On Jul 2, 12:28 pm, David Harvey <[EMAIL PROTECTED]> wrote:
Hi David,
> Returning to a slightly old thread.
I meant to answer earlier ...
> I have implemented a new algorithm for computing large Bernoulli
> numbers.
>
> Running on 10 cores for 5.5 days, I computed B_k for k = 10^8, wh
Returning to a slightly old thread.
I have implemented a new algorithm for computing large Bernoulli
numbers.
Running on 10 cores for 5.5 days, I computed B_k for k = 10^8, which
I believe is a new record. (Recall the Mathematica blog post from
April was for k = 10^7.)
Essentially it'
On Tue, May 6, 2008 at 11:55 AM, David Harvey <[EMAIL PROTECTED]> wrote:
>
>
> On May 6, 2008, at 2:19 PM, Mike Hansen wrote:
>
> >> Probably not so cool, since it would be like 50 machines vs one
> >> machine.
> >
> > Sure, but the Mathematica blog post is scalablity: "In Mathematica, a
> >
On May 6, 2008, at 2:19 PM, Mike Hansen wrote:
>> Probably not so cool, since it would be like 50 machines vs one
>> machine.
>
> Sure, but the Mathematica blog post is scalablity: "In Mathematica, a
> core principle is that everything should be scalable. So in my job of
> creating algorithms
I agree, I think demonstrating a distributed algorithm would be very
cool. From what I can tell of processor trends, we won't see enormous
gains in speed but we might see an awful lot of processors (like
Intel's prototype 80-core chip).
On May 6, 12:19 pm, "Mike Hansen" <[EMAIL PROTECTED]> wrote
> Probably not so cool, since it would be like 50 machines vs one machine.
Sure, but the Mathematica blog post is scalablity: "In Mathematica, a
core principle is that everything should be scalable. So in my job of
creating algorithms for Mathematica I have to make sure that
everything I produce
On May 6, 2008, at 2:12 PM, Mike Hansen wrote:
> I think a blog post with PARI timings and then timings for a modular
> dsage approach would be cool.
Probably not so cool, since it would be like 50 machines vs one machine.
david
--~--~-~--~~~---~--~~
To post t
I think a blog post with PARI timings and then timings for a modular
dsage approach would be cool.
--Mike
On Tue, May 6, 2008 at 11:08 AM, William Stein <[EMAIL PROTECTED]> wrote:
>
>
> On Tue, May 6, 2008 at 10:57 AM, David Harvey <[EMAIL PROTECTED]> wrote:
> >
> >
> >
> > On May 6, 2008,
On Tue, May 6, 2008 at 10:57 AM, David Harvey <[EMAIL PROTECTED]> wrote:
>
>
>
> On May 6, 2008, at 1:18 PM, William Stein wrote:
>
> >
> > On Tue, May 6, 2008 at 10:15 AM, <[EMAIL PROTECTED]> wrote:
> >>
> >> William has mentioned some congruence tests that we can perform
> >> -- I'd like
On May 6, 2008, at 1:18 PM, William Stein wrote:
>
> On Tue, May 6, 2008 at 10:15 AM, <[EMAIL PROTECTED]> wrote:
>>
>> William has mentioned some congruence tests that we can perform
>> -- I'd like to make sure that I got the right answer before we pat
>> ourselves on the back too much.
>>
On Tue, May 6, 2008 at 10:15 AM, <[EMAIL PROTECTED]> wrote:
>
> William has mentioned some congruence tests that we can perform -- I'd like
> to make sure that I got the right answer before we pat ourselves on the back
> too much.
>
>
David Harvey's congruence tests would be pretty good. Jus
William has mentioned some congruence tests that we can perform -- I'd like to
make sure that I got the right answer before we pat ourselves on the back too
much.
On Tue, 6 May 2008, mhampton wrote:
>
> That certainly merits a blog post somewhere - ?
>
> On May 5, 2:02 pm, [EMAIL PROTECTED] w
On May 6, 2008, at 12:53 PM, mhampton wrote:
>
> That certainly merits a blog post somewhere - ?
>
> On May 5, 2:02 pm, [EMAIL PROTECTED] wrote:
>> My computation of bernoulli(10^7+4) using GP version 2.3.3 has
>> completed in 217417011 miliseconds. That's about 2 days, 12
>> hours. Anybod
That certainly merits a blog post somewhere - ?
On May 5, 2:02 pm, [EMAIL PROTECTED] wrote:
> My computation of bernoulli(10^7+4) using GP version 2.3.3 has completed in
> 217417011 miliseconds. That's about 2 days, 12 hours. Anybody know how I
> can print the thing to file?
