Pure ANSI C Implementation of DGEMM

In the following we present the cache optimized implementation of the matrix-matrix product. The function dgemm_nn can compute operations of the form \(C \leftarrow \beta C + \alpha A B\). All matrices can have arbitrary row and column stride. That in particular means each matrix can be row or column wise stored. It further means that we also can use the function for the computation of \(C \leftarrow \beta C + \alpha A^T B\), \(C \leftarrow \beta C + \alpha A B^T\) and \(C \leftarrow \beta C + \alpha A^T B^T\).

Select the demo-pure-c Branch

Before switching a branch do a make clean first:

$shell> cd ulmBLAS                                                       
$shell> make clean                                                       
for dir in src refblas test bench; do make -C $dir clean; done
rm -f  auxiliary/xerbla.o  level1/dasum.o level1/daxpy.o level1/dcopy.o level1/ddot.o level1/dnrm2.o level1/drot.o level1/drotg.o level1/drotm.o level1/drotmg.o level1/dscal.o level1/dswap.o level1/idamax.o  level3/dgemm.o level3/dgemm_nn.o level3/dsymm.o level3/stubs.o
rm -f  auxiliary/atl_xerbla.o  level1/atl_dasum.o level1/atl_daxpy.o level1/atl_dcopy.o level1/atl_ddot.o level1/atl_dnrm2.o level1/atl_drot.o level1/atl_drotg.o level1/atl_drotm.o level1/atl_drotmg.o level1/atl_dscal.o level1/atl_dswap.o level1/atl_idamax.o  level3/atl_dgemm.o level3/atl_dgemm_nn.o level3/atl_dsymm.o level3/atl_stubs.o
rm -f ../libulmblas.a
rm -f ../libatlulmblas.a
rm -f caxpy.o ccopy.o cdotc.o cdotu.o cgbmv.o cgemm.o cgemv.o cgerc.o cgeru.o chbmv.o chemm.o chemv.o cher.o cher2.o cher2k.o cherk.o chpmv.o chpr.o chpr2.o crotg.o cscal.o csrot.o csscal.o cswap.o csymm.o csyr2k.o csyrk.o ctbmv.o ctbsv.o ctpmv.o ctpsv.o ctrmm.o ctrmv.o ctrsm.o ctrsv.o dasum.o daxpy.o dcabs1.o dcopy.o ddot.o dgbmv.o dgemm.o dgemv.o dger.o dnrm2.o drot.o drotg.o drotm.o drotmg.o dsbmv.o dscal.o dsdot.o dspmv.o dspr.o dspr2.o dswap.o dsymm.o dsymv.o dsyr.o dsyr2.o dsyr2k.o dsyrk.o dtbmv.o dtbsv.o dtpmv.o dtpsv.o dtrmm.o dtrmv.o dtrsm.o dtrsv.o dzasum.o dznrm2.o icamax.o idamax.o isamax.o izamax.o lsame.o sasum.o saxpy.o scabs1.o scasum.o scnrm2.o scopy.o sdot.o sdsdot.o sgbmv.o sgemm.o sgemv.o sger.o snrm2.o srot.o srotg.o srotm.o srotmg.o ssbmv.o sscal.o sspmv.o sspr.o sspr2.o sswap.o ssymm.o ssymv.o ssyr.o ssyr2.o ssyr2k.o ssyrk.o stbmv.o stbsv.o stpmv.o stpsv.o strmm.o strmv.o strsm.o strsv.o xerbla.o xerbla_array.o zaxpy.o zcopy.o zdotc.o zdotu.o zdrot.o zdscal.o zgbmv.o zgemm.o zgemv.o zgerc.o zgeru.o zhbmv.o zhemm.o zhemv.o zher.o zher2.o zher2k.o zherk.o zhpmv.o zhpr.o zhpr2.o zrotg.o zscal.o zswap.o zsymm.o zsyr2k.o zsyrk.o ztbmv.o ztbsv.o ztpmv.o ztpsv.o ztrmm.o ztrmv.o ztrsm.o ztrsv.o
rm -f ../librefblas.a
rm -f  dblat1_ref  dblat3_ref  dblat1_ulm  dblat3_ulm *.SUMM
rm -f xdl1blastst libtstatlas.a l1blastst.o  ATL_cputime.o  ATL_epsilon.o  ATL_f77amax.o  ATL_f77asum.o  ATL_f77axpy.o  ATL_f77copy.o  ATL_f77dot.o  ATL_f77gemm.o  ATL_f77nrm2.o  ATL_f77rot.o  ATL_f77rotg.o  ATL_f77rotm.o  ATL_f77rotmg.o  ATL_f77scal.o  ATL_f77swap.o  ATL_f77symm.o  ATL_f77syr2k.o  ATL_f77syrk.o  ATL_f77trmm.o  ATL_f77trsm.o  ATL_flushcache.o  ATL_gediffnrm1.o  ATL_gegen.o  ATL_genrm1.o  ATL_infnrm.o  ATL_rand.o  ATL_set.o  ATL_synrm.o  ATL_trnrm1.o  ATL_vdiff.o  ATL_zero.o  ATL_df77wrap.o

Now we switch by checking out the demo-pure-cbranch:

