Unizor - Derivatives - Newton's Method of Finding Zeros of a Function
Channel: Zor Shekhtman on UNIZOR Education
Date: 2017-01-09

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Notes to a video lecture on http://www.unizor.com

Derivatives - Newton's Method to Find Zeros of Function

The purpose of this lecture is to address a specific problem of finding an approximate solution of equations f(x)=0 using some methodology suggested by Sir Isaac Newton a few centuries ago.
Obviously, it makes sense to apply this methodology when there is no analytical solution of a given equation, or this analytical solution requires too much efforts.

A word of warning should be given up front. This method might sometimes fail to find an approximate solution. After describing this method it will be obvious that under some circumstances the process of finding a solution is not converging to any number.

Assume, we need to find zeros of some smooth function f(x).

Step 0
Start with some approximation of a value of an argument x0 that is relatively close to the one where our function equals to zero.
There is no universal recipe for this approximation, but the better your approximation is - the sooner you will approach the real zero of f(x).
And, to be noted, if your starting value is too far from the real zero, the Newton's method might never converge to real zero of our function.

Step 1
From the chosen value x0 calculate the next value of a sequence:
x1 = x0 − f(x0) / f'(x0)
This and subsequent steps are straightforward calculations.

Step 2, 3 etc.
Check two things:
(a) the value of f(xn) - is it close to zero within your chosen precision?
(b) the difference between xn and xn−1 - is it smaller than your chosen precision?
If both checks are satisfactory, stop the process. The last xn is your final result.
If your precision requirements are not satisfied yet, continue.
Repeat the previous step according to this recursive formula:
xn+1 = xn − f(xn) / f'(xn)

As we noted, Newton's method is not universal. A lot depends on properties of function, whose zeros we are trying to find, and on a choice of the first approximation x0.