Calculating the approximate solution to a nonlinear equations of the form
is an important computational task in its own right. This problems cannot be solved via direct methods. Iterative methods are required.
A root of Equation 1.2 of 
is a number 
such that 
.
This lesson considers the case of a differentiable real
function 
of a single real variable x such as the one given by
Fermat's principle in Equation 1.1.
Newton's method should be familiar from calculus.
Consider the situation in Figure 1.2.
Approximate the function 
by a straight line 
tangent to
at the most recent approximation 
.
The next approximation 
is the root of 
.
Equivalently, Newton's method can be derived by approximating 
by
the first order Taylor's expansion about 
,
This suggests solving
for 
to approximate 
if 
.
Hence,
This iteration needs to be stopped when two consecutive iterations are within acceptable tolerance (x-convergence), or when the function value is sufficiently small (f-convergence). These convergences are set through input tolerance parameters.. The method can be summarized in the following Algorithm.
This algorithm requires also a way to calculate the function 
and
its derivative 
which bring us again to the original problem:
Fermat's principle of least time traveled.
