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Newton's Method for Finding Roots of Functions

Date: 06/03/2003 at 17:21:56
From: Melissa
Subject: Newton's method for finding roots

Hi Dr. Math,

I was recently assigned the task of teaching the class about Newton's 
Method for Finding Roots; however, I am having problems learning it 
myself. Is there an easy way to learn and understand this method or an 
example that would allow me to understand the steps taken to work 
through the problem? Your help is greatly appreciated.

Date: 06/04/2003 at 08:39:03
From: Doctor George
Subject: Re: Newton's method for finding roots

Hi Melissa,

Thanks for writing to Doctor Math.

Newton's method starts with a guess at the correct solution for the 
root. Let's say you have f(x) and our initial guess is that f(x) = 0 
when x = x_0.

The process is to use this initial x_0 to find a better estimate that 
we will call x_1. The way we do this is to draw a line tangent to f(x) 
at the point (x_0, f(x_0)). Call the line g(x).

To find g(x) we need to compute f'(x) at x = x_0, or f'(x_0). Using 
the point-slope equation we can write g(x) like this.

   g(x) - f(x_0) = f'(x_0) (x - x_0)

Now we ask where g(x) = 0. This happens where

   x = x_0 - f(x_0) / f'(x_0).

This x is our x_1. Now repeat the process by constructing the tangent 
to f(x) at x_1, and finding x_2 where g(x) = 0 for this new tangent 
line. Try drawing these tangent lines to see what is happening.

In general

   x_(i+1) = x_i - f(x_i) / f'(x_i)

We repeat this equation until we converge. Each time we use this 
equation is called an iteration.

Now let's do an example. Let's say that f(x) = x^2 - 4, and pretend 
that we do not know that the roots are x = 2 and -2. By checking 
f(1.8) and f(2.2) we can tell that there must be a root between 
1.8 and 2.2.

Let's set x_0 = 2.1 as our guess and compute f'(x) = 2x.

The equation that we interate is

   x_(i+1) = x_i - f(x_i) / f'(x_i).

           = x_i - [(x_i)^2 - 4] / [2(x_i)]

From this we compute

       x_1 = 2.1 - [(2.1)^2 - 4] / (2*2.1)

           = 2.00238

       x_2 = 2.00238 - [(2.00238)^2 - 4] / (2*2.00238)

           = 2.00119

       x_3 = 2.00119 - [(2.00119)^2 - 4] / (2*2.00119)

           = 2.00000...

If we had started with a guess of -2.1 we would have found the other 
root of the equation.

As long as the function is well-behaved in the region between the 
initial guess and the root, Newton's method will converge nicely. 
Coming up with the initial guess is an important step in getting 
Newton's method to work correctly.

That's a start on Newton's method. Write again if you need more help.

- Doctor George, The Math Forum 
Associated Topics:
High School Functions

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