The Newton-Raphson method is one of the most widely used root-finding methods. It starts with an initial guess of the root and iteratively finds better approximations by calculating the intersection of the tangent line from the previous guess with the x-axis. This process is repeated, using the previous approximation as the new starting point, until the difference between guesses is within a specified tolerance.

2. The Newton-Raphson method
and its modification is
probably the most widely
used of all root-finding
methods. Starting with an
initial guess x1 at the root,
the next guess x2 is the
intersection of the tangent
from the point [x1, f(x1)] to
the x-axis. The next guess
x3 is the intersection of the
tangent from the point [x2,
f(x2)] to the x-axis as shown
in Figure . The process can
be repeated until the desired
tolerance is attained.
THE NEWTON-RAPHSON METHOD
Graphical depiction of the Newton-
Raphson method.

4. MATLAB CODE