Augment kernel basis functionalities

[vneiger]

• provide a verification function is_kernel (which still lacks some generation check) and make testing more complete for kernel basis functions
• add a function for kernel via approximation at large order (useful for very unbalanced shifts)
• review recent code for ZLS algorithm and provide an "enhanced" version (see below)
• add a profile file to benchmark performance
• add an interface that provides a convenient choice for the orientation (left kernel, right kernel, and conventions for pivots in weak Popov form)

Enhanced because the modified version:

• returns a basis in a stronger form (ordered weak Popov form, rather than reduced)
• is often faster (sometimes marginally), thanks to the use of more aggressive parameters (in particular for the order when calling approximant basis).

[vneiger]

Some timings.
The column flint is for nmod_poly_mat_nullspace: this has a weaker output, by computing a basis of the rational nullspace (which most often does not generate the polynomial kernel via polynomial combinations), and not ensuring any kind of reducedness.

*****************************************************
*  left kernel of rdim x cdim matrix of degree deg  *
*****************************************************

bits rdim cdim deg     approx   zls(now) flint    zls(before)
50   5    1    2       2.94e-06 3.37e-06 1.77e-07 7.31e-06
50   5    1    5       9.05e-06 6.36e-06 1.84e-07 2.13e-05
50   5    1    10      2.36e-05 1.41e-05 2.02e-07 6.02e-05
50   5    1    20      6.84e-05 3.41e-05 2.03e-07 1.62e-04
50   5    1    40      1.66e-04 8.47e-05 2.09e-07 3.35e-04
50   5    1    80      4.77e-04 2.43e-04 2.33e-07 8.44e-04
50   5    1    160     1.24e-03 7.73e-04 4.48e-07 1.95e-03
50   5    1    320     3.18e-03 2.15e-03 6.98e-07 4.65e-03
50   5    1    640     7.68e-03 5.57e-03 1.27e-06 1.06e-02
50   5    1    1280    1.86e-02 1.47e-02 2.45e-06 2.40e-02
50   5    1    2560    4.63e-02 3.99e-02 4.82e-06 5.68e-02
50   5    1    5120    1.12e-01 9.05e-02 1.00e-05 1.31e-01
50   5    1    10240   2.59e-01 2.29e-01 2.21e-05 2.96e-01
50   5    4    2       1.08e-05 1.52e-05 3.25e-06 4.77e-05
50   5    4    5       5.22e-05 4.12e-05 8.20e-06 1.07e-04
50   5    4    10      2.11e-04 1.06e-04 2.38e-05 2.34e-04
50   5    4    20      7.88e-04 2.97e-04 6.20e-05 7.58e-04
50   5    4    40      2.36e-03 1.06e-03 1.59e-04 2.13e-03
50   5    4    80      6.72e-03 2.37e-03 3.91e-04 5.81e-03
50   5    4    160     1.80e-02 5.71e-03 9.51e-04 1.56e-02
50   5    4    320     4.78e-02 1.57e-02 2.43e-03 3.94e-02
50   5    4    640     1.37e-01 3.51e-02 5.11e-03 1.03e-01
50   5    4    1280    3.13e-01 8.79e-02 1.06e-02 2.54e-01
50   5    4    2560    7.71e-01 2.11e-01 2.20e-02 5.96e-01
50   5    4    5120    1.82e+00 5.13e-01 4.45e-02 1.44e+00
50   5    4    10240   4.37e+00 1.19e+00 1.12e-01 3.40e+00
50   100  1    2       4.91e-04 4.01e-04 7.33e-06 1.59e-03
50   100  1    5       1.51e-03 7.70e-04 7.53e-06 4.14e-03
50   100  1    10      4.57e-03 1.42e-03 7.68e-06 1.32e-02
50   100  1    20      3.57e-02 2.70e-03 7.75e-06 7.94e-02
50   100  1    40      1.38e-01 8.03e-03 8.61e-06 3.15e-01
50   100  1    80      4.43e-01 1.87e-02 1.08e-05 9.59e-01
50   100  1    160     1.28e+00 4.43e-02 1.51e-05 2.81e+00
50   100  1    320     3.48e+00 1.16e-01 2.05e-05 7.49e+00
50   100  50   2       2.58e-01 5.30e-03 1.84e-01 1.16e-02
50   100  50   5       2.35e+00 1.85e-02 1.02e+00 3.58e-02
50   100  50   10      7.19e+00 4.89e-02 2.71e+00 1.07e-01
50   100  50   20      2.08e+01 2.24e-01 6.64e+00 4.44e-01
50   100  50   40      5.70e+01 8.21e-01 1.61e+01 1.55e+00
50   100  50   80      xxxxxxxx 2.98e+00 3.58e+01 5.85e+00
50   100  50   160     xxxxxxxx 1.42e+01 xxxxxxxx 2.11e+01
50   100  99   2       8.96e-01 1.79e-02 6.48e-01 6.86e-02
50   100  99   5       1.08e+01 9.50e-02 3.72e+00 3.35e-01
50   100  99   10      4.02e+01 3.10e-01 1.04e+01 1.21e+00
50   100  99   20      1.25e+02 9.74e-01 2.75e+01 4.33e+00
50   100  99   40      xxxxxxxx 3.08e+00 6.71e+01 1.41e+01

