Numerical Methods

1
Numerical Methods

• Lesson 2
• Numerical Integration

2
Mid-Point Trapezium

• There is a link between the mid-point rule and
  the trapezium rule.
• Given by the formula
• Prove the following

3
Mid-Point Trapezium
4
Upper and Lower bounds

• Lets look at a general curve y f(x).
• You can see that the curve is concave in this
  example
• We want to estimate the area between two bounds a
  b.

• If we use T1 we can clearly see that the
  approximation will be an over estimate.
• Now lets look what would happen if we use M1.
• Its difficult to see what is happening here so
  lets look at the area in a different way.
• We can change the area to a trapezium by cutting
  of a triangle.
• Here we can see that the approximation is an
  under estimate.

5
Upper and Lower bounds

• Now lets add back in to the diagram the
  approximation for T1.
• From this you can see that the Trapezium rule is
  an over estimate for the area and the midpoint
  rule is an under estimate.
• That is

• Draw a similar diagram to show that if f(x) is
  convex then

6
Upper and Lower bounds

• In fact for most well behaved functions, this
  will also apply to Tn Mn.
• This is important for two reasons.
• i) It is possible to produce upper and lower
  bounds for the exact area.
• (this means that we can give the area to a
  specified degree of accuracy)
• ii) It leads to the development of another
  method for approximating area.
• This is known as Simpsons rule.

7
Upper and Lower bounds

• In fact with the example we have looked at in
  lesson 1 Mn remains an over estimate and Tn an
  under estimate.
• From the table below it is clear that using M32
  T32 gives us an estimate for the area correct to
  3 d.p.

8
Different Example

• Given that
• a) Use the trapezium rule with 3 interval to
  estimate I.
• b) Evaluate I exactly using integration.
• c) Can you explain why you have got the
  results shown in parts a) b).

• Answers
• a) 45 units2.
• b) 45 units2.
• c)

9
Simpsons Rule

• Lets look again at the question we looked at in
  Lesson 1.
• If we calculate the exact value of this integral
  we get 0.913 986 to 6 d.p.
• Here again are the estimates that we calculated
  in lesson 1.

10
Simpsons Rule

• As we have discussed already in this example the
  Mid-point gives an over estimate and the
  Trapezium rule gives an under estimate.
• Next lets look at the absolute error in each
  estimation.
• First thing to notice is that the error quarters
  each time you double the intervals.
• (This concept will be useful in your coursework).

11
Simpsons Rule

• The second thing to notice is that the absolute
  error for T1 is roughly twice as big as the
  absolute error for M1.
• In fact this is true for all Tn Mn.
• This concept can be shown on a number line.

12
Simpsons Rule

• This weighted result is the basis for Simpsons
  rule.
• Simpsons rule takes into account that the actual
  value of the area is twice as close to Mn as it
  is to Tn.
• Simpsons rule is calculated by using the
  following weighted formula.

13
Simpsons Rule

• Lets look again at the number line for T1 M1.
• If you take the mean of these two values you can
  see that is not that close to the actual value.
• However if we use Simpsons rule we can see that
  Sn is a far better approximation.

14
Simpsons Rule

• Also if we look at the diagrams from the start of
  the lesson we can also see that the Mid-point
  rule estimate is closer to the real answer than
  the Trapezium rule estimate.

15
Question

• Use Simpsons rule to calculate the values of S1,
  S2, S4, S8 for the example.

16
Simpsons Rule

• Simpsons rule approximation can be calculated
  directly from f(x).
• The Trapezium rule uses n strips and evaluates
  f(x) on the edge of those intervals.
• The Mid-point rule uses n strips and evaluates
  f(x) in the middle of each interval.
• As Simpsons rule uses both of the above it must
  be evaluated with 2n strips.
• When you use Simpsons rule divide the interval
  up in to 2n strips and label from the left hand
  of the first strip to the right hand of the last
  strip.
• Using f0, f1,, f2n, we have

17
Example

• Find S4 for

18
Question

• Find to 6 decimal places, the approximations S1,
  S2, S4 for the integral