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