Numerical Methods

1
Numerical Methods

• Lesson 1
• Numerical Integration

2
Numerical Integration

• In mathematics it is very useful to calculate the
  area under the curve.
• This process is known as integration and you have
  already met this in your main A level.
• However there are often occasions where we are
  unable to integrate certain functions to
  calculate the area.
• This is were we would use a numerical method.
• We need to estimate the area underneath the
  curve.
• There are three methods that we are going to
  study.
• Mid-Point rule
• Trapezium rule
• Simpsons rule
• We will start with the Mid-Point rule.

3
Mid-Point Rule

• The Mid-Point rule uses rectangles to estimate
  the area under the curve.
• You use the mid-point of the interval you are
  trying to integrate, go up to the curve and draw
  a rectangle.

• This rectangle is used as an estimate for the
  area under the curve.
• You can see from the diagram that the more
  rectangles you use the better the estimate
  becomes.

4
Mid-Point Rule

• Lets look at an example with n 4.
• We are looking to integrate between the limits a
  b.
• We divide the interval up into 4 equal bars.
• Each bar will have a width of h. Where

• The height of each bar will be f(mn)
• Now the estimate for the area will be area 1
  area 2 area 3 area 4.
• Area hf(m1) hf(m2) hf(m3) hf(m4)
• h(f(m1) f(m2) f(m3) f(m4) )

yf(x)
f(m2)
f(m3)
f(m4)
f(m1)
2
3
4
1
h
a
b
m1
m2
m3
m4
5
Mid-Point Rule

• To calculate the areas of the bars we also need
  to find the values of m1 , m2 , m3 , m4,
• M1 a h/2
• M2 a 3h/2
• M3 a 5h/2
• M4 a 7h/2

yf(x)
f(m2)
f(m3)

• This estimate for the area is known as M4 as it
  uses four bars.

f(m4)
f(m1)
2
3
4
1
h
a
b
m1
m2
m3
m4
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Mid-Point Rule

• So in general
• Mn hf(m1) f(m2) f(m3) .. f(mn)
• Where the mn are the midpoints of each bar and h
  is the width of each bar.
• m1 a h/2
• m2 a 3h/2
• m3 a 5h/2
• ..
• mn a (2n 1)h/2

yf(x)
f(m2)
f(m3)
f(m4)
f(m1)
2
3
4
1
h
a
b
m1
m2
m3
m4
7
Example 1

• Use the Mid-Point rule to find an estimate for
  the area under the curve f(x) and the limits 0
  and 1 and n 1,2,4,8.
• Where

8
Trapezium Rule

• Another method for estimating the area under a
  curve is the trapezium rule.
• You need to be able to calculate the area of a
  trapezium.

• Calculate the area of the trapezium below.

5cm
7cm
6cm
9cm
9
Trapezium Rule

• With the trapezium rule you basically divide the
  area under the curve into trapeziums of equal
  height.
• You can then use the combined area of the
  trapeziums as an estimate for the area under the
  curve.

• The more trapeziums there are under the curve
  then the more accurate the estimate becomes.
• What will the estimate for this example be in
  comparison to the real answer?
• In this case you would have an under estimate as
  all the trapeziums are below the curve.

10
Trapezium Rule

• We are looking to find the area between the curve
  the x-axis and two limits a and b.
• For this example we are going to estimate the
  area by breaking it up in to 4 trapeziums.
• Each trapezium will have an equal height (h).
• What will be the formula for h?

2
3

• Now Area area 1 area 2 area 3 area 4.
• How do we calculate the area of each trapezium?
• We need to know the lengths of each of the orange
  bars.

4
1
h
a
b
11
Trapezium Rule

• What will be the x values for each of the bars?

2
3
4
1
h
a
b
ah
a2h
a3h
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Trapezium Rule

• The equation of the curve is y f(x).
• So the height of the first bar will be f(a).
• The height of the next will be f(ah) and so on.
• From this and the area of a trapezium we get

yf(x)
f(a2h)
f(ah)
f(a3h)
f(b)
2
3
4
f(a)
1
h
a
b
ah
a2h
a3h
13
Trapezium Rule

• This is getting a bit messy so lets change the
  notation a bit.
• Let f(a) f0, f(ah) f1, ., f(anh) f(b)
  fn

yf(x)
f(a2h)
f(ah)
f(ah)
f(b)
2
3
4
f(a)
1
h
a
b
ah
a2h
a3h
14
Trapezium Rule

• This is getting a bit messy so lets change the
  notation a bit.
• Let f(a) f0, f(ah) f1, ., f(anh) f(b)
  fn

yf(x)
f2
f1
f3
f4
2
3
4
f0
1
h
a
b
ah
a2h
a3h
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Trapezium Rule
yf(x)
f2
f1
f3

• First we sum

f4
2
3
4
f0
1
h
a
b
ah
a2h
a3h
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Trapezium Rule
yf(x)
f2
f1

• Finally, we could look at the case for n
  trapeziums.

f3
f4
2
3
4
f0
1
h
a
b
ah
a2h
a3h
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Example 2

• Use the Trapezium rule to find an estimate for
  the area under the curve f(x) and the limits 0
  and 1 and n 1,2,4,8.
• Where