  Numerical Searching Method - Newton-Ralphson
Newton-Ralphson method - or simply the Newton's method is one of the most commonly used numerical searching method for solving equations. Usually Newton's method converges well and quickly, but the convergence is not guaranteed. Newton's method requires an initial value. This values can determine the way the search is converged. The major challenge to using this method is that the first differential (first derivative) of the equation is required as an input for the search precedure. Sometimes, it may be difficult or impossible to derive that.

The example below demostrates the application using Newton-Ralphson method to for solve 2 equations (both equations are set to zero) individually. The first equation has one root (outcome) and the second has two (see the charts). The method returns 0.56714 as the value of the root for the first equation. We then plug this value into the equation in cell D13 for checking and get a value of zero - just as what we should be getting. It takes 4 steps (loops or iterations) to converge.

Since equation 2 has two roots, we need to run the search precedure twice - one for each of the first root and the second root. By setting the initial value of 1 for first root search precedure, the precudure converges after 5 steps and returns a velue of 1.79129. The initial vlaue of the second root is set to -1 and the search precedure converges after 7 steps with a returned value of -2.79129. Both returned values are then plug into equation 2 for checking and both checks return zero. The first chart below shows the function line intercepts the x-axis ( f(x)=0 ), at approximate 0.55. The seconds chart shows the function line (curve) inptercepts the x-axis twice, at around 1.8 and -2.8.    