Bernoulli numbers and large factorials
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02-12-2014, 08:56 PM
Post: #8
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RE: Bernoulli numbers and large factorials
(02-09-2014 09:44 PM)Paul Dale Wrote: The 34S is computing the Bernoulli numbers from the zeta function. A short series expansion like you've got here will be faster I expect. Well, this series expansion, i.e. the sum in the formula, actually is the Zeta function. ;-) \( \begin{align*} B_n &= 2 n! {(2\pi)^{-n}} \sum\limits_{i=1}^{\infty}i^{-n}\\ &= 2 n! {(2\pi)^{-n}} \zeta (n) \end{align*} \) The 35s program works so fast because the larger n gets, the less terms are required. The following table shows the number of terms needed for an error of at most 0,1 ULP in Zeta: Code: n 10 12 16 34 digits This also explains why up to n = 8 the result is given directly. Otherwise the number of required terms would increase rapidly. I tried a program with the same algorithm on the 34s. In SP mode it is much faster than the internal Bernoulli function. For large n the result appears within a fraction of a second. As you will expect after a look at the above table, DP mode with 34 digit precision is a different story, at least for small n. ;-) Dieter |
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Messages In This Thread |
Bernoulli numbers and large factorials - Dieter - 02-09-2014, 04:59 PM
RE: Bernoulli numbers and large factorials - Tugdual - 02-09-2014, 07:47 PM
RE: Bernoulli numbers and large factorials - Marcus von Cube - 02-09-2014, 08:01 PM
RE: Bernoulli numbers and large factorials - Dieter - 02-09-2014, 09:26 PM
RE: Bernoulli numbers and large factorials - Dieter - 02-09-2014, 08:43 PM
RE: Bernoulli numbers and large factorials - Bunuel66 - 02-09-2014, 08:42 PM
RE: Bernoulli numbers and large factorials - Paul Dale - 02-09-2014, 09:44 PM
RE: Bernoulli numbers and large factorials - Dieter - 02-12-2014 08:56 PM
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