[gilvillard]

See Issue #32

[vneiger]

(cancelled workflows because they are expected to hang, cf #39 )

[vneiger]

Recent commits significantly enhance the variety of tests performed for kernel bases. In the latest commits, the tests fail for left kernel on input:

<5 x 2 matrix over Z/nZ[x]>
[[0, 1],
[0, 0],
[0, 1],
[0, 0],
[0, 1]]

with modulus 2. This is in fact good news, because it looks similar to a failing case revealed in #32 , which was until now not detected by the testing (which was testing "too generic" cases only).

[vneiger]

The last commits bring fixes to the early exit, and various tests show no error so hopefully this is now correct in all cases. Efficiency hasn't changed on random input, see below for left kernels (first three columns are timings with uniform degree, last three columns with row degrees chosen at random between 0 and bound):

bits rdim cdim deg      flint   approx  zls-app  u_flint    u_app  u_zls-a
50   5    1    2     1.91e-07 2.88e-06 3.10e-06 1.55e-07 1.64e-07 2.19e-06
50   5    1    5     2.01e-07 8.90e-06 5.80e-06 2.02e-07 4.60e-06 4.70e-06
50   5    1    10    2.01e-07 2.36e-05 1.36e-05 1.81e-07 2.00e-05 1.03e-05
50   5    1    20    2.32e-07 6.87e-05 3.22e-05 2.05e-07 6.28e-05 2.89e-05
50   5    1    40    2.24e-07 1.66e-04 8.54e-05 1.93e-07 5.40e-05 5.33e-05
50   5    1    80    2.66e-07 4.51e-04 2.22e-04 1.88e-07 3.82e-04 2.46e-04
50   5    1    160   4.84e-07 1.26e-03 7.43e-04 2.74e-07 6.08e-04 4.07e-04
50   5    1    320   7.44e-07 3.22e-03 2.05e-03 3.50e-07 2.94e-03 1.45e-03
50   5    1    640   1.57e-06 7.76e-03 5.61e-03 6.62e-07 3.66e-03 3.87e-03
50   5    1    1280  2.63e-06 1.91e-02 1.41e-02 8.03e-07 1.22e-02 8.72e-03
50   5    1    2560  5.18e-06 4.76e-02 3.69e-02 3.88e-06 1.69e-02 2.63e-02
50   5    1    5120  9.94e-06 1.16e-01 9.67e-02 4.41e-06 9.79e-02 5.71e-02
50   5    1    10240 2.51e-05 2.78e-01 2.34e-01 6.01e-06 1.38e-01 9.22e-02
50   5    4    2     3.48e-06 1.08e-05 1.29e-05 9.76e-07 2.18e-06 6.71e-06
50   5    4    5     8.65e-06 5.08e-05 4.05e-05 2.89e-06 3.77e-05 1.74e-05
50   5    4    10    2.49e-05 2.14e-04 1.06e-04 7.86e-06 1.81e-04 5.02e-05
50   5    4    20    6.55e-05 8.04e-04 2.87e-04 3.00e-05 6.05e-04 1.97e-04
