1. “Teach A Level Maths” Vol. 2: A2 Core Modules
19: Newton-Raphson Iteration
© Christine Crisp

2. Module C3
MEI/OCR
"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

3. It isn’t always possible to find iterative formulae of the type
that will find the solution of every equation.
Another iterative method that is useful is called the Newton-Raphson method.

4. Suppose we want to find an approximate solution to the equation
To see how the method works, we’ll sketch
using
The root lies between 1 and 2.
We’ll zoom in near

5. Suppose our first estimate is given by .
We draw the tangent to the curve at
The point where the tangent meets the x-axis we call .
Repeating . . .
Each point , , is closer to .

6. To carry out the iteration we need to find the points where the tangents meet the x-axis.
The grad. of the tangent

7. We have and we need to find .
Using and in the formula isn’t very convenient, so, since we have
and
Then,
Rearranging:

8. So,
We just need to alter the subscripts to find :
Generalising gives
We don’t need a diagram to use this formula but we must know how to differentiate
Convergence is often very fast.

9. e.g. Use the Newton-Raphson method with
to find the root of the equation
correct to 4 d.p.
Solution:
Let
Differentiate:
Using a calculator we need:
Then,

10. SUMMARY
To use the Newton-Raphson method to estimate a root of an equation:
rearrange the equation into the form
differentiate to find
substitute and into the formula
choose a suitable starting value for
use a calculator to iterate
Tip: It saves a lot of errors if, before you type the formula into your calculator, you write the formula with ANS replacing every x.

11. Exercise
1. (a) Use the Newton-Raphson method to estimate the root of the following equation to 6 d.p. using the starting value given:
(b) What happens if you use ?
(c) Use your calculator or a graph plotter to sketch the graph of
(d) What is special about the graph at and why does it explain the answer to (b) ?
2. Use the Newton-Raphson method to estimate one root of to 4 d.p. using

12. (a)
Solution: Let

13. (b) What happens if you use ?
The iteration fails immediately.
(c)
At x = 0, there is a stationary point.
At a stationary point
so in the formula we are dividing by 0.
We also notice that the tangent never meets the x-axis.

14. 2. Use the Newton-Raphson method to estimate one root of to 4 d. p
2. Use the Newton-Raphson method to estimate one root of to 4 d.p. using
Solution:
Let
Radians!

15. The Newton-Raphson method will fail if
i.e. at a stationary point
It will also sometimes fail to give the expected root if the initial value is close to a stationary point. Can you draw a graph to show what could happen in this case?
This is one example.

16. With the iteration gives the root
instead of the closer root

18. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

19. It isn’t always possible to find iterative formulae of the type
that will find the solution of every equation.
Another iterative method that is useful is called the Newton-Raphson method.

20. SUMMARY
To use the Newton-Raphson method to estimate a root of an equation:
rearrange the equation into the form
choose a suitable starting value for
substitute and into the formula
differentiate to find
Tip: It saves a lot of errors if, before you type the formula into your calculator, you write the formula with ANS replacing every x.
use a calculator to iterate

21. e.g. Use the Newton-Raphson method with
to find the root of the equation
correct to 4 d.p.
Solution:
Let
Differentiate:
Using a calculator we need:
Then,

22. The Newton-Raphson method will fail if
i.e. at a stationary point
It will also sometimes fail if the initial value is close to a stationary point.