|
|
In a Nut Shell:
Newton's
Method provides an iterative method, involving successive
approximations, to find
one or more roots of equations of the form, f(x) = 0. One key
is in selecting the
initial value for the root. It is
possible that each successive approximation
gets further and further
away from the actual root.
|
|
Newton's Method: If the nth approximation is given by xn, and if
f ' (xn) ≠
0, then
the next approximation
for the root is:
xn+1 = xn ˗ f(xn) / f ' (xn)
|
|
Strategy:
|
1. |
Write down the
expression for f(x).
|
|
2. |
Set up a table with
columns for xn ,
f(xn) , f ' (xn)
, f(xn) /f ' (xn),
and xn+1 .
|
|
3. |
By trial, identify the
starting value, x1. Note:
The value for x1
should not be such
that f ' (x1) =
0. Start the first
iteration using x1 .
|
|
4. |
Once the value for the
root, xn+1 is found, it represents the new
value for xn .
Then repeat the iteration until the desired accuracy
is met.
|
|
|
Applications using Newton's Method
Two common applications
using Newton's Method include:
1. Find an approximate value for a given
number such as the cube root of 31
by converting to an equation and then
using the value 31.
2. Find one or more roots of the
equations, f(x) =
0
|
|
Click here for examples.
|
|