$shell> git branch -a                                                    
* demo-pure-c
  master
  remotes/origin/HEAD -> origin/master
  remotes/origin/bench-atlas
  remotes/origin/bench-blis
  remotes/origin/bench-eigen
  remotes/origin/bench-mkl
  remotes/origin/blis-avx-microkernel
  remotes/origin/demo-naive-avx-with-intrinsics
  remotes/origin/demo-naive-sse-with-intrinsics
  remotes/origin/demo-naive-sse-with-intrinsics-unrolled
  remotes/origin/demo-pure-c
  remotes/origin/demo-sse-all-asm
  remotes/origin/demo-sse-all-asm-try-prefetching
  remotes/origin/demo-sse-all-asm-try-prefetching-v2
  remotes/origin/demo-sse-all-asm-with-prefetching
  remotes/origin/demo-sse-asm
  remotes/origin/demo-sse-asm-for-AB-loop
  remotes/origin/demo-sse-asm-unrolled
  remotes/origin/demo-sse-asm-unrolled-v2
  remotes/origin/demo-sse-asm-unrolled-v3
  remotes/origin/demo-sse-asm-unrolled-with-prefetch
  remotes/origin/demo-sse-intrinsics
  remotes/origin/demo-sse-intrinsics-for-AB-loop
  remotes/origin/demo-sse-intrinsics-v2
  remotes/origin/demo-sse-intrinsics-v3
  remotes/origin/demo-with-sse-intrinsics
  remotes/origin/master
  remotes/origin/trsm-assignment
  remotes/origin/trsm-pure-c
$shell> git checkout -B demo-pure-c remotes/origin/demo-pure-c           
Reset branch 'demo-pure-c'
Branch demo-pure-c set up to track remote branch demo-pure-c from origin.
Your branch is up-to-date with 'origin/demo-pure-c'.

