Opened 12 years ago
Closed 9 years ago
#10170 closed enhancement (fixed)
Speed up the computation of Bell numbers
Reported by: | gerbicz | Owned by: | sage-combinat |
---|---|---|---|
Priority: | major | Milestone: | sage-5.10 |
Component: | combinatorics | Keywords: | bell number |
Cc: | sage-combinat | Merged in: | sage-5.10.beta1 |
Authors: | Robert Gerbicz, Travis Scrimshaw | Reviewers: | Ben Salisbury |
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | Stopgaps: |
Description (last modified by )
For large n values use the formula: bell(n) = exp(-1)*sum(k=0, infinity, k**n/k!)
Currently my code beats all known implementations, also sage's code which wraps GAP's Bell is slow and using lots of memory, and unable to compute bell_number(n) if n is large.
Some timings on Amd 2.1 GHz:
n time (Wall time)
300 0.01 sec.
1000 0.03 sec.
3000 0.18 sec.
10000 2.88 sec.
30000 46.79 sec.
100000 819.37 sec.
Compare these with another programs: http://fredrik-j.blogspot.com/2009/03/computing-generalized-bell-numbers.html
Also exposes mpmath's bell()
as an alternative algorithm.
Attachments (3)
Change History (19)
Changed 12 years ago by
Changed 12 years ago by
comment:1 follow-up: ↓ 2 Changed 12 years ago by
- Status changed from new to needs_info
comment:2 in reply to: ↑ 1 ; follow-up: ↓ 3 Changed 12 years ago by
- Status changed from needs_info to needs_review
Replying to jbandlow:
Hello and thanks for the patch! I'm curious, have you compared your solution with wrapping mpmath directly? For example,
sage: import mpmath sage: mpmath.bell?
Yes, but times are not saved I will rerun them: here are the timings for mpmath:
n time (Wall time)
300 0.02 sec.
1000 0.06 sec.
3000 0.41 sec.
10000 10.26 sec.
30000 168.27 sec.
100000 (still computing)
So here mpmath is about 2-4 times slower, and seems that using more memory than my code.
comment:3 in reply to: ↑ 2 Changed 12 years ago by
comment:4 Changed 12 years ago by
- Status changed from needs_review to needs_work
Patch will not apply to sage 4.5.3
comment:5 Changed 9 years ago by
- Cc sage-combinat added
- Description modified (diff)
- Milestone set to sage-5.9
- Status changed from needs_work to needs_review
comment:6 Changed 9 years ago by
I've folded in the changes from #14247 and marked that as a duplicate since there's no need for a separate ticket.
For patchbot:
Apply: trac_10170-bell_number_improvements-ts.patch
comment:7 Changed 9 years ago by
Hmmm... Looks nice, but shouldn't this be in that module ?
http://www.sagemath.org/doc/reference/combinat/sage/combinat/expnums.html
Nathann
comment:8 Changed 9 years ago by
- Description modified (diff)
From my 10 minutes of googling, no, it should not be. The exponential numbers seem to be a generalization of the Bell numbers and I don't know that this implementation can be generalized to that (and hence go into that module). My guess is that it's probably yes, but I'll leave that to someone with more expertise to answer.
comment:9 Changed 9 years ago by
At the line where you define E_1
and E_2
they appear to be perfectly equal. Is that intended ? O_o
Nathann
comment:10 follow-up: ↓ 11 Changed 9 years ago by
That is not true, in fact: E_2=exp(-1)*E_1 as you can read in my original bell_proof.txt
About exponential numbers: that is a different problem. Here we compute only a single Bell number not the first n Bell numbers and there the algorithm is optimal (O(n*n) time), the same question for Bell numbers is an open problem.
comment:11 in reply to: ↑ 10 Changed 9 years ago by
That is not true, in fact:
I have no idea, but this is what is written in the patch.
Nathann
comment:12 Changed 9 years ago by
Yes, there is a missing parentheses in line 255 in the patch.
comment:13 Changed 9 years ago by
Fixed.
comment:14 Changed 9 years ago by
- Milestone changed from sage-5.9 to sage-5.10
- Reviewers set to Ben Salisbury
- Status changed from needs_review to positive_review
comment:15 Changed 9 years ago by
- Description modified (diff)
comment:16 Changed 9 years ago by
- Merged in set to sage-5.10.beta1
- Resolution set to fixed
- Status changed from positive_review to closed
Hello and thanks for the patch! I'm curious, have you compared your solution with wrapping mpmath directly? For example,
Also, this ticket is basically a dup of #7812, so one of these (maybe that one, since this has a patch) should be closed.
Finally, don't forget to set the status as 'needs_review' once you've added a patch!
Thanks, Jason