>
> Machine:
>
On Mon, May 5, 2008 at 1:02 PM, <[EMAIL PROTECTED]> wrote:
>
> My computation of bernoulli(10^7+4) using GP version 2.3.3 has completed in
> 217417011 miliseconds. That's about 2 days, 12 hours. Anybody know how I
> can print the thing to file?
>
So PARI is already over twice as fast as Mat
My computation of bernoulli(10^7+4) using GP version 2.3.3 has completed in
217417011 miliseconds. That's about 2 days, 12 hours. Anybody know how I can
print the thing to file?
Machine:
Quad-core 2.0Ghz Xeon, 1333MHz FSB, 32GB RAM.
Currently, my gp session is using 4GB of RAM.
--~--~-
I probably also mean:
Then the error in the the zeta function
not:
Then the error in the inverse of the zeta function
Bill.
On 3 May, 13:12, Bill Hart <[EMAIL PROTECTED]> wrote:
> I think I nearly understand what Pari does.
>
> The value of B_k is given by zeta(n)*(2*n!)/(2^n pi^n). However,
Sorry, this:
The log of this expression is never more than 0.1 of the log of n!.
should read:
The log of this expression is always within 0.1 of the log of n!.
Bill.
On 3 May, 13:12, Bill Hart <[EMAIL PROTECTED]> wrote:
> I think I nearly understand what Pari does.
>
> The value of B_k is giv
I think I nearly understand what Pari does.
The value of B_k is given by zeta(n)*(2*n!)/(2^n pi^n). However,
zeta(n) is *very* close to 1 for large n. So one starts by computing
zeta to a precision given by the size of (2*n!)/(2^n pi^n) (which is
basically the size of B_k) with 3 added to the pre
Actually, it might be n/log(n) steps, so the time might be something
like n^2 though there are other terms involved.
Bill.
On 3 May, 00:30, Bill Hart <[EMAIL PROTECTED]> wrote:
> The theoretical complexity of all the algorithms that rely on
> recurrences is supposed to be n^2. But this doesn't t
The theoretical complexity of all the algorithms that rely on
recurrences is supposed to be n^2. But this doesn't take into account
the size of the numbers themselves. When you do this they are all
about n^3 as far as I can see. You can use Ramanujan identities, the
Akiyama-Tanigawa algorithm, the
I did some computations using von Staudt's theorem and up to 40 no
errors. Of course that doesn't prove anything for much larger n.
Bill.
On 2 May, 21:04, "William Stein" <[EMAIL PROTECTED]> wrote:
> On Fri, May 2, 2008 at 12:55 PM, David Harvey <[EMAIL PROTECTED]> wrote:
>
> > On May 2, 20
On May 2, 10:34 pm, "didier deshommes" <[EMAIL PROTECTED]> wrote:
> Here is some more information about the machine used to compute this:
Hi,
> Hi Didier,
>
> I used Linux, with 64 bit AMD processor:
>
> AMD Opteron(tm) Processor 250
> cpu MHz : 1000.000
> cache size : 1024 KB
Here is some more information about the machine used to compute this:
-- Forwarded message --
From: Oleksandr Pavlyk <[EMAIL PROTECTED]>
Date: Fri, May 2, 2008 at 4:29 PM
Subject: Re: Today We Broke the Bernoulli Record: From the Analytical
Engine to Mathematica
To: didier deshomm
One more data point (2.6GHz opteron):
sage: time x = bernoulli(6)
Wall time: 3.79
sage: time x = bernoulli(12)
Wall time: 16.97
sage: time x = bernoulli(24)
Wall time: 118.24
sage: time x = bernoulli(48)
Wall time: 540.25
sage: time x = bernoulli(96)
Wall time: 2436.06
Th
Sorry, the y-axis in the lower plot is log(time in seconds).