Then we compile the project

$shell> make                                                             
make -C src
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o auxiliary/xerbla.o auxiliary/xerbla.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level1/dasum.o level1/dasum.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level1/daxpy.o level1/daxpy.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level1/dcopy.o level1/dcopy.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level1/ddot.o level1/ddot.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level1/dnrm2.o level1/dnrm2.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level1/drot.o level1/drot.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level1/drotg.o level1/drotg.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level1/drotm.o level1/drotm.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level1/drotmg.o level1/drotmg.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level1/dscal.o level1/dscal.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level1/dswap.o level1/dswap.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level1/idamax.o level1/idamax.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level3/dgemm.o level3/dgemm.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level3/dgemm_nn.o level3/dgemm_nn.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level3/dsymm.o level3/dsymm.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -c -o level3/stubs.o level3/stubs.c
ar cru ../libulmblas.a  auxiliary/xerbla.o  level1/dasum.o level1/daxpy.o level1/dcopy.o level1/ddot.o level1/dnrm2.o level1/drot.o level1/drotg.o level1/drotm.o level1/drotmg.o level1/dscal.o level1/dswap.o level1/idamax.o  level3/dgemm.o level3/dgemm_nn.o level3/dsymm.o level3/stubs.o
ranlib ../libulmblas.a
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o auxiliary/atl_xerbla.o auxiliary/xerbla.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level1/atl_dasum.o level1/dasum.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level1/atl_daxpy.o level1/daxpy.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level1/atl_dcopy.o level1/dcopy.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level1/atl_ddot.o level1/ddot.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level1/atl_dnrm2.o level1/dnrm2.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level1/atl_drot.o level1/drot.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level1/atl_drotg.o level1/drotg.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level1/atl_drotm.o level1/drotm.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level1/atl_drotmg.o level1/drotmg.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level1/atl_dscal.o level1/dscal.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level1/atl_dswap.o level1/dswap.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level1/atl_idamax.o level1/idamax.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level3/atl_dgemm.o level3/dgemm.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level3/atl_dgemm_nn.o level3/dgemm_nn.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level3/atl_dsymm.o level3/dsymm.c
gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level3/atl_stubs.o level3/stubs.c
ar cru ../libatlulmblas.a  auxiliary/atl_xerbla.o  level1/atl_dasum.o level1/atl_daxpy.o level1/atl_dcopy.o level1/atl_ddot.o level1/atl_dnrm2.o level1/atl_drot.o level1/atl_drotg.o level1/atl_drotm.o level1/atl_drotmg.o level1/atl_dscal.o level1/atl_dswap.o level1/atl_idamax.o  level3/atl_dgemm.o level3/atl_dgemm_nn.o level3/atl_dsymm.o level3/atl_stubs.o
ranlib ../libatlulmblas.a
make -C refblas
gfortran -fimplicit-none -O3 -c -o caxpy.o caxpy.f
gfortran -fimplicit-none -O3 -c -o ccopy.o ccopy.f
gfortran -fimplicit-none -O3 -c -o cdotc.o cdotc.f
gfortran -fimplicit-none -O3 -c -o cdotu.o cdotu.f
gfortran -fimplicit-none -O3 -c -o cgbmv.o cgbmv.f
gfortran -fimplicit-none -O3 -c -o cgemm.o cgemm.f
gfortran -fimplicit-none -O3 -c -o cgemv.o cgemv.f
gfortran -fimplicit-none -O3 -c -o cgerc.o cgerc.f
gfortran -fimplicit-none -O3 -c -o cgeru.o cgeru.f
gfortran -fimplicit-none -O3 -c -o chbmv.o chbmv.f
gfortran -fimplicit-none -O3 -c -o chemm.o chemm.f
gfortran -fimplicit-none -O3 -c -o chemv.o chemv.f
gfortran -fimplicit-none -O3 -c -o cher.o cher.f
gfortran -fimplicit-none -O3 -c -o cher2.o cher2.f
gfortran -fimplicit-none -O3 -c -o cher2k.o cher2k.f
gfortran -fimplicit-none -O3 -c -o cherk.o cherk.f
gfortran -fimplicit-none -O3 -c -o chpmv.o chpmv.f
gfortran -fimplicit-none -O3 -c -o chpr.o chpr.f
gfortran -fimplicit-none -O3 -c -o chpr2.o chpr2.f
gfortran -fimplicit-none -O3 -c -o crotg.o crotg.f
gfortran -fimplicit-none -O3 -c -o cscal.o cscal.f
gfortran -fimplicit-none -O3 -c -o csrot.o csrot.f
gfortran -fimplicit-none -O3 -c -o csscal.o csscal.f
gfortran -fimplicit-none -O3 -c -o cswap.o cswap.f
gfortran -fimplicit-none -O3 -c -o csymm.o csymm.f
gfortran -fimplicit-none -O3 -c -o csyr2k.o csyr2k.f
gfortran -fimplicit-none -O3 -c -o csyrk.o csyrk.f
gfortran -fimplicit-none -O3 -c -o ctbmv.o ctbmv.f
gfortran -fimplicit-none -O3 -c -o ctbsv.o ctbsv.f
gfortran -fimplicit-none -O3 -c -o ctpmv.o ctpmv.f
gfortran -fimplicit-none -O3 -c -o ctpsv.o ctpsv.f
gfortran -fimplicit-none -O3 -c -o ctrmm.o ctrmm.f
gfortran -fimplicit-none -O3 -c -o ctrmv.o ctrmv.f
gfortran -fimplicit-none -O3 -c -o ctrsm.o ctrsm.f
gfortran -fimplicit-none -O3 -c -o ctrsv.o ctrsv.f
gfortran -fimplicit-none -O3 -c -o dasum.o dasum.f
gfortran -fimplicit-none -O3 -c -o daxpy.o daxpy.f
gfortran -fimplicit-none -O3 -c -o dcabs1.o dcabs1.f
