Bernoulli numbers and large factorials
02-12-2014, 08:56 PM
Post: #8
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
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 ---------------------------------------  8      17      31     100     17782 10      10      15      39      2511 12       6      10      21       681 14       5       7      13       268 16       4       5      10       133 20       3       3       6        50 24       2       3       4        26 30       2       2       3        13 40       1       1       2         7

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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