Blocking gemm

[mgates3]

[Based on PR #99.]

Implements a basic blocked, templated gemm. This improves performance, particularly for transposed cases. Still a long ways from optimized libraries like OpenBLAS.

Compare performance after:

blaspp/test> setenv OMP_NUM_THREADS 1
blaspp/test> ./tester --dim 1000 --type s,d,c,z --transA n,t --transB n,t gemm
BLAS++ version 2024.10.26, id e29bac75,  OpenBLAS 0.3.29
input: ./tester --dim '1000,1024' --type 's,d,c,z' --transA 'n,t' --transB 'n,t' gemm

type  layout   transA   transB       m       n       k      alpha       beta     error   time (s)       gflop/s  ref time (s)   ref gflop/s  status
   s     col  notrans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  6.23e-09     0.0453        44.104        0.0215        93.127  pass
   s     col  notrans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  6.24e-09     0.0501        39.929        0.0204        97.962  pass
   s     col    trans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  6.24e-09     0.0527        37.946        0.0204        97.914  pass
   s     col    trans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  6.22e-09     0.0583        34.295        0.0204        97.924  pass

   d     col  notrans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  9.11e-18     0.0967        20.682        0.0413        48.417  pass
   d     col  notrans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  9.11e-18     0.0999        20.020        0.0411        48.668  pass
   d     col    trans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  9.11e-18      0.102        19.681        0.0407        49.162  pass
   d     col    trans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  9.10e-18      0.105        19.023        0.0405        49.416  pass

   c     col  notrans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  1.75e-09      0.631        12.680        0.0997        80.226  pass
   c     col  notrans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  1.75e-09      0.634        12.621        0.0996        80.287  pass
   c     col    trans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  1.75e-09      0.646        12.383         0.101        79.401  pass
   c     col    trans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  1.75e-09      0.646        12.378        0.0997        80.251  pass

   z     col  notrans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  2.81e-18      0.712        11.230         0.201        39.831  pass
   z     col  notrans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  2.81e-18      0.744        10.752         0.200        39.939  pass
   z     col    trans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  2.81e-18      0.651        12.291         0.201        39.810  pass
   z     col    trans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  2.81e-18      0.663        12.071         0.200        39.902  pass

with before:

blaspp/test> ./tester --dim 1000,1024 --type s,d,c,z --transA n,t --transB n,t gemm
BLAS++ version 2024.10.26, id 952c1da9,  OpenBLAS 0.3.29
input: ./tester --dim '1000,1024' --type 's,d,c,z' --transA 'n,t' --transB 'n,t' gemm

type  layout   transA   transB       m       n       k      alpha       beta     error   time (s)       gflop/s  ref time (s)   ref gflop/s  status
   s     col  notrans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  1.23e-08     0.0894        22.360        0.0209        95.841  pass
   s     col  notrans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  1.23e-08     0.0912        21.938        0.0204        97.843  pass
   s     col    trans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  1.05e-08      0.807         2.478        0.0205        97.347  pass
   s     col    trans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  1.05e-08      1.148         1.742        0.0206        97.120  pass

   d     col  notrans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  2.18e-17      0.170        11.785        0.0407        49.111  pass
   d     col  notrans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  2.17e-17      0.173        11.556        0.0406        49.311  pass
   d     col    trans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  1.94e-17      1.162         1.721        0.0407        49.151  pass
   d     col    trans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  1.94e-17      1.150         1.740        0.0405        49.358  pass

   c     col  notrans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  4.15e-09      0.714        11.211        0.0998        80.166  pass
   c     col  notrans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  4.15e-09      0.719        11.121        0.0997        80.278  pass
   c     col    trans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  3.72e-09      0.918         8.717        0.0999        80.045  pass
   c     col    trans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  3.72e-09      0.920         8.697        0.0996        80.329  pass

   z     col  notrans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  7.55e-18      0.730        10.966         0.204        39.144  pass
   z     col  notrans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  7.55e-18      0.765        10.462         0.200        39.936  pass
   z     col    trans  notrans    1000    1000    1000   3.1+1.4i   2.7+1.7i  6.74e-18      0.927         8.627         0.201        39.745  pass
   z     col    trans    trans    1000    1000    1000   3.1+1.4i   2.7+1.7i  6.72e-18      0.936         8.546         0.200        39.906  pass