On Fri, 2 May 2008, David Harvey wrote:
>
>
> On May 2, 2008, at 4:08 PM, [EMAIL PROTECTED] wrote:
>
>> Funny this should come up. William just gave a take-home midterm
>> in which we had to predict the runtime for various computatio
On May 2, 2008, at 4:08 PM, [EMAIL PROTECTED] wrote:
> Funny this should come up. William just gave a take-home midterm
> in which we had to predict the runtime for various computations, so
> I wrote some generic code to help. According to my code, and some
> liberal assumptions, it shou
Funny this should come up. William just gave a take-home midterm in which we
had to predict the runtime for various computations, so I wrote some generic
code to help. According to my code, and some liberal assumptions, it should
take 5.1 days. I've attached the plots that show the curves I f
On Fri, May 2, 2008 at 12:55 PM, David Harvey <[EMAIL PROTECTED]> wrote:
>
>
> On May 2, 2008, at 3:45 PM, William Stein wrote:
>
> > The complexity mostly depends on the precision one uses in
> > computing a certain Euler product approximation to zeta
> > and also the number of factors in the
On May 2, 2008, at 3:45 PM, William Stein wrote:
> The complexity mostly depends on the precision one uses in
> computing a certain Euler product approximation to zeta
> and also the number of factors in the product. If you look
> at the PARI source code the comments do *not* inspire confidence
On Fri, May 2, 2008 at 12:41 PM, David Harvey <[EMAIL PROTECTED]> wrote:
>
>
> On May 2, 2008, at 3:40 PM, William Stein wrote:
>
> > Also, when I tried
> >
> > bernoulli(10^7+2)
> >
> > directly in Sage there were a couple of issues that arose, since
> > that command
> > is much more
On May 2, 2008, at 3:43 PM, Bill Hart wrote:
> I think the asymptotics aren't going to go our way if we use pari. It
> takes 11s for 10^5 and I've been sitting here for quite a few minutes
> and didn't get 10^6 yet.
So far I have on a 2.6GHz opteron:
sage: time x = bernoulli(6)
Wall time:
On Fri, May 2, 2008 at 12:10 PM, John Cremona <[EMAIL PROTECTED]> wrote:
>
> ok, so the docstring reaveals (1) that the pari version is "by far the
> fastest" as I suspected, but also that for n>5 that we use a gp
> interface rather than the pari library " since the C-library interface
> t
I think the asymptotics aren't going to go our way if we use pari. It
takes 11s for 10^5 and I've been sitting here for quite a few minutes
and didn't get 10^6 yet.
I think pari uses the zeta function to compute bernoulli numbers.
If I'm reading the code right it first computes 1/zeta(n) using t
On Fri, May 2, 2008 at 3:40 PM, William Stein <[EMAIL PROTECTED]> wrote:
>
> On Fri, May 2, 2008 at 11:34 AM, Fredrik Johansson
> <[EMAIL PROTECTED]> wrote:
> >
>
> > Oleksandr Pavlyk reports on the Wolfram Blog that he has computed the
> > 10 millionth Bernoulli number using Mathematica:
>
On May 2, 2008, at 3:40 PM, William Stein wrote:
> Also, when I tried
>
> bernoulli(10^7+2)
>
> directly in Sage there were a couple of issues that arose, since
> that command
> is much more designed for smaller input. I fixed those small issues.
> I guess we'll see in a week ..
I hope
On Fri, May 2, 2008 at 11:34 AM, Fredrik Johansson
<[EMAIL PROTECTED]> wrote:
>
> Oleksandr Pavlyk reports on the Wolfram Blog that he has computed the
> 10 millionth Bernoulli number using Mathematica:
>
> http://blog.wolfram.com/2008/04/29/today-we-broke-the-bernoulli-record-from-the-analyti
On Fri, May 2, 2008 at 11:34 AM, Fredrik Johansson
<[EMAIL PROTECTED]> wrote:
>
> Oleksandr Pavlyk reports on the Wolfram Blog that he has computed the
> 10 millionth Bernoulli number using Mathematica:
>
> http://blog.wolfram.com/2008/04/29/today-we-broke-the-bernoulli-record-from-the-analyti
ok, so the docstring reaveals (1) that the pari version is "by far the
fastest" as I suspected, but also that for n>5 that we use a gp
interface rather than the pari library " since the C-library interface
to PARI
is limited in memory for individual operations" -- whatever that means!
I might take a look at this, as there are some ways fo computing B nos
which are very much faster tha others, and not everyone knows them.
Pari has something respectable, certainly.
John
2008/5/2 mhampton <[EMAIL PROTECTED]>:
>
> It takes about 30 seconds on my machine to get the 10^5 Bernoulli
On May 2, 2008, at 2:56 PM, mhampton wrote:
> It takes about 30 seconds on my machine to get the 10^5 Bernoulli
> number. The mathematica blog says it took a "development" version of
> mathematica 6 days to do the 10^7 calc. So it would probably take
> some work, but we are not that badly off
It takes about 30 seconds on my machine to get the 10^5 Bernoulli
number. The mathematica blog says it took a "development" version of
mathematica 6 days to do the 10^7 calc. So it would probably take
some work, but we are not that badly off as is.
-M. Hampton
On May 2, 12:34 pm, Fredrik Joha
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