Flu

• almost defeated mine

M4RI

Pronounciation

• It is pronounced "MARI" now.

PLUQ Factorisation of Dense GF(2) Matrices

• learned a lot from Clement

• still work in progress, some initial code is written
• nothing to be shown yet, but will keep working
• work-in-progress, alpha, not working code will be released in a few days with the standard M4RI library

M4RI Improvements

• autodetection of L1 and L2 cache
• switch over to Strassen-Winograd Multiplication by default in Sage
• learned a potential further improvement to multiplication from Bill Hart (needs to be implemented)

• Performance on Core2Duo improved:

   1 sage: A = random_matrix(GF(2),3.2*10^4,3.2*10^4)
   2 sage: B = random_matrix(GF(2),3.2*10^4,3.2*10^4)
   3 sage: time C = A._multiply_strassen(B,cutoff=4092) #Old
   4 CPU times: user 51.32 s, sys: 0.14 s, total: 51.46 s
   5 Wall time: 51.86
   6 
   7 sage: time C = A._multiply_strassen(B,cutoff=8192) #New
   8 CPU times: user 44.93 s, sys: 0.15 s, total: 45.08 s
   9 Wall time: 45.32

   1 sage: A = random_matrix(GF(2),3.2*10^4,3.2*10^4)
   2 sage: time A.echelonize() #Old
   3 CPU times: user 53.67 s, sys: 0.05 s, total: 53.71 s
   4 Wall time: 53.99
   5 
   6 sage: A = random_matrix(GF(2),3.2*10^4,3.2*10^4)
   7 sage: time A.echelonize() #New
   8 CPU times: user 44.25 s, sys: 0.03 s, total: 44.29 s
   9 Wall time: 44.50

• RAM consumption for elimination seems lower than Magma, since we don't use any temporaries due to the lack of asymptotically fast elimination. (after you substract the static Sage RAM).

  • Magma: Total time: 340.579 seconds, Total memory usage: 1934.02MB (for 64000² / 8 / 1024.0² = 488.281MB)

   1 sage: A = random_matrix(GF(2),6.4*10^4,6.4*10^4)
   2 sage: time A.echelonize()
   3 CPU times: user 357.87 s, sys: 1.26 s, total: 359.12 s
   4 Wall time: 362.16

> A:=RandomMatrix(GF(2),64*10^3, 64*10^3);
> time E:=EchelonForm(A);
Time: 336.350

Parallel M4RI

• Tried to implement parallel elimination and failed

• If it worked however it would enable in the only parallel Gröbner basis engine (PolyBoRi) in commutative polynomial rings I'm aware of.

Review Process

• Editor Meetings
• Reviews