gfortran -fimplicit-none -O3 -c -o dcopy.o dcopy.f
gfortran -fimplicit-none -O3 -c -o ddot.o ddot.f
gfortran -fimplicit-none -O3 -c -o dgbmv.o dgbmv.f
gfortran -fimplicit-none -O3 -c -o dgemm.o dgemm.f
gfortran -fimplicit-none -O3 -c -o dgemv.o dgemv.f
gfortran -fimplicit-none -O3 -c -o dger.o dger.f
gfortran -fimplicit-none -O3 -c -o dnrm2.o dnrm2.f
gfortran -fimplicit-none -O3 -c -o drot.o drot.f
gfortran -fimplicit-none -O3 -c -o drotg.o drotg.f
gfortran -fimplicit-none -O3 -c -o drotm.o drotm.f
gfortran -fimplicit-none -O3 -c -o drotmg.o drotmg.f
gfortran -fimplicit-none -O3 -c -o dsbmv.o dsbmv.f
gfortran -fimplicit-none -O3 -c -o dscal.o dscal.f
gfortran -fimplicit-none -O3 -c -o dsdot.o dsdot.f
gfortran -fimplicit-none -O3 -c -o dspmv.o dspmv.f
gfortran -fimplicit-none -O3 -c -o dspr.o dspr.f
gfortran -fimplicit-none -O3 -c -o dspr2.o dspr2.f
gfortran -fimplicit-none -O3 -c -o dswap.o dswap.f
gfortran -fimplicit-none -O3 -c -o dsymm.o dsymm.f
gfortran -fimplicit-none -O3 -c -o dsymv.o dsymv.f
gfortran -fimplicit-none -O3 -c -o dsyr.o dsyr.f
gfortran -fimplicit-none -O3 -c -o dsyr2.o dsyr2.f
gfortran -fimplicit-none -O3 -c -o dsyr2k.o dsyr2k.f
gfortran -fimplicit-none -O3 -c -o dsyrk.o dsyrk.f
gfortran -fimplicit-none -O3 -c -o dtbmv.o dtbmv.f
gfortran -fimplicit-none -O3 -c -o dtbsv.o dtbsv.f
gfortran -fimplicit-none -O3 -c -o dtpmv.o dtpmv.f
gfortran -fimplicit-none -O3 -c -o dtpsv.o dtpsv.f
gfortran -fimplicit-none -O3 -c -o dtrmm.o dtrmm.f
gfortran -fimplicit-none -O3 -c -o dtrmv.o dtrmv.f
gfortran -fimplicit-none -O3 -c -o dtrsm.o dtrsm.f
gfortran -fimplicit-none -O3 -c -o dtrsv.o dtrsv.f
gfortran -fimplicit-none -O3 -c -o dzasum.o dzasum.f
gfortran -fimplicit-none -O3 -c -o dznrm2.o dznrm2.f
gfortran -fimplicit-none -O3 -c -o icamax.o icamax.f
gfortran -fimplicit-none -O3 -c -o idamax.o idamax.f
gfortran -fimplicit-none -O3 -c -o isamax.o isamax.f
gfortran -fimplicit-none -O3 -c -o izamax.o izamax.f
gfortran -fimplicit-none -O3 -c -o lsame.o lsame.f
gfortran -fimplicit-none -O3 -c -o sasum.o sasum.f
gfortran -fimplicit-none -O3 -c -o saxpy.o saxpy.f
gfortran -fimplicit-none -O3 -c -o scabs1.o scabs1.f
gfortran -fimplicit-none -O3 -c -o scasum.o scasum.f
gfortran -fimplicit-none -O3 -c -o scnrm2.o scnrm2.f
gfortran -fimplicit-none -O3 -c -o scopy.o scopy.f
gfortran -fimplicit-none -O3 -c -o sdot.o sdot.f
gfortran -fimplicit-none -O3 -c -o sdsdot.o sdsdot.f
gfortran -fimplicit-none -O3 -c -o sgbmv.o sgbmv.f
gfortran -fimplicit-none -O3 -c -o sgemm.o sgemm.f
gfortran -fimplicit-none -O3 -c -o sgemv.o sgemv.f
gfortran -fimplicit-none -O3 -c -o sger.o sger.f
gfortran -fimplicit-none -O3 -c -o snrm2.o snrm2.f
gfortran -fimplicit-none -O3 -c -o srot.o srot.f
gfortran -fimplicit-none -O3 -c -o srotg.o srotg.f
gfortran -fimplicit-none -O3 -c -o srotm.o srotm.f
gfortran -fimplicit-none -O3 -c -o srotmg.o srotmg.f
gfortran -fimplicit-none -O3 -c -o ssbmv.o ssbmv.f
gfortran -fimplicit-none -O3 -c -o sscal.o sscal.f
gfortran -fimplicit-none -O3 -c -o sspmv.o sspmv.f
gfortran -fimplicit-none -O3 -c -o sspr.o sspr.f
gfortran -fimplicit-none -O3 -c -o sspr2.o sspr2.f
gfortran -fimplicit-none -O3 -c -o sswap.o sswap.f
gfortran -fimplicit-none -O3 -c -o ssymm.o ssymm.f
gfortran -fimplicit-none -O3 -c -o ssymv.o ssymv.f
gfortran -fimplicit-none -O3 -c -o ssyr.o ssyr.f
gfortran -fimplicit-none -O3 -c -o ssyr2.o ssyr2.f
gfortran -fimplicit-none -O3 -c -o ssyr2k.o ssyr2k.f
gfortran -fimplicit-none -O3 -c -o ssyrk.o ssyrk.f
gfortran -fimplicit-none -O3 -c -o stbmv.o stbmv.f
gfortran -fimplicit-none -O3 -c -o stbsv.o stbsv.f
gfortran -fimplicit-none -O3 -c -o stpmv.o stpmv.f
gfortran -fimplicit-none -O3 -c -o stpsv.o stpsv.f
gfortran -fimplicit-none -O3 -c -o strmm.o strmm.f
gfortran -fimplicit-none -O3 -c -o strmv.o strmv.f
gfortran -fimplicit-none -O3 -c -o strsm.o strsm.f
gfortran -fimplicit-none -O3 -c -o strsv.o strsv.f
gfortran -fimplicit-none -O3 -c -o xerbla.o xerbla.f
gfortran -fimplicit-none -O3 -c -o xerbla_array.o xerbla_array.f
gfortran -fimplicit-none -O3 -c -o zaxpy.o zaxpy.f
gfortran -fimplicit-none -O3 -c -o zcopy.o zcopy.f
gfortran -fimplicit-none -O3 -c -o zdotc.o zdotc.f
gfortran -fimplicit-none -O3 -c -o zdotu.o zdotu.f
gfortran -fimplicit-none -O3 -c -o zdrot.o zdrot.f
gfortran -fimplicit-none -O3 -c -o zdscal.o zdscal.f
gfortran -fimplicit-none -O3 -c -o zgbmv.o zgbmv.f
gfortran -fimplicit-none -O3 -c -o zgemm.o zgemm.f
gfortran -fimplicit-none -O3 -c -o zgemv.o zgemv.f
gfortran -fimplicit-none -O3 -c -o zgerc.o zgerc.f
gfortran -fimplicit-none -O3 -c -o zgeru.o zgeru.f
gfortran -fimplicit-none -O3 -c -o zhbmv.o zhbmv.f
gfortran -fimplicit-none -O3 -c -o zhemm.o zhemm.f
gfortran -fimplicit-none -O3 -c -o zhemv.o zhemv.f
gfortran -fimplicit-none -O3 -c -o zher.o zher.f
gfortran -fimplicit-none -O3 -c -o zher2.o zher2.f
gfortran -fimplicit-none -O3 -c -o zher2k.o zher2k.f
gfortran -fimplicit-none -O3 -c -o zherk.o zherk.f
gfortran -fimplicit-none -O3 -c -o zhpmv.o zhpmv.f
gfortran -fimplicit-none -O3 -c -o zhpr.o zhpr.f
gfortran -fimplicit-none -O3 -c -o zhpr2.o zhpr2.f
gfortran -fimplicit-none -O3 -c -o zrotg.o zrotg.f
gfortran -fimplicit-none -O3 -c -o zscal.o zscal.f
gfortran -fimplicit-none -O3 -c -o zswap.o zswap.f
gfortran -fimplicit-none -O3 -c -o zsymm.o zsymm.f
gfortran -fimplicit-none -O3 -c -o zsyr2k.o zsyr2k.f
gfortran -fimplicit-none -O3 -c -o zsyrk.o zsyrk.f
gfortran -fimplicit-none -O3 -c -o ztbmv.o ztbmv.f
gfortran -fimplicit-none -O3 -c -o ztbsv.o ztbsv.f
gfortran -fimplicit-none -O3 -c -o ztpmv.o ztpmv.f
gfortran -fimplicit-none -O3 -c -o ztpsv.o ztpsv.f
gfortran -fimplicit-none -O3 -c -o ztrmm.o ztrmm.f
