Post on 06-Oct-2015
description
Newtons Method
Lets find a root of the equation
That means finding a number such that . Such number is also
called a zero of the function .
Newtons Approach:
If is differentiable near the root, then tangent lines can be used to produce
a sequence of approximations to the root that approaches the root quite
quickly.
Make an initial guess at the root, say . The tangent to the curve
at is given by
Let be the zero of the tangent. Then, we have
Similar formulas produce from , then from , and so on. As a
generalization, we can write
which is known as the Newtons Method Formula.
Tools that can be used to calculate the successive approximations .
Calculator (Ex. http://web2.0calc.com/ )
Spreadsheet Software (Ex. Microsoft Excel)
Interactive Graphing Software (Ex. GeoGebra)
After trying all these tools, I found that GeoGebra is the most convenient
and flexible tool for solving problems of this type. Moreover, we can observe whether these approximations appear to converge to a limit.
The number will be a zero if exists, and if is continuous near
, then must be a zero of . However, convergence will not occur if the
graph of has horizontal or vertical tangent at any of the numbers in the
sequence. This method is known as Newtons Method or The Newton-
Raphson Method.
Example 1:
Use Newtons method to find all roots of the equation
correct to ten decimal places.
Solution:
To find the roots of the equation
That is, if we take
we need to approximate all zeroes of correct to six decimal places.
Now,
The Newtons Method Formula is
Let the initial guess be . Now, and . Thus,
Now, a question arises how to do all these calculations. Here, we have to
use a worksheet software, like Excel or GeoGebra Spreadsheet View.
Step 1:
Open GeoGebra
Step 2:
Input: f(x) = x^4 x-1. Graph of f will be displaced.
Step 3:
Input: Derivative[f]. Graph of f will be displayed
Step 4:
Options Rounding (Choose Number of Decimal Places)
Step 5:
View Spreadsheet (Select)
Step 6:
A Spreadsheet View appears
Step 7:
Enter initial guess 1 in the cell A1, =f(A1) in the cell B1, = f(A1) in
the cell C1 as shown below:
Step 8:
Enter =(A1)-((B1)/(C1)) in the cell A2, =f(A2) in the cell B2, =f(A2) in the
cell C2.
Then, select the cells A2, B2, and C2 and drag down.
Step 9:
This looks like the following after dragging down:
Step 10:
We can see that a zero near to x= 1 and corrected to 10 decimal places
would be
Note:
Sketching the graph would be useful in determining an initial guess . Even
a rough sketch of the graph of can show you how many roots the
equation has and approximately where they are. Usually, the closer
the initial approximation is to the actual root, the smaller the number of
iterations needed to achieve the desired precision.
The graph of the above function is:
We can see that there is another root near to . Now, we take
and just change its value in the cell A1 of the Spreadsheet View.
We can see that
is another root to the given equation.
Idea Once you are comfortable in using GeoGebra for solving equations using
Newtons Method, you can use the same GeoGebra file (save it as
newton.ggb on desktop) by just changing the function, number of decimal
places, and initial guess.
Example 2
Let us solve the equation to 11 decimal places.
Step 1:
Set number of decimal places to 15.
Step 2:
Change the function by double clicking on it in Algebraic View.
Step 3:
Change initial guess (Cell A1) in the Spreadsheet View. Let us take
based on the graph.
We can see that
First Warning
Before you try to use Newtons Method to find a real root of a function , you
should make sure that a real root actually exists. If you use the method of
starting with a real initial guess, but the function has no real root nearby,
the successive approximations can exhibit strange behavior.
Example 3:
Consider tehf unction . It is clear that has no real roots though
it does have complex roots . Here,
And hence, the Newtons Method formula for is
Let us take the initial guess , and iterate the formula for several times.
Here, let us plot the resulting points using GeoGebra as shown below.
Second Warning
Newtons Method does not always work as well as it does in the first two
examples. A single iteration of the formula can take us from quite close to
the root to quite far away
If the first derivative is very small near the root, or
If the second derivative is very large near the root.
Example 4 (Divergent Oscillations):
Let us apply Newtons Method to with
Solution:
Here,
And,
Now, the Newtons Method formula is
Now,
Here, the first derivative is very small near the root.
Example 5 (Convergent Oscillations):
Let us take the function
with initial guess
Now,
The Newtons Method formula is given by
Thus,
, , , , ,
Example 6 (Oscillation):
Let us take and
Now,
The Newtons Method formula is
Given that