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Nächste Überarbeitung | Vorhergehende ÜberarbeitungNächste ÜberarbeitungBeide Seiten der Revision | ||
analyticalengine:bernoullinumbercalculation [2015-04-21 08:53] – created rainer | analyticalengine:bernoullinumbercalculation [2015-05-06 18:17] – rainer | ||
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- | ====== Note G and the calculation of Bernoulli numbers ====== | + | __false__ |
- | See [[wp> | ||
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- | They may have been selected as an example because: | ||
- | * They cannot be enumerated using the // | ||
- | * The values seem to be fairly random | ||
- | * The values require a machine that can handle very large numbers | ||
- | * In approximations of trigonometric functions, they allow less calculating steps than other approximations known at that time. Note that Tchebysheff' | ||
- | * They are a sequence where the n-th number can be calculated from the previous numbers | ||
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- | Note that variants for the indexing and sign for Bernoulli numbers were proposed since that time; the later forms allow shorter writing of the formulas. | ||
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- | In //Note G//, equation 8 shows the recursion formula used by AAL: | ||
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- | < | ||
- | `0 = -1/2 * (2n-1)/ | ||
- | </ | ||
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- | and in terms of the number to be calculated: | ||
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- | < | ||
- | `-B_(2n-1) = -1/2 * (2n-1)/ | ||
- | </ | ||
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- | Using `A_i` as abbreviation for the factors gives | ||
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- | < | ||
- | `B_(2n-1) = A_0(n) + A_1(n) * B_1 + A_3(n) * B_3 + ... + A_5(n) * B_(2n-3)` | ||
- | </ | ||
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- | In the text, AAL wrotes: // | ||
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- | Thus, the coefficients -- which are not given in //Note G//, are: | ||
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- | `A_0(n) = 1/2 * (2n-1)/ | ||
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- | `A_1(n) = -(2n) / 2` | ||
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- | `A_3(n) = A_1(n) * (2n-1)/3 * (2n-2)/4` | ||
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- | `A_5(n) = A_3(n) * (2n-3)/5 * (2n-4)/6` | ||
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- | This makes clear, that for `A_5` and the following the calculation steps are structurally equal and thus can be calculated within a loop. | ||
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- | Note the signs in contrast to //Note G//. | ||
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- | (to be continued) | ||
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