Approximation and Second Derivatives

Newton-Raphson Method

Often we are interested in finding the roots (or zeros) of a function, but we do not want to do all that work, so we ask a computer to do it for us. The problem? Computers cannot solve equations. So we have to provide them with the steps to come up with an approximate solution. Here is one method:

The Formula

Moving from the initial guess to the next x-value can be accomplished by using the formula:

$x_2=x_1-\frac{f(x_1)}{f^{\prime }(x_1)}$

A closer examination reveals that this is simply the original formula of a derivative/slope rearranged!

$f^{\prime }(x_1)=\frac{0-f(x_1)}{x_2-x_1}$

We use a 0 here because we are interested in the roots of the function, or when it crosses y = 0. This is our $y_2$.

We can test to see that the formula is true by referring to the graph on the previous page. With an initial guess of 1, the slope of the tangent line$,f^{\prime }(x),$ at that point was equal to 2, and the function value at 1 was equal to -1. Plugging those values into the equation gives

$x_2=1-\frac{-1}{2}$

or 1.5, which as it turns out, is the point at which the blue tangent line crosses the x-axis! Let’s try our hand at some problems using the Newton-Raphson Method.

Example Problem 3.3.1

Compute two iterations to get a numerical approximation of the root of the function $f(x)=x^2-2$ starting at x = 2.

Note: Notice that we reach the second iteration when x’s value is 3, or 1 greater than our desired number of iterations. This is why sometimes the Newton-Raphson formula is written as

$x_1=x_0-\frac{f(x_0)}{f^{\prime }(x_0)}$ .

Limitations of the Newton-Raphson Method

The Newton Raphson Method often only works when the initial guess is close to the solution. Otherwise, we could run into a few inconveniences, as shown in Gallery 3.3.1. So remember, make a good initial guess.