1. Numerical Solution Of Equations

2. This is only examined through coursework and will not appear on an exam paper.
If you can solve equations through the use of algebra then you should do.
Only if algebraic methods are not available should you use numerical methods.
It is useful to use Autograph to sketch the functions for which you are trying to solve.

3. Method 1:Change of Sign
We will look at roots of the equation f(x)=0, meaning where y=f(x) crosses the x axis.
Above the x axis y co-ordinates are positive, and below are negative. When the curve crosses the x axis we observe a change of sign.
Once we have located an interval in which f(x) changes sign we know that the interval must contain a root.

4. Intervals written as [a,b] includes a and b.
Intervals written as (a,b) don’t include a and b.

5. Decimal Search y=2x5-5x2+2 has roots in the intervals
[-1,0],[0,1] and [1,2].
To locate the roots you can use a decimal search.
This is almost like trial and error.

6. Excel is used for this – eliminates most errors.
With this method you take increments in x of size 0.1 within say [0,1]. We work out the value of f(x) until there is a change of sign.
Excel is used for this – eliminates most errors.
This process should be done to at least 5 decimal places x
f(x) 2
0.1
0.2
0.3
0.4
0.5
0.8125
0.6
0.7
0.6
0.61
0.62
0.63
0.64
0.65
0.66
0.67
0.68
Then take x to 2dp, that is intervals of 0.01

7. Failure + Decimal Search
Curve just touches the x axis – no change of sign.
Several roots close together – a decimal search is stopped when 1 root is found.
Discontinuity of f(x) – y=1/(x-3) has no roots but a decimal search would converge on a false root of x=3.

8. What to do? Find your own equations which have no integer solutions.
You need 1 that works and 1 that fails.
Failure should be where you have 2 roots together.
Show decimal search graphically – 3 diagrams zooming in.
Error bounds shown + checked.
You need to show which formulas you use in Excel.
Say why change of sign methods may not be good.