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 ID: 580514
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Slide1
Numerical MethodsSlide2
IntroductionThis chapter focuses on using some numerical methods to solve problems
We will look at finding the region where a root liesWe will learn what iteration is and how it solves equationsSlide3
Teachings for Exercise 4ASlide4
Numerical Methods You need to be able to show approximations for roots graphically and numerically
Show that the equation:x
3
- 3x
2
+ 3x – 4 = 0
has a root between 2 and 3
Draw the equation
4A
1
2
3
4
-1
-4
-3
-2
1
2
3
4
5
-1
-4
-3
-2
There is a root between 2 and 3
IMPORTANT POINT:
Either side of a root,
the y value changes sign…Slide5
Numerical Methods You need to be able to show approximations for roots graphically and numerically
Show numerically that the equation:
e
x
+ 2x – 3 = 0
has a root between x = 0.5 and x = 0.6…
4A
f(x) = e
x
+ 2x - 3
f(0.5) = e
0.5
+ 2(0.5) - 3
f(0.6) = e0.6 + 2(0.6) - 3
f(0.5) =-0.351…
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 = 0.6Slide6
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
4A
y =
1
/
xSlide7
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
4A
y =
1
/x
y = lnx
Only 1 intersectionSlide8
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
4A
lnx =
1
/x
lnx - 1/x = 0
ln(1.7) - 1/1.7
ln(1.8) - 1/1.8
Subtract
1/x
= -0.0576
= 0.0322
The sign has changed from negative to positive, so the root must be in this interval!Slide9
Teachings for Exercise 4BSlide10
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
4B
…But how would you work out square roots?
You could use a calculator, but what is the calculator doing?Slide11
Numerical Methods
Calculating Square RootsYour 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
4B
This formula will calculate a square root
x
n
is the first guess
a is the number being rooted
x
n + 1
is the answer which will be put in again
So if we want to calculate
√3…Slide12
Numerical Methods
Calculating Square RootsYour 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
4B
Using iteration to find √3
a = 3, we can use 2 as our ‘first guess’ (x
0
)
Type into the calculator (this would once have been done by hand!)
a = 3, we now use 1.75 (x
1
)
Type into the calculator (this would once have been done by hand!)Slide13
Numerical Methods
Calculating Square RootsYour 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
4B
Using iteration to find √3
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!Slide14
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:
4B
Add 4x, Subtract 1
Divide by xSlide15
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 x
0
= 3
4B
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 2dpSlide16
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…
4BSlide17
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:
4B
When solving this, we are looking at the intersections of two different graphs…Slide18
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…
4B
The point to understand from this is that
solving a rearranged equation is the same as solving the original one
!Slide19
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…
4B
We will now zoom in on part of the graph…Slide20
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 !
4B
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…Slide21
Numerical Methods
4BSlide22
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
4B
Add 5x, Add 3
Square root
Add 5x
Divide by 5Slide23
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 x
0
= 5
4B
x
4
is usually enough unless specified!Slide24
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 x
0
= 5
4B
Sometimes you will have to go further in order to find answers to a given number of decimal places…Slide25
SummaryWe have looked at various numerical methods that can be used to approximate solutions to equations
We have seen how to rearrange formulae for iterationWe have seen how iteration works…