Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
Beide Seiten der vorigen RevisionVorhergehende ÜberarbeitungNächste Überarbeitung | Vorhergehende ÜberarbeitungNächste ÜberarbeitungBeide Seiten der Revision | ||
analyticalengine:bernoullinumbercalculation [2015-04-21 22:23] – tim | analyticalengine:bernoullinumbercalculation [2015-05-06 18:29] – add remark on using equation 8 over 2 or 3 rainer | ||
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Zeile 15: | Zeile 15: | ||
In //Note G//, equation 8 shows the recursion formula used by AAL: | In //Note G//, equation 8 shows the recursion formula used by AAL: | ||
- | < | + | < |
and in terms of the number to be calculated: | and in terms of the number to be calculated: | ||
Zeile 29: | Zeile 29: | ||
Thus, the coefficients – which are not given in //Note G// – are: | Thus, the coefficients – which are not given in //Note G// – are: | ||
- | `A_0(n) = 1/2 < | + | `A_0(n) = 1/2 * (2n-1)/ |
`A_1(n) = -(2n) / 2` | `A_1(n) = -(2n) / 2` | ||
- | `A_3(n) = A_1(n) | + | `A_3(n) = A_1(n) * (2n-1)/3 * (2n-2)/4` |
- | `A_5(n) = A_3(n) | + | `A_5(n) = A_3(n) * (2n-3)/5 * (2n-4)/6` |
As each `A_i(n)` depends only on the previous | As each `A_i(n)` depends only on the previous | ||
Zeile 44: | Zeile 44: | ||
Note that the signs are different to those in //Note G//. | Note that the signs are different to those in //Note G//. | ||
+ | |||
+ | ====== Why not equation 2 or 3? ====== | ||
+ | |||
+ | On page 725, AAL shows two commonly used formulas in equation 2 and 3, that do calculate the n-th Bernoulli number just as a series without recursion to other Bernoulli numbers, i.e. as a function of n alone. Then she writes: | ||
+ | > As however our object is not simplicity or facility of computation, | ||
+ | |||
+ | Thus she probably delibrately took into account the indexing problem. | ||
+ | |||
====== The problem in line 21 ====== | ====== The problem in line 21 ====== | ||
- | The tabular programme in the //Diagram accompanying Translator' | + | The tabular programme in the //Diagram accompanying Translator' |
To calculate `B_7`, the lines 13 to 23 are executed once, then in line 24 and 25 the result is stored. | To calculate `B_7`, the lines 13 to 23 are executed once, then in line 24 and 25 the result is stored. | ||
Zeile 55: | Zeile 63: | ||
Howerver, in line 21 during the second round, the value to be multiplied is not `V_22`, containing `B_5`, but `V_23`, contining the just calculated `B_7`. | Howerver, in line 21 during the second round, the value to be multiplied is not `V_22`, containing `B_5`, but `V_23`, contining the just calculated `B_7`. | ||
- | AAL has evidently see that problem and writes on page 729 bottom: >The only exception to a //perfect identity// | + | AAL has evidently see that problem and writes on page 729 bottom: |
+ | >The only exception to a //perfect identity// | ||
However, while it is correctly assumed that this may be solved, there is no hint of how this is //easily provided//, that the loom of variable cards are changed with each round. | However, while it is correctly assumed that this may be solved, there is no hint of how this is //easily provided//, that the loom of variable cards are changed with each round. | ||
Zeile 65: | Zeile 74: | ||
However, to calculate millions of Bernoulli numbers, the AE would need a store of millions of numbers; this may be a place where she did not fully understand the problem. | However, to calculate millions of Bernoulli numbers, the AE would need a store of millions of numbers; this may be a place where she did not fully understand the problem. | ||
- | Alan Bromley in his 1998 article on page 44 left column, second to last paragraph, writes: >With hindsight, we can note that in the Analytical Engine (at least until 1840), Babbage did not posess the variable-address concept; that is, there was no mechanism by which the machine could, as a result of a calculation, | + | Alan Bromley in his 1998 article on page 44 left column, second to last paragraph, writes: |
+ | >With hindsight, we can note that in the Analytical Engine (at least until 1840), Babbage did not posess the variable-address concept; that is, there was no mechanism by which the machine could, as a result of a calculation, | ||
He refers this to solving matrices, and thus to a much much more complex problem. | He refers this to solving matrices, and thus to a much much more complex problem. | ||
Zeile 71: | Zeile 81: | ||
====== Actual Calculation ====== | ====== Actual Calculation ====== | ||
- | < | + | `A_0(1) = 1/2 * 1/3 = 1/6` |
`B_1 = A_0(1) = 1/6` | `B_1 = A_0(1) = 1/6` | ||
Zeile 77: | Zeile 87: | ||
`A_0(2) = 3/10` | `A_0(2) = 3/10` | ||
- | < | + | `A_1(2) = -2` |
- | `B_3 = A_0(2) + A_1(2) | + | `B_3 = A_0(2) + A_1(2) * B_1 = 3/10 - 2/6 = (9 - 10)/30 = - 1/30` |
`A_0(3) = 5/14` | `A_0(3) = 5/14` | ||
Zeile 85: | Zeile 95: | ||
`A_1(3) = -3` | `A_1(3) = -3` | ||
- | `A_3(3) = -3 < | + | `A_3(3) = -3 * 5/3 * 4/4 = -5` |
`B_5 = 5/14 - 3/6 + 5/30 = (5 - 7)/14 + 1/6 = -1/7 + 1/6 = 1/42` | `B_5 = 5/14 - 3/6 + 5/30 = (5 - 7)/14 + 1/6 = -1/7 + 1/6 = 1/42` | ||
Zeile 93: | Zeile 103: | ||
`A_1(4) = -4` | `A_1(4) = -4` | ||
- | `A_3(4) = -4 < | + | `A_3(4) = -4 * 7/3 * 6/4 = -14` |
- | `A_5(4) = -14 < | + | `A_5(4) = -14 * 5/5 * 4/6 = - 28/3` |
- | `B_7 = 7/18 - 4/6 + 14/30 - 28/(3< | + | `B_7 = 7/18 - 4/6 + 14/30 - 28/(3*42) = (7 - 12)/18 + 7/15 - 14/63 = -5/18 + 7/15 - 14/63 = (-25 + 28)/90 + 14/63 = 17/90 -14/63 = (119 - 140 ) / (7*9*10) = - 21/(7*9*10) = - 1/30` |