Numerical Methods

1
Numerical Methods

• Lesson 11
• - Approximating Functions -

2
Finite difference tables.

• Have a look at the difference table below.

3
Truncating

• In this example none of the differences end up as
  zero.
• From the table since the ?4fi value is not equal
  to zero then a quartic function is required for
  an exact fit for this data.
• However the second difference is almost constant.
• Hence the third differences and so on are almost
  zero.
• Here using the polynomial up to the quadratic
  term will be sufficient as an approximation for
  the points.
• The technical term for ignoring the end of the
  polynomial is known as truncating.
• Lets look at what happens with the example from
  the first slide.
• First we need to find the polynomial up to the
  quadratic term.

4
Example

• Task Find the truncating polynomial up to the
  quadratic term.
• First we need the formula up to the x2 term.

• Next the key values from the table are
• f0 36.000 40
• ?f0 24.005 40
• ?2f0 8.031 91

5
Example

• Now plug the numbers in to the formula.
• Lets see what happens if we plug the original
  values back in to the quadratic we have just
  calculated.

6
Example

• If we plug the x-values from the original table
  in to the function we have just calculated then
  we get the following results.
• Now lets compare them to the original values.
• You can see that the quadratic is a pretty close
  approximation over the interval 5 (2) 13.

7
Question

• What degree of polynomial is required for a good
  approximation to the following data values?

• Solution
• Here truncating the third difference and using a
  quadratic will be a sufficient approximation.

8
Further Points

• Sometimes it may not be necessary to calculate a
  polynomial that goes through all of the data
  points.
• If you are interested in calculating values
  between two limits then you only need to
  calculate a polynomial over that range.

9
Further Points

• Have a look at the example below.
• The graph shows a quartic equation over the
  interval 0,7.
• Lets say that we only needed to look at values
  over the interval 3,6.
• Then a quadratic over this interval would be a
  good enough approximation.

10
Further Points

• Note You can always fit a polynomial to however
  many data points you have.
• It may not be the exact polynomial but it will
  always work for however many data points you
  have.

11
Further Points

• Lets have a look at how to find an approximate
  polynomial for part of your data.
• You can take the interpolating polynomial about
  any point.
• We have just been starting all of our examples at
  x0.
• You can adjust the formula for starting at any
  position through your data.
• Here is what the formula looks like if we
  calculate the polynomial from x2.
• Notice that it is exactly the same formula as
  before except all of the sub-scripts have been
  raised by 2.
• (Alternatively you could just adjust your
  notation to make x2, x0.)

12
Further Points

• Look at the example below.

f2
?f2
?2f2
?3f2
13
Further Points
f2
?f2
?2f2
?3f2
14
Further Points

• The graph below shows the shape of the polynomial
  formed from all of the points over the interval
  2,7.
• Now lets look at the polynomial through the
  points over the interval 4,7.

• If we combine the two on the same graph we can
  see that using the approximation over the
  interval is perfectly adequate.

15
Notation

• If you truncate the polynomial to f(x) f0 then
  this is referred to as a constant approximation
  taken about x0.
• If you truncate the polynomial to f(x) then
  this is referred to as a linear approximation
  taken about x0.
• If you truncate the polynomial to f(x) then
  this is referred to as a quadratic approximation
  taken about x0.
• If you truncate the polynomial to f(x) then
  this is referred to as a cubic approximation
  taken about x0.