Mathematics and Best Estimate
below shows some of the formulae entered to generate the spreadsheet above. Extrapolation in terms of a diagram and geometric progressions
T8 T16 “T32”“ T64”
X
According to the theory derived earlier 32 16 16 8
1 ( -
4
T T≈ + T T ) This gives us the so called “extrapolated” value 32 16 16 8
1 " " ( -). 4
T T TT = +
Note, this is exactly how “T32” was calculated on the previous page. And then
2 2
64 32 16 8 16 16 8 16 8
1 11 " "" " ( -) ( -) ( - 4 44
T T TT T TT TT ⎛ ⎞ ⎛ ⎞ = + =+ + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ) This is exactly how “T64” was calculated on the previous page. We can obviously continue to do this as many times as we like. In fact if we do this several times (infinitely many times in fact) we are reminded of the expression below which involves a geometric progression.Formula When to use
Midpoint Rule: M (f ( ) f ( ) ... f ( )) n n = + ++ hm m m 1 2 , where b a h n − = .
Obvious
Trapezium Law: T f 2(f f f ... f ) f [ ] 0 123 1 2 n n n h = + ++++ + − , where b a n − h = .
Obvious
Simpson’s Law: 2M T S
3
n n n + =
When you have Mn and
Tn this is the quickest way to get Sn
Simpson’s Law: S f 4(f f f ... f ) 2(f f ... f ) f [ ] 0 1 3 5 21 2 4 22 2 3 n n h = + ++++ + +++ + − −n n where
2
b a h n − = . Not often used, this is another way to get Sn, but often it is quicker to calculate Mn and Tn and then use the previous formulae 2 2 n n
T M T + = n for example, 4 4
8 2
T M T + = A quick way to get T2n when you have Mn and
Tn
2 4
3
n n T T S − = n , for example 32 16
16
4
3
T T S − = A quick way to get Sn when you have T2n and
Tn
Best estimate based on calculated valuesMn and M2nobtained from a G.P.
2 4
3
M n − Mn "Best midpoint" = , e.g. 16 8 4
3
M − M
Use this for a quick best estimate when you have calculated two successive midpoint values like M8 and M16
T8 T16 “T32”“ T64”
X
According to the theory derived earlier 32 16 16 8
1 ( -
4
T T≈ + T T ) This gives us the so called “extrapolated” value 32 16 16 8
1 " " ( -). 4
T T TT = +
Note, this is exactly how “T32” was calculated on the previous page. And then
2 2
64 32 16 8 16 16 8 16 8
1 11 " "" " ( -) ( -) ( - 4 44
T T TT T TT TT ⎛ ⎞ ⎛ ⎞ = + =+ + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ) This is exactly how “T64” was calculated on the previous page. We can obviously continue to do this as many times as we like. In fact if we do this several times (infinitely many times in fact) we are reminded of the expression below which involves a geometric progression.Formula When to use
Midpoint Rule: M (f ( ) f ( ) ... f ( )) n n = + ++ hm m m 1 2 , where b a h n − = .
Obvious
Trapezium Law: T f 2(f f f ... f ) f [ ] 0 123 1 2 n n n h = + ++++ + − , where b a n − h = .
Obvious
Simpson’s Law: 2M T S
3
n n n + =
When you have Mn and
Tn this is the quickest way to get Sn
Simpson’s Law: S f 4(f f f ... f ) 2(f f ... f ) f [ ] 0 1 3 5 21 2 4 22 2 3 n n h = + ++++ + +++ + − −n n where
2
b a h n − = . Not often used, this is another way to get Sn, but often it is quicker to calculate Mn and Tn and then use the previous formulae 2 2 n n
T M T + = n for example, 4 4
8 2
T M T + = A quick way to get T2n when you have Mn and
Tn
2 4
3
n n T T S − = n , for example 32 16
16
4
3
T T S − = A quick way to get Sn when you have T2n and
Tn
Best estimate based on calculated valuesMn and M2nobtained from a G.P.
2 4
3
M n − Mn "Best midpoint" = , e.g. 16 8 4
3
M − M
Use this for a quick best estimate when you have calculated two successive midpoint values like M8 and M16