Suppose a number x₀ is known to be fairly close to the root. This means that y₀ = f(x₀) is fairly close to 0. The number x₀ is the initial guess of the root. Then under most circumstances, Newton's method lets one calculate a new, much better, guess for the root.

Specifically, starting from the point (x₀, y₀) on the graph of the equation, let L be the line tangent to the graph at that point. You follow this line back to the x-axis, and the value of x where it intersects the x-axis is the new, better guess: x₁.

Here is one way to calculate the equation of this line: a point (x, y) lies on this line, if the slope of the line through (x, y) and (x₀, y₀) is equal to f'(x₀). But the slope of the line through two points is just the difference of the x values divided by the difference of the y values. Plugging this in, one gets the following equation for the line:

  y - y0
 --------  = f'(x0)
  x - x0

Rewriting in the form y = m*x + b gives:

  y = f'(x0)*x + (y0 - x0*f'(x0)

Now where this line intersects the x-axis, one has y = 0. So setting y = 0, and set y₀ = f(x₀) in the equation, and solving for x, one gets:

  x = x0 - f(x0)/f'(x0)

This value of x is the new, better approximation for the root, that was called x₁ above, so we have the magic formula:

x1 = x0 - f(x0)/f'(x0)

Plugging in f(x) = 2 - x² and f'(x) = -2x into the magic formula, we get a specific formula for this example:

x1 = x0/2 + 1/x0

Here is a picture of this example:

Here is a table showing what happens when we use the magic formula over and over again,

Newton's Method: f(x) = 2 - x2 (sqrt(2) in red)
x0: Old guess	x1 = x0/2 + 1/x0: New guess
1.000000000000000	1.500000000000000 = 3/2
1.500000000000000	1.416666666666666 = 17/12
1.416666666666666	1.414215686274509 = 577/408
1.414215686274509	1.414213562374689 = 665857/470832
1.414213562374689	1.414213562373095 = 886731088897/627013566048

So, put the formula into a loop. Start with x₀ = 1. In the loop calculate the new, better value x₁ = 1. Finally, set x₀ = x₁, so the next time the loop is executed, the new guess will be the old guess again.

You might try terminating the loop when fabs(x₀ - x₁) < 0.00000000001, perhaps using the break statement that we discussed in class.