OK, apparently the Dedekind sums should be done using an algorithm due
to Apostol. I haven't tracked it down yet, but this is clearly what we
need.
Bill.
On 26 Jul, 23:37, Bill Hart <[EMAIL PROTECTED]> wrote:
> On 26 Jul, 23:18, Jonathan Bober <[EMAIL PROTECTED]> wrote:
>
> > s(h,k) = h(k - 1)(2k - 1)/(6k) - (k-1)/4 -
> > (1/k) sum_{j=1}^{k-1} j*floor(hj/k).
>
> > (To compute the floor in the inner sum, you can just use integer
> > division.)
>
> Yes, this will be faster. Of course integer division is not faster
> than floating point division, but since we are doing a sum from j = 1
> to k-1 we only need to know when floor(hj/k) increases by 1. This is
> simply a matter of adding h each iteration and seeing if we have gone
> over k (subtracting k if we do). If so, floor(hj/k) has increased by 1
> and so on.
>
> This gets the inner computation down to about 8 cycles or so on
> average. Not enough to beat Mathematica though, unless I made a
> mistake in my back of the envelope computation somewhere.
>
> Bill.
--~--~---------~--~----~------------~-------~--~----~
To post to this group, send email to sage-devel@googlegroups.com
To unsubscribe from this group, send email to [EMAIL PROTECTED]
For more options, visit this group at http://groups.google.com/group/sage-devel
URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/
-~----------~----~----~----~------~----~------~--~---
