Improve algorithm selection in fmpz_mat_rref

[fredrik-johansson]

Should fix #2129.

Based on the following profiling data with randtest matrices. For each row, the initial number is the number of rows r. For c = 1, 2, ... columns we print f when fmpz_mat_fflu is faster and M when fmpz_mat_rref_mul if faster. The fraction on the right is the ratio c / r of the largest c where M wins.

We see that M basically wins exactly when c <= r for r up to about 20 and in an extended triangular region for r <= 100. Presumably fflu still wins for r > 100 for ratios much larger than 2 but this doesn't seem particularly worth optimizing for (and the profiles take a long time to run).

bits = 10

   1 fff       0.000
   2 ffffff       0.000
   3 fffffff       0.000
   4 ffffffff       0.000
   5 fffffffff       0.000
   6 fffffMffff       1.000
   7 ffffffMffff       1.000
   8 fffffMMMffff       1.000
   9 fffffMMMMffff       1.000
  10 fffffMMMMMffff       1.000
  20 ffffMMMMMMMMMMMMMMMMffff       1.000
  30 ffffMMMMMMMMMMMMMMMMMMMMMMMMMMMffff       1.033
  40 ffffMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfff       1.025
  50 ffffMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMff       1.040
  60 ffffMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfff       1.117
  70 ffffMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMffM       1.200
  80 ffffMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM       1.262
  90 ffffMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfM       1.267
 100 ffffMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfMfMffff       1.880
 110 fffMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfMMMfMMfMMMMfMfMfMMffMfMfMff       2.355
 120 ffffMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfMMMMMMMMMMMMfMMMMMMMMMMMMMMMMMfMMMMMfMMMfMMMMMMfMffMMfMMfMfffMffff       2.525

bits = 100

   1 fff       0.000
   2 fffff       0.000
   3 ffffff       0.000
   4 ffMMfff       1.000
   5 ffMMMfff       1.000
   6 fMMMMMfff       1.000
   7 ffMMMMMfff       1.000
   8 fMMMMMMMfff       1.000
   9 fMMMMMMMMfff       1.000
  10 fMMMMMMMMMfff       1.000
  20 fMMMMMMMMMMMMMMMMMMMfff       1.000
  30 fMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfff       1.033
  40 fMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfff       1.025
  50 fMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfff       1.080
  60 fMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMff       1.233
  70 MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfMfff       1.529
  80 MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfMMMMMMMMffMMMfff       1.712
  90 MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfff       1.856
 100 MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfMMMMMMMfff       2.210

bits = 1000

   1 fff       0.000
   2 fffff       0.000
   3 ffMMMMMMMMM       3.667
   4 fMMMfff       1.000
   5 fMMMMfff       1.000
   6 fMMMMMfff       1.000
   7 fMMMMMMfff       1.000
   8 fMMMMMMMfff       1.000
   9 fMMMMMMMMfff       1.000
  10 fMMMMMMMMMfff       1.000
  20 fMMMMMMMMMMMMMMMMMMMfff       1.000
  30 fMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfff       1.033
  40 fMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfff       1.150
  50 fMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfff       1.260
  60 fMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfff       1.300
  70 fMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMfff       1.314

[fredrik-johansson]

Old and new timings for the 10000-bit random matrices in the original issue:

r = 1 c = 1        old: 0.000002 seconds           new: 0.000001 seconds
r = 2 c = 2        old: 0.000010 seconds           new: 0.000010 seconds
r = 3 c = 3        old: 0.000089 seconds           new: 0.000087 seconds
r = 4 c = 4        old: 0.000376 seconds           new: 0.000207 seconds
r = 5 c = 5        old: 0.001129 seconds           new: 0.000007 seconds
r = 6 c = 6        old: 0.002702 seconds           new: 0.000009 seconds
r = 7 c = 7        old: 0.005647 seconds           new: 0.000011 seconds
r = 8 c = 8        old: 0.010621 seconds           new: 0.000023 seconds
r = 9 c = 9        old: 0.017647 seconds           new: 0.000022 seconds
r = 10 c = 10        old: 0.027957 seconds           new: 0.000024 seconds
r = 11 c = 11        old: 0.042708 seconds           new: 0.000090 seconds
r = 12 c = 12        old: 0.060910 seconds           new: 0.000034 seconds
r = 13 c = 13        old: 0.086553 seconds           new: 0.000042 seconds
r = 14 c = 14        old: 0.118287 seconds           new: 0.000039 seconds
r = 15 c = 15        old: 0.159068 seconds           new: 0.000055 seconds
r = 16 c = 16        old: 0.209279 seconds           new: 0.000080 seconds
r = 17 c = 17        old: 0.269945 seconds           new: 0.000074 seconds
r = 18 c = 18        old: 0.343680 seconds           new: 0.000103 seconds
r = 19 c = 19        old: 0.431984 seconds           new: 0.000114 seconds
r = 20 c = 20        old: 0.536749 seconds           new: 0.000139 seconds
r = 21 c = 21        old: 0.000057 seconds           new: 0.000059 seconds
r = 22 c = 22        old: 0.000065 seconds           new: 0.000065 seconds
r = 23 c = 23        old: 0.000071 seconds           new: 0.000071 seconds
r = 24 c = 24        old: 0.000077 seconds           new: 0.000076 seconds
r = 25 c = 25        old: 0.000083 seconds           new: 0.000083 seconds
r = 26 c = 26        old: 0.000091 seconds           new: 0.000089 seconds
r = 27 c = 27        old: 0.000097 seconds           new: 0.000098 seconds
r = 28 c = 28        old: 0.000103 seconds           new: 0.000103 seconds
r = 29 c = 29        old: 0.000112 seconds           new: 0.000111 seconds
r = 30 c = 30        old: 0.000119 seconds           new: 0.000118 seconds
r = 31 c = 31        old: 0.000140 seconds           new: 0.000128 seconds
r = 32 c = 32        old: 0.000145 seconds           new: 0.000135 seconds
r = 33 c = 33        old: 0.000144 seconds           new: 0.000143 seconds
r = 34 c = 34        old: 0.000197 seconds           new: 0.000175 seconds
r = 35 c = 35        old: 0.000162 seconds           new: 0.000161 seconds
r = 36 c = 36        old: 0.000171 seconds           new: 0.000169 seconds
r = 37 c = 37        old: 0.000174 seconds           new: 0.000174 seconds
r = 38 c = 38        old: 0.000191 seconds           new: 0.000190 seconds
r = 39 c = 39        old: 0.000194 seconds           new: 0.000193 seconds
r = 40 c = 40        old: 0.000204 seconds           new: 0.000208 seconds

[albinahlback]

Very nice! Make FLINT fast again!