gfortran -fimplicit-none -O3 -c -o ztrmv.o ztrmv.f
gfortran -fimplicit-none -O3 -c -o ztrsm.o ztrsm.f
gfortran -fimplicit-none -O3 -c -o ztrsv.o ztrsv.f
ar cru ../librefblas.a caxpy.o ccopy.o cdotc.o cdotu.o cgbmv.o cgemm.o cgemv.o cgerc.o cgeru.o chbmv.o chemm.o chemv.o cher.o cher2.o cher2k.o cherk.o chpmv.o chpr.o chpr2.o crotg.o cscal.o csrot.o csscal.o cswap.o csymm.o csyr2k.o csyrk.o ctbmv.o ctbsv.o ctpmv.o ctpsv.o ctrmm.o ctrmv.o ctrsm.o ctrsv.o dasum.o daxpy.o dcabs1.o dcopy.o ddot.o dgbmv.o dgemm.o dgemv.o dger.o dnrm2.o drot.o drotg.o drotm.o drotmg.o dsbmv.o dscal.o dsdot.o dspmv.o dspr.o dspr2.o dswap.o dsymm.o dsymv.o dsyr.o dsyr2.o dsyr2k.o dsyrk.o dtbmv.o dtbsv.o dtpmv.o dtpsv.o dtrmm.o dtrmv.o dtrsm.o dtrsv.o dzasum.o dznrm2.o icamax.o idamax.o isamax.o izamax.o lsame.o sasum.o saxpy.o scabs1.o scasum.o scnrm2.o scopy.o sdot.o sdsdot.o sgbmv.o sgemm.o sgemv.o sger.o snrm2.o srot.o srotg.o srotm.o srotmg.o ssbmv.o sscal.o sspmv.o sspr.o sspr2.o sswap.o ssymm.o ssymv.o ssyr.o ssyr2.o ssyr2k.o ssyrk.o stbmv.o stbsv.o stpmv.o stpsv.o strmm.o strmv.o strsm.o strsv.o xerbla.o xerbla_array.o zaxpy.o zcopy.o zdotc.o zdotu.o zdrot.o zdscal.o zgbmv.o zgemm.o zgemv.o zgerc.o zgeru.o zhbmv.o zhemm.o zhemv.o zher.o zher2.o zher2k.o zherk.o zhpmv.o zhpr.o zhpr2.o zrotg.o zscal.o zswap.o zsymm.o zsyr2k.o zsyrk.o ztbmv.o ztbsv.o ztpmv.o ztpsv.o ztrmm.o ztrmv.o ztrsm.o ztrsv.o
ranlib ../librefblas.a
make -C test
gfortran dblat1.f -L.. -lrefblas -o dblat1_ref
dblat1.f:215.44:
               CALL STEST1(DNRM2(N,SX,INCX),STEMP,STEMP,SFAC)           
                                            1
Warning: Rank mismatch in argument 'strue1' at (1) (scalar and rank-1)
dblat1.f:219.44:
               CALL STEST1(DASUM(N,SX,INCX),STEMP,STEMP,SFAC)           
                                            1
Warning: Rank mismatch in argument 'strue1' at (1) (scalar and rank-1)
gfortran dblat3.f -L.. -lrefblas -o dblat3_ref
gfortran dblat1.f -L.. -lulmblas -o dblat1_ulm
dblat1.f:215.44:
               CALL STEST1(DNRM2(N,SX,INCX),STEMP,STEMP,SFAC)           
                                            1
Warning: Rank mismatch in argument 'strue1' at (1) (scalar and rank-1)
dblat1.f:219.44:
               CALL STEST1(DASUM(N,SX,INCX),STEMP,STEMP,SFAC)           
                                            1
Warning: Rank mismatch in argument 'strue1' at (1) (scalar and rank-1)
gfortran dblat3.f -L.. -lulmblas -o dblat3_ulm
make -C bench
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o l1blastst.o l1blastst.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_cputime.o ATL_cputime.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_epsilon.o ATL_epsilon.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77amax.o ATL_f77amax.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77asum.o ATL_f77asum.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77axpy.o ATL_f77axpy.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77copy.o ATL_f77copy.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77dot.o ATL_f77dot.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77gemm.o ATL_f77gemm.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77nrm2.o ATL_f77nrm2.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77rot.o ATL_f77rot.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77rotg.o ATL_f77rotg.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77rotm.o ATL_f77rotm.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77rotmg.o ATL_f77rotmg.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77scal.o ATL_f77scal.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77swap.o ATL_f77swap.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77symm.o ATL_f77symm.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77syr2k.o ATL_f77syr2k.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77syrk.o ATL_f77syrk.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77trmm.o ATL_f77trmm.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77trsm.o ATL_f77trsm.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_flushcache.o ATL_flushcache.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_gediffnrm1.o ATL_gediffnrm1.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_gegen.o ATL_gegen.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_genrm1.o ATL_genrm1.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_infnrm.o ATL_infnrm.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_rand.o ATL_rand.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_set.o ATL_set.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_synrm.o ATL_synrm.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_trnrm1.o ATL_trnrm1.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_vdiff.o ATL_vdiff.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_zero.o ATL_zero.c
gfortran   -c -o ATL_df77wrap.o ATL_df77wrap.f
ar r libtstatlas.a  ATL_cputime.o  ATL_epsilon.o  ATL_f77amax.o  ATL_f77asum.o  ATL_f77axpy.o  ATL_f77copy.o  ATL_f77dot.o  ATL_f77gemm.o  ATL_f77nrm2.o  ATL_f77rot.o  ATL_f77rotg.o  ATL_f77rotm.o  ATL_f77rotmg.o  ATL_f77scal.o  ATL_f77swap.o  ATL_f77symm.o  ATL_f77syr2k.o  ATL_f77syrk.o  ATL_f77trmm.o  ATL_f77trsm.o  ATL_flushcache.o  ATL_gediffnrm1.o  ATL_gegen.o  ATL_genrm1.o  ATL_infnrm.o  ATL_rand.o  ATL_set.o  ATL_synrm.o  ATL_trnrm1.o  ATL_vdiff.o  ATL_zero.o  ATL_df77wrap.o
ar: creating archive libtstatlas.a
ranlib libtstatlas.a
gfortran -o xdl1blastst l1blastst.o libtstatlas.a ../libatlulmblas.a ../librefblas.a
gfortran -o xdl3blastst l3blastst.o libtstatlas.a ../libatlulmblas.a ../librefblas.a

