Opened 8 years ago
Last modified 8 years ago
#11036 closed enhancement
improve solve_mod performance — at Version 10
Reported by: | dsm | Owned by: | burcin |
---|---|---|---|
Priority: | minor | Milestone: | sage-4.7.2 |
Component: | symbolics | Keywords: | modular arithmetic, sd32 |
Cc: | kcrisman | Merged in: | |
Authors: | Douglas McNiel, Maarten Derickx | Reviewers: | John Cremona, Simon Spicer |
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | Stopgaps: |
Description (last modified by )
Currently solve_mod(x^2==41, 100)
breaks this down into solving modulus 2^2 and 5^2
, but it doesn't do further reductions, so it actually loops over 4 and 25 possibilities directly. This works well for small exponents:
sage: time solve_mod(x^2 == 41, 10^2) CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s Wall time: 0.01 s [(29,), (21,), (79,), (71,)]
but performs poorly on larger ones, even if the primes involved are small:
sage: time solve_mod(x^2 == 41, 10^6) CPU times: user 0.86 s, sys: 0.01 s, total: 0.87 s Wall time: 0.99 s [(703821,), (452429,), (47571,), (796179,), (203821,), (952429,), (547571,), (296179,)]
Apply only trac_11036-solve-mod.patch to the Sage library.
Change History (12)
comment:1 Changed 8 years ago by
- Cc kcrisman added
comment:2 Changed 8 years ago by
This patch doesn't attempt to do anything sophisticated, it simply reaches for the low-hanging fruit:
(1) It modifies solve_mod_enumerate so that if the modulus is 5^100
, instead of iterating over 5^100
possibilities, it finds solutions mod 5, then uses those to find solutions mod 5^2
, etc.
(2) It modifies solve_mod to short-circuit in some cases.
(3) It also makes an unrelated change: currently solve_mod(x==1, 1) returns [()], but should probably return [(0,)].
The change has little effect for small moduli but wins easily on anything nontrivial:
#OLD sage: time solve_mod(x^2==41, 10^7) CPU times: user 4.21 s, sys: 0.05 s, total: 4.27 s Wall time: 4.33 s [(3703821,), (1452429,), (3547571,), (1296179,), (8703821,), (6452429,), (8547571,), (6296179,)]
# NEW sage: time solve_mod(x^2==41, 10^7) CPU times: user 0.04 s, sys: 0.00 s, total: 0.04 s Wall time: 0.04 s [(3703821,), (1452429,), (3547571,), (1296179,), (8703821,), (6452429,), (8547571,), (6296179,)]
It still behaves very badly for large primes, but a lot better than the current approach on powers of small primes. This includes my use case of powers of 10-- there was an OEIS sequence I was extending which I had to do manually because solve_mod was too slow, and it irritated me that Mma could do it quickly..
Beyond the doctests I've tested it for small-coefficient univariate polynomials of order <= 5 with modulus <= 12, and for lots of random multivariate polynomials. Tests were against a four-line brute force solver. Probably I've missed something, though, so more tests would be appreciated!
comment:3 Changed 8 years ago by
- Status changed from new to needs_review
Changed 8 years ago by
comment:4 Changed 8 years ago by
- Status changed from needs_review to needs_work
trac_11036_improve_solve_mod_perf.patch does not apply to sage-4.7.1.alpha2 on sknet/taurus (x86_64-Linux-nehalem-fc):
taurus% ./sage ---------------------------------------------------------------------- | Sage Version 4.7.1.alpha2, Release Date: 2011-06-07 | | Type notebook() for the GUI, and license() for information. | ---------------------------------------------------------------------- ********************************************************************** * * * Warning: this is a prerelease version, and it may be unstable. * * * ********************************************************************** sage: hg_sage.apply("/home/mariah/trac_11036_improve_solve_mod_perf.patch") cd "/home/mariah/sage/sage-4.7.1.alpha2-x86_64-Linux-nehalem-fc-review-11036/devel/sage" && hg status cd "/home/mariah/sage/sage-4.7.1.alpha2-x86_64-Linux-nehalem-fc-review-11036/devel/sage" && hg status cd "/home/mariah/sage/sage-4.7.1.alpha2-x86_64-Linux-nehalem-fc-review-11036/devel/sage" && hg import "/home/mariah/trac_11036_improve_solve_mod_perf.patch" applying /home/mariah/trac_11036_improve_solve_mod_perf.patch patching file sage/symbolic/relation.py Hunk #3 FAILED at 735 Hunk #4 succeeded at 810 with fuzz 2 (offset 44 lines). Hunk #5 FAILED at 821 Hunk #6 FAILED at 830 3 out of 6 hunks FAILED -- saving rejects to file sage/symbolic/relation.py.rej abort: patch failed to apply sage:
Perhaps it needs to be rebased?
comment:5 Changed 8 years ago by
- Keywords modular arithmetic added
- Milestone set to sage-4.7.2
- Reviewers set to John Cremona
- Status changed from needs_work to needs_review
I did a rebase but also have managed to mess up my 4.7.1.rc[01] builds while testing other tickets, so no guarantees...
The code seems to assume that the expressions is a polynomial (in one or more variables), so there should definitely be a more efficient way to do the enumeration of solutions modulo prime powers -- this should be a linear (algebra mod p) problem once the solutions mod p are known, right?
comment:6 Changed 8 years ago by
- Reviewers changed from John Cremona to John Cremona, Simon Spicer
- Status changed from needs_review to positive_review
Looks fine; can't find any bugs, tests work, speedups as advertised. Revised patch works with 4.7.2alpha2.
comment:7 Changed 8 years ago by
If there were things I missed -- either preexisting bugs I didn't notice or bugs I introduced -- then we should probably add doctests to make sure they stay fixed.
comment:8 Changed 8 years ago by
the bugs I fixed where all in the existing doctests, I don't know how you got the timings because the code wasn't working at all
comment:9 Changed 8 years ago by
- Keywords sd32 added
comment:10 Changed 8 years ago by
- Description modified (diff)
Agreed that this is a place we need better work. It's been fairly low priority, as you can see. Ideas? Hensel's Lemma?