank you so much! – Amnesia Smith Oct 20 '21 at 18:11

score 1 · Answer 2 · answered Oct 20 '21 at 17:58

1

I can see at least 2 issues with your code -

You don't update x
if x0<x then the loop will end but they can differ in more the 10^-8
Wrong conditionn

To sum this up I would change the loop -

while abs(x0-x) >= 10**-8:
    new_x = newton_step(f, fp, (x0))
    x = x0
    x0 = new_x
    count +=1
    print(x0, x)

answered Oct 20 '21 at 17:58

Tom Ron

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I keep getting "NameError: name 'x0' is not defined" – Amnesia Smith Oct 20 '21 at 18:00
I assumed it was defined earlier in the code – Tom Ron Oct 20 '21 at 18:01
The thing I'm struggling with is whether they're asking me for a finite value/the actual root for this equation or if they're asking to me call the function again such that x0 = (f, fp, (x0)) – Amnesia Smith Oct 20 '21 at 18:03
Thank you so much! – Amnesia Smith Oct 20 '21 at 18:12

score 1 · Answer 3 · answered Oct 20 '21 at 17:58

1

Your while loop runs as long as the difference is smaller, but it should run as long as the difference is bigger. Change while (x0 - x) <= 10**-8 to while (x0 - x) >= 10**-8.

answered Oct 20 '21 at 17:58

Anteino

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Thank you so much! – Amnesia Smith Oct 20 '21 at 18:11

How to loop over a condition until the values have converged? (Newton Raphson Method)

3 Answers3