50   5    4    40    1.75e-04 2.37e-03 8.30e-04 3.96e-05 1.42e-03 3.18e-04
50   5    4    80    4.06e-04 6.83e-03 2.30e-03 1.87e-04 1.45e-03 8.21e-04
50   5    4    160   1.01e-03 1.95e-02 5.66e-03 6.83e-04 1.58e-02 4.38e-03
50   5    4    320   2.56e-03 5.03e-02 1.41e-02 8.59e-04 3.73e-02 1.20e-02
50   5    4    640   5.14e-03 1.34e-01 3.61e-02 1.09e-03 7.55e-02 2.20e-02
50   5    4    1280  1.07e-02 3.20e-01 8.70e-02 4.05e-03 1.97e-01 4.29e-02
50   5    4    2560  2.31e-02 7.74e-01 2.16e-01 7.66e-03 6.88e-01 1.37e-01
50   5    4    5120  4.59e-02 1.89e+00 5.03e-01 1.06e-02 7.88e-01 2.36e-01
50   5    4    10240 1.14e-01 4.52e+00 1.17e+00 3.10e-02 1.48e+00 8.77e-01
50   100  1    2     7.55e-06 4.20e-04 4.27e-04 6.23e-06 1.38e-04 2.90e-04
50   100  1    5     7.99e-06 1.51e-03 7.83e-04 6.64e-06 1.10e-03 6.75e-04
50   100  1    10    8.18e-06 5.01e-03 1.46e-03 6.75e-06 4.08e-03 1.45e-03
50   100  1    20    8.68e-06 3.65e-02 2.65e-03 6.84e-06 3.27e-02 2.58e-03
50   100  1    40    9.46e-06 1.41e-01 8.57e-03 6.86e-06 1.35e-01 8.24e-03
50   100  1    80    1.12e-05 4.61e-01 1.99e-02 7.97e-06 4.52e-01 1.96e-02
50   100  1    160   1.48e-05 1.32e+00 4.63e-02 8.66e-06 1.24e+00 5.16e-02
50   100  1    320   2.17e-05 3.61e+00 1.19e-01 1.55e-05 3.57e+00 1.32e-01
50   100  50   2     1.91e-01 2.61e-01 2.07e-03 4.34e-03 5.27e-04 1.15e-03
50   100  50   5     1.06e+00 2.48e+00 1.02e-02 2.68e-01 1.66e+00 2.10e-02
50   100  50   10    2.84e+00 7.41e+00 3.24e-02 6.94e-01 6.14e+00 3.13e-02
50   100  50   20    6.78e+00 2.12e+01 1.78e-01 2.68e+00 1.91e+01 1.84e-01
50   100  50   40    1.64e+01 5.81e+01 7.82e-01 6.50e+00 5.57e+01 7.68e-01
50   100  99   2     6.77e-01 8.90e-01 1.31e-02 1.16e-02 8.15e-04 5.87e-03
50   100  99   5     3.87e+00 1.08e+01 8.57e-02 7.31e-01 4.28e+00 3.20e-02
50   100  99   10    1.11e+01 3.84e+01 2.79e-01 3.37e+00 1.70e+01 1.04e-01
50   100  99   20    2.85e+01 1.20e+02 9.27e-01 1.08e+01 5.51e+01 3.97e-01

As indicated in #31 and #36 , improvements are to be expected when the number of columns is very small compared to the number of rows. (Still, e.g. for 1 column, this will likely not reach the efficiency of flint's nullspace function, whose output kernel basis is not minimal.)