Newton's Method: A Computer Project
Newton's Method is used to find the root of an equation provided that the
function f[x] is equal to zero. Newton Method is an equation created before the days of
calculators and was
used to find approximate roots to numbers. The roots of the function are where the
function crosses the x axis. The basic principle behind Newton's Method is that the root
can be found by subtracting the
function divided by its derivative from the initial guess of the root.
Newtons Method worked well because an initial guess was given to put into the equation.
This is important because a wrong initial guess may give you the wrong root for the
function.
With Mathematica, a program for Newton's method can be produced and a graph of the
function can be made. From the graph, the a good initial guess can be made.
Although Newton's Method works to find roots for many functions, it does have its
disadvantages. The root sometimes cannot be found by using Newton's Method. The reason
it
sometimes cannot be found is because when the function is equal to zero, there is no
slope to the tangent line.
As seen in experimentation's, it is important to select an initial guess close to the
root because some functions have multiple roots. Failure to choose an initial value that
is close to the root
could result in finding a the wrong root or wasting a lot of time doing multiple
iterations while getting close to the actual root.
On some occasions, the program cannot find a root to an initial guess that is placed
into the program. In some instances Mathmatica could not find the root to the function,
like if it is a
parabola with its vertex is placed right on the y axis with its roots an equal distance
away in both directions. In a case like this, the computer could not decide which root
to work towards so it gave an
indeterminate answer.
Although Newton's Method does have its disadvantages, it is very effective for finding
the roots of most equations. The advantages definitely outweigh the slight
disadvantages, and that is
why it is still used to this day.
|