[with patch] Make Sage's LLL faster: Magma seems to totally blow Sage (i.e., NTL) away for large-ish problems.

[was]

Magma's LLL is... way way way faster than the one in NTL. E.g., (this requires patch #325)

sage: a = random_matrix(ZZ,200)
sage: time b=a.lll()
CPU times: user 8.74 s, sys: 0.08 s, total: 8.83 s
Wall time: 8.85

sage: m = magma(a)
sage: time c=m.LLL()
Wall time: 1.02

sage: a = random_matrix(ZZ,400)
sage: time b=a.lll()
CPU times: user 202.89 s, sys: 1.54 s, total: 204.44 s
Wall time: 206.16
sage: time c=magma(a)
CPU times: user 0.24 s, sys: 0.02 s, total: 0.26 s
Wall time: 0.38
sage: time d=c.LLL()
Wall time: 13.23

It would also be good to benchmark PARI's LLL and compare. Make sure to use the 1 option, so it knows the matrix is integral:

sage: a = random_matrix(ZZ,100)
sage: time b=a.lll()
CPU times: user 0.53 s, sys: 0.00 s, total: 0.53 s
Wall time: 0.53
sage: c = pari(a)
sage: time d=c.qflll(1)
CPU times: user 0.47 s, sys: 0.00 s, total: 0.47 s
Wall time: 0.47
sage: a = random_matrix(ZZ,200)
sage: time b=a.lll()
CPU times: user 9.02 s, sys: 0.06 s, total: 9.08 s
Wall time: 9.14
sage: c = pari(a)
sage: time d=c.qflll(1)
CPU times: user 9.88 s, sys: 0.05 s, total: 9.93 s
Wall time: 9.95

[was]

David Kohel had by far the most interesting response on sage-devel to this ticket:

David Kohel <drkohel@gmail.com> 		 hide details	 6:00 am (6 hours ago) 
	reply-to		sage-devel@googlegroups.com	 
	to		sage-devel <sage-devel@googlegroups.com>	 
	date		Sep 21, 2007 6:00 AM	 
	subject		[sage-devel] Re: LLL	 
	mailed-by		googlegroups.com	 

Since LLL and LLLGram in Magma V2.13 are written by Damien Stehle,
using his asymptotically better algorithm.  The previous version was
not
even mathematically correct (adhoc "improvements" or post-processing
could destroy the LLL reduction condition).

Damien provides C code under GPL and can be found on his web page:

http://perso.ens-lyon.fr/damien.stehle/english.html

It might take some art to decide on optimal parameters, but linking it
into
SAGE should provide the same asymptotic performance as in Magma.

--David
- Show quoted text -

[malb]

The attached patch exposes NTL's floating point LLL implementations which are significantly faster than the exact one. However, these are approximates only.

[malb]

after patch:

double precision vs. MAGMA:

sage: a = random_matrix(ZZ,200)
sage: time b=a.LLL(algorithm="NTL:LLL_FP")
Time: CPU 0.91 s, Wall: 0.93 s

sage: m = magma(a)
sage: time c=m.LLL()
Wall: 1.33 s

sage: a = random_matrix(ZZ,400)
sage: time b=a.LLL(algorithm="NTL:LLL_FP")
Time: CPU 6.75 s, Wall: 6.94 s

sage: c=magma(a)
sage: time d=c.LLL()
Wall: 20.41 s

quad float precision vs. MAGMA:

sage: a = random_matrix(ZZ,200)
sage: time b=a.LLL(algorithm="NTL:LLL_QP")
Time: CPU 2.56 s, Wall: 2.60 s

sage: m = magma(a)
sage: time c=m.LLL()
Wall: 1.34 s

sage: a = random_matrix(ZZ,400)
sage: time b=a.LLL(algorithm="NTL:LLL_QP")
Time: CPU 31.36 s, Wall: 32.00 s

sage: c=magma(a)
sage: time d=c.LLL()
Wall: 20.13 s

However:

sage: B = MatrixSpace(IntegerRing(), 50, 51)(0
sage: for i in range(50): B[i,0] = ZZ.random_element(2**10000)
sage: for i in range(50): B[i,i+1] = 1;  

sage: B.LLL(algorithm="NTL:LLL_FP")
LLL_FP: numbers too big...use LLL_XD
...
<type 'exceptions.RuntimeError'>

sage: time B.LLL(algorithm="NTL:LLL_XD")
50 x 51 dense matrix over Integer Ring
CPU time: 43.55 s,  Wall time: 44.73 s

sage: M = magma(B)
sage: time C = M.LLL()
Wall: 15.53 s

[malb]

fpLLL as mentioned above is patched into SAGE by the patch found at

 http://sage.math.washington.edu/home/malb/fplll.patch

. Using this code, we can achieve a quite good performance.

sage:  = MatrixSpace(IntegerRing(), 50, 51)(0)
sage: for i in range(50): B[i,0] = ZZ.random_element(2**10000)
....:
sage: for i in range(50): B[i,i+1] = 1
....:
sage: time C = B.LLL('fpLLL:fast')
CPU times: user 9.54 s, sys: 0.00 s, total: 9.54 s
Wall time: 9.56

sage: C.is_LLL_reduced()
True

sage: BM = B._magma_()
sage: time CM = BM.LLL()
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 15.20

sage: time magma.eval("CM := LLL(%s:Fast:=1)"%BM.name())
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 11.68

[was]

This patch has fplll.c, which is Cython autogenerated code. Please make a clean new patch that doesn't have fplll.c or any other autogenerated code in it.

[was]

I *have* applied ntls-fp-lll.patch. The patch that isn't clean is

 http://sage.math.washington.edu/home/malb/fplll.patch

[was]

The Damien part has been moved to ticket 929.