Building Blocks of dgemm_nn

pack_MRxk and pack_A are copying row panels from matrix blocks of A into the buffer _A. Details are described in GEMM Packing Matrix A.
pack_kxNR and pack_B are copying col panels from matrix blocks of A into the buffer _B. Details are described in GEMM Packing Matrix B.
dgemm_micro_kernel multiplies a row panel with a col panel. Details are given in GEMM Micro Kernel.
dgemm_macro_kernel multiplies a row panel with a col panel. Details are given in GEMM Macro Kernel.
dgemm_nn computes \(C \leftarrow \beta C + \alpha A B\) as described in GEMM.

The Micro Kernel Algorithm

For the sake of simplicity we assume \(m_r = n_r = 4\) in the description of the algorithm. Note that the pure C implementation works as long as \(m_r\) is a divisor or \(m_c\) and \(n_r\) a divisor of \(n_c\).

Recall that A points to a packed (maybe zero padded) row panel of height four and width \(k_c\). That means we have the column wise stored panel

\[A=\begin{pmatrix}a_0 & a_4 & \dots & a_{4 k_c -4} \\a_1 & a_5 & \dots & a_{4 k_c -3} \\a_2 & a_6 & \dots & a_{4 k_c -2} \\a_3 & a_7 & \dots & a_{4 k_c -1} \\\end{pmatrix}\]

Further B points to a column panel of height \(k_c\) and width four. That can be illustrated as a row wise stored panel

\[B=\begin{pmatrix}b_0 & b_1 & b_2 & b_3 \\b_4 & b_5 & b_6 & b_7 \\\vdots & \vdots & \vdots & \vdots \\b_{4 k_c-4} & b_{4 k_c-3} & b_{4 k_c-2} & b_{4 k_c-1} \\\end{pmatrix}\]

