1. Numerical Methods

2. Introduction
This chapter focuses on using some numerical methods to solve problems
We will look at finding the region where a root lies
We will learn what iteration is and how it solves equations

3. Teachings for Exercise 4A

4. Numerical Methods
You need to be able to show approximations for roots graphically and numerically
Show that the equation:
x3 - 3x2 + 3x – 4 = 0
has a root between 2 and 3
 Draw the equation 5 4 3 2 1 -4 -3 -2 -1 1 2 3 4 -1 -2 -3 -4
There is a root between 2 and 3
IMPORTANT POINT:
 Either side of a root, the y value changes sign… 4A

5. Numerical Methods f(x) = ex + 2x - 3 f(0.5) = e0.5 + 2(0.5) - 3
You need to be able to show approximations for roots graphically and numerically
Show numerically that the equation:
ex + 2x – 3 = 0
has a root between x = 0.5 and x = 0.6…
f(x) = ex + 2x - 3
f(0.5) = e (0.5) - 3
f(0.5) =-0.351…
f(0.6) = e (0.6) - 3
f(0.6) = 0.022
The sign has swapped so the function has passed through the x-axis
 Hence, there is a root between x = 0.5 and x = 0.6 4A

6. Numerical Methods
You need to be able to show approximations for roots graphically and numerically
In General, if there is an interval where f(x) changes sign, there is a root in that interval.
The only exception is if there is a discontinuity in f(x), such as on the graph y = 1/x
On this graph, there is a missing value where the sign change would take place
y = 1/x 4A

7. Numerical Methods
You need to be able to show approximations for roots graphically and numerically
a) On the same set of axes, sketch the graphs of y = lnx and y = 1/x.
Hence show why the equation:
lnx = 1/x
has only one root.
b) Show that this root lies in the interval:
1.7 < x < 1.8
y = 1/x
y = lnx
Only 1 intersection 4A

8. Numerical Methods lnx = 1/x lnx - 1/x = 0 ln(1.7) - 1/1.7
You need to be able to show approximations for roots graphically and numerically
a) On the same set of axes, sketch the graphs of y = lnx and y = 1/x.
Hence show why the equation:
lnx = 1/x
has only one root.
b) Show that this root lies in the interval:
1.7 < x < 1.8
lnx = 1/x
Subtract 1/x
lnx - 1/x = 0
ln(1.7) - 1/1.7
ln(1.8) - 1/1.8 = =
The sign has changed from negative to positive, so the root must be in this interval! 4A

9. Teachings for Exercise 4B

10. Numerical Methods
You are able to work out lots of different types of sum:
For example, Multiplying. You could use the grid method
Addition or Subtraction. You use the column method
You can divide using the bus shelter
You can also deal with powers and fractions
…But how would you work out square roots?
You could use a calculator, but what is the calculator doing? 4B

11. Numerical Methods Calculating Square Roots
Your calculator will use a method called ‘iteration’
This involves putting a ‘guess’ number into a formula, and getting an answer
The answer is then put back into the formula, to get another answer
This process is repeated many times, and each time, the answer becomes more accurate
This formula will calculate a square root
xn is the first guess
xn + 1 is the answer which will be put in again
a is the number being rooted
So if we want to calculate √3… 4B

12. Using iteration to find √3
Numerical Methods
Using iteration to find √3
Calculating Square Roots
Your calculator will use a method called ‘iteration’
This involves putting a ‘guess’ number into a formula, and getting an answer
The answer is then put back into the formula, to get another answer
This process is reapeated many times, and each time, the answer becomes more accurate
a = 3, we can use 2 as our ‘first guess’ (x0)
Type into the calculator (this would once have been done by hand!)
a = 3, we now use 1.75 (x1)
Type into the calculator (this would once have been done by hand!) 4B

13. Using iteration to find √3
Numerical Methods
Using iteration to find √3
Calculating Square Roots
Your calculator will use a method called ‘iteration’
This involves putting a ‘guess’ number into a formula, and getting an answer
The answer is then put back into the formula, to get another answer
This process is reapeated many times, and each time, the answer becomes more accurate
a = 3, use the previous answer again
Make use of the Ans button on your calculator. After this you can effectively just keep pressing the = sign!
Keep going like this and you get closer to the exact answer to √3! 4B

14. can be written in the form:
Numerical Methods
You can use iteration to find approximations for f(x) = 0, to any desired degree of accuracy…
a) Show that:
can be written in the form:
Add 4x, Subtract 1
Divide by x 4B

15. Numerical Methods
You can use iteration to find approximations for f(x) = 0, to any desired degree of accuracy…
b) Use the iteration formula:
to find, to 2 decimal places, a root of the equation:
Use x0 = 3
You do not need to write this out every single time, but a few examples are needed
After a few iterations, the first few decimal places stop changing
 3.73 to 2dp 4B

16. But why does this work? What exactly is going on…
Numerical Methods
You can use iteration to find approximations for f(x) = 0, to any desired degree of accuracy…
But why does this work? What exactly is going on…
When we find the roots of this equation, we are really finding where 2 equations cross each other… 4B

17. Numerical Methods
You can use iteration to find approximations for f(x) = 0, to any desired degree of accuracy…
But why does this work? What exactly is going on…
This equation was rearranged to:
When solving this, we are looking at the intersections of two different graphs… 4B

18. Look at the x values that solve each pair of equations…
Numerical Methods
You can use iteration to find approximations for f(x) = 0, to any desired degree of accuracy…
Look at the x values that solve each pair of equations…
The point to understand from this is that solving a rearranged equation is the same as solving the original one! 4B

19. We will now zoom in on part of the graph…
Numerical Methods
You can use iteration to find approximations for f(x) = 0, to any desired degree of accuracy…
Now we know that solving the rearranged equation will still give the correct values, we can see what is happening through iteration…
We will now zoom in on part of the graph… 4B

20. Numerical Methods
You can use iteration to find approximations for f(x) = 0, to any desired degree of accuracy…
 It is important to realise that different starting numbers can result in convergence to different roots
 Different rearrangements for iteration can also yield different results
 It is also possible that this will not work and the answers diverge away from any roots. In this case, the correction is to change the starting number !
We assumed x = 3 (Left side) on our first guess
 This gave us a value of 3.66 on the right side
We then took this for our new x-value (Left side)
 Putting into the right side gave us a new value of 3.72
The process causes the x-values to converge at a root… 4B

21. Numerical Methods 4B

22. Can be written either as:
Numerical Methods
You can use iteration to find approximations for f(x) = 0, to any desired degree of accuracy…
Show that:
Can be written either as: or
Add 5x, Add 3
Square root
Add 5x
Divide by 5 4B

23. Numerical Methods
You can use iteration to find approximations for f(x) = 0, to any desired degree of accuracy…
Show that the iteration formulae:
Give different roots of the equation:
Use x0 = 5
x4 is usually enough unless specified! 4B

24. Numerical Methods
You can use iteration to find approximations for f(x) = 0, to any desired degree of accuracy…
Show that the iteration formulae:
Give different roots of the equation:
Use x0 = 5
Sometimes you will have to go further in order to find answers to a given number of decimal places… 4B

25. Summary
We have looked at various numerical methods that can be used to approximate solutions to equations
We have seen how to rearrange formulae for iteration
We have seen how iteration works…