The Newton Raphson method does not need a change of sign, but instead uses the tangent to the graph at a known point to provide a better estimate for the root of the equation.
Here our new estimate for the root is found using the iteration:
Note: f'(x) is the differential of the function f(x).
(This equation is essentially saying you must divide the y-value by the gradient, and subtract this from the previous estimate.)
Repeat the process until the root is found to the desired degree of accuracy.
Lets take the equation f(x) = ex − 3x.
Differentiating this gives f'(x) = ex - 3.
Let's estimate that the root to this equation is 0.6.
Then a better estimation of the answer will be given by the Newton Raphson equation:
We can take this better estimation and put it through the Newton Raphson equation again to get an even more accurate solution:
As this gives the same solution, it must be the correct root for the equation (to 3 decimal places).
Note: If the gradient = 0 at the starting point then no solution can be found by this method (because we cannot divide by 0).