For the product \(A \cdot B\) this means

\[\begin{eqnarray*}A \cdot B&=&\begin{pmatrix}a_0 \\a_1 \\a_2 \\a_3 \\\end{pmatrix}\begin{pmatrix}b_0 & b_1 & b_2 & b_3\end{pmatrix}+\begin{pmatrix}a_4 \\a_5 \\a_6 \\a_7 \\\end{pmatrix}\begin{pmatrix}b_4 & b_5 & b_6 & b_7\end{pmatrix}+\dots+\begin{pmatrix}a_{4 k_c-4} \\a_{4 k_c-3} \\a_{4 k_c-2} \\a_{4 k_c-1} \\\end{pmatrix}\begin{pmatrix}b_{4 k_c-4} & b_{4 k_c-3} & b_{4 k_c-2} & b_{4 k_c-1}\end{pmatrix}\\[1cm]&=&\begin{pmatrix}a_0 b_0 & a_0 b_1 & a_0 b_2 & a_0 b_3 \\a_1 b_0 & a_1 b_1 & a_1 b_2 & a_1 b_3 \\a_2 b_0 & a_2 b_1 & a_2 b_2 & a_2 b_3 \\a_3 b_0 & a_3 b_1 & a_3 b_2 & a_3 b_3 \\\end{pmatrix}+\begin{pmatrix}a_4 b_4 & a_4 b_5 & a_4 b_6 & a_4 b_7 \\a_5 b_4 & a_5 b_5 & a_5 b_6 & a_5 b_7 \\a_6 b_4 & a_6 b_5 & a_6 b_6 & a_6 b_7 \\a_7 b_4 & a_7 b_5 & a_7 b_6 & a_7 b_7 \\\end{pmatrix}+\dots+\begin{pmatrix}a_{4 k_c-4} b_{4 k_c-4} & a_{4 k_c-4} b_{4 k_c-3} & a_{4 k_c-4} b_{4 k_c-2} & a_{4 k_c-4} b_{4 k_c-1} \\a_{4 k_c-3} b_{4 k_c-4} & a_{4 k_c-3} b_{4 k_c-3} & a_{4 k_c-3} b_{4 k_c-2} & a_{4 k_c-3} b_{4 k_c-1} \\a_{4 k_c-2} b_{4 k_c-4} & a_{4 k_c-2} b_{4 k_c-3} & a_{4 k_c-2} b_{4 k_c-2} & a_{4 k_c-2} b_{4 k_c-1} \\a_{4 k_c-1} b_{4 k_c-4} & a_{4 k_c-1} b_{4 k_c-3} & a_{4 k_c-1} b_{4 k_c-2} & a_{4 k_c-1} b_{4 k_c-1} \\\end{pmatrix}\end{eqnarray*}\]

We compute \(\mathbf{AB} = A \cdot B\) sequentially:

Initialize: \(\mathbf{AB} \leftarrow \mathbf{0}_{4 \times 4}\)
For \(l = 0, \dots, k_c-1\) update:
- \(\mathbf{AB} \leftarrow \mathbf{AB} + \begin{pmatrix} a_{4l} \\ a_{4l+1} \\ a_{4l+2} \\ a_{4l+3}\end{pmatrix} \begin{pmatrix} b_{4l} & b_{4l+1} & b_{4l+2} & b_{4l+3}\end{pmatrix} \)
  
  (Note that this loop is the computational hotspot. In principal the overall performance only depends on the efficient update of \(\mathbf{AB}\))

Afterwards we merely update the left hand side micro block \(\tilde{C}\):

\(\tilde{C} \leftarrow \beta \tilde{C}\)
\(\tilde{C} \leftarrow \tilde{C} + \alpha \mathbf{AB}\)

The dgemm_nn Code (less than 450 lines!)

ulmBLAS/src/level3/dgemm_nn.c

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#include <ulmblas.h>

#include <stdio.h>

#define MC  384

#define KC  384

#define NC  4096

#define MR  4

#define NR  4

//

//  Local buffers for storing panels from A, B and C

//

static double _A[MC*KC];

static double _B[KC*NC];

static double _C[MR*NR];

//

//  Packing complete panels from A (i.e. without padding)

//

static void

pack_MRxk(int k, const double *A, int incRowA, int incColA,

          double *buffer)

{

    int i, j;

    for (j=0; j<k; ++j) {

        for (i=0; i<MR; ++i) {

            buffer[i] = A[i*incRowA];

        }

        buffer += MR;

        A      += incColA;

    }

}

//

//  Packing panels from A with padding if required

//

static void

pack_A(int mc, int kc, const double *A, int incRowA, int incColA,

       double *buffer)

{

    int mp  = mc / MR;

    int _mr = mc % MR;

    int i, j;

    for (i=0; i<mp; ++i) {

        pack_MRxk(kc, A, incRowA, incColA, buffer);

        buffer += kc*MR;

        A      += MR*incRowA;

    }

    if (_mr>0) {

        for (j=0; j<kc; ++j) {

            for (i=0; i<_mr; ++i) {

                buffer[i] = A[i*incRowA];

            }

            for (i=_mr; i<MR; ++i) {

                buffer[i] = 0.0;

            }

            buffer += MR;

            A      += incColA;

        }

    }

}

//

//  Packing complete panels from B (i.e. without padding)

//

static void

pack_kxNR(int k, const double *B, int incRowB, int incColB,

          double *buffer)

{

    int i, j;

    for (i=0; i<k; ++i) {

        for (j=0; j<NR; ++j) {

            buffer[j] = B[j*incColB];

        }

        buffer += NR;

        B      += incRowB;

    }

}

//

//  Packing panels from B with padding if required

//

static void

pack_B(int kc, int nc, const double *B, int incRowB, int incColB,

       double *buffer)

{

    int np  = nc / NR;

    int _nr = nc % NR;

    int i, j;

    for (j=0; j<np; ++j) {

        pack_kxNR(kc, B, incRowB, incColB, buffer);

        buffer += kc*NR;

        B      += NR*incColB;

    }

    if (_nr>0) {

        for (i=0; i<kc; ++i) {

            for (j=0; j<_nr; ++j) {

                buffer[j] = B[j*incColB];

            }

            for (j=_nr; j<NR; ++j) {

                buffer[j] = 0.0;

            }

            buffer += NR;

            B      += incRowB;

        }

    }

}

//

//  Micro kernel for multiplying panels from A and B.

//

static void

dgemm_micro_kernel(int kc,

                   double alpha, const double *A, const double *B,

                   double beta,

                   double *C, int incRowC, int incColC)

{

    double AB[MR*NR];

    int i, j, l;

//

//  Compute AB = A*B

//

    for (l=0; l<MR*NR; ++l) {

        AB[l] = 0;

    }

    for (l=0; l<kc; ++l) {

        for (j=0; j<NR; ++j) {

            for (i=0; i<MR; ++i) {

                AB[i+j*MR] += A[i]*B[j];

            }

        }

        A += MR;

        B += NR;

    }

//

//  Update C <- beta*C

//

    if (beta==0.0) {

        for (j=0; j<NR; ++j) {

            for (i=0; i<MR; ++i) {

                C[i*incRowC+j*incColC] = 0.0;

            }

        }

    } else if (beta!=1.0) {

        for (j=0; j<NR; ++j) {

            for (i=0; i<MR; ++i) {

                C[i*incRowC+j*incColC] *= beta;

            }

        }

    }

//

//  Update C <- C + alpha*AB (note: the case alpha==0.0 was already treated in

//                                  the above layer dgemm_nn)

//

    if (alpha==1.0) {

        for (j=0; j<NR; ++j) {

            for (i=0; i<MR; ++i) {

                C[i*incRowC+j*incColC] += AB[i+j*MR];

            }

        }

    } else {

        for (j=0; j<NR; ++j) {

            for (i=0; i<MR; ++i) {

                C[i*incRowC+j*incColC] += alpha*AB[i+j*MR];

            }

        }

    }

}

//

//  Compute Y += alpha*X

//

static void

dgeaxpy(int           m,

        int           n,

        double        alpha,

        const double  *X,

        int           incRowX,

        int           incColX,

        double        *Y,

        int           incRowY,

        int           incColY)

{

    int i, j;

    if (alpha!=1.0) {

        for (j=0; j<n; ++j) {

            for (i=0; i<m; ++i) {

                Y[i*incRowY+j*incColY] += alpha*X[i*incRowX+j*incColX];

            }

        }

    } else {

        for (j=0; j<n; ++j) {

            for (i=0; i<m; ++i) {

                Y[i*incRowY+j*incColY] += X[i*incRowX+j*incColX];

            }

        }

    }

}

//

//  Compute X *= alpha

//

static void

dgescal(int     m,

        int     n,

        double  alpha,

        double  *X,

        int     incRowX,

        int     incColX)

{

    int i, j;

    if (alpha!=0.0) {

        for (j=0; j<n; ++j) {

            for (i=0; i<m; ++i) {

                X[i*incRowX+j*incColX] *= alpha;

            }

        }

    } else {

        for (j=0; j<n; ++j) {

            for (i=0; i<m; ++i) {

                X[i*incRowX+j*incColX] = 0.0;

            }

        }

    }

}

//

//  Macro Kernel for the multiplication of blocks of A and B.  We assume that

//  these blocks were previously packed to buffers _A and _B.

//

static void

dgemm_macro_kernel(int     mc,

                   int     nc,

                   int     kc,

                   double  alpha,

                   double  beta,

                   double  *C,

                   int     incRowC,

                   int     incColC)

{

    int mp = (mc+MR-1) / MR;

    int np = (nc+NR-1) / NR;

    int _mr = mc % MR;

    int _nr = nc % NR;

    int mr, nr;

    int i, j;

    for (j=0; j<np; ++j) {

        nr    = (j!=np-1 || _nr==0) ? NR : _nr;

        for (i=0; i<mp; ++i) {

            mr    = (i!=mp-1 || _mr==0) ? MR : _mr;

            if (mr==MR && nr==NR) {

                dgemm_micro_kernel(kc, alpha, &_A[i*kc*MR], &_B[j*kc*NR],

                                   beta,

                                   &C[i*MR*incRowC+j*NR*incColC],

                                   incRowC, incColC);

            } else {

                dgemm_micro_kernel(kc, alpha, &_A[i*kc*MR], &_B[j*kc*NR],

                                   0.0,

                                   _C, 1, MR);

                dgescal(mr, nr, beta,

                        &C[i*MR*incRowC+j*NR*incColC], incRowC, incColC);

                dgeaxpy(mr, nr, 1.0, _C, 1, MR,

                        &C[i*MR*incRowC+j*NR*incColC], incRowC, incColC);

            }

        }

    }

}

//

//  Compute C <- beta*C + alpha*A*B

//

void

dgemm_nn(int            m,

         int            n,

         int            k,

         double         alpha,

         const double   *A,

         int            incRowA,

         int            incColA,

         const double   *B,

         int            incRowB,

         int            incColB,

         double         beta,

         double         *C,

         int            incRowC,

         int            incColC)

{

    int mb = (m+MC-1) / MC;

    int nb = (n+NC-1) / NC;

    int kb = (k+KC-1) / KC;

    int _mc = m % MC;

    int _nc = n % NC;

    int _kc = k % KC;

    int mc, nc, kc;

    int i, j, l;

    double _beta;

    if (alpha==0.0 || k==0) {

        dgescal(m, n, beta, C, incRowC, incColC);

        return;

    }

    for (j=0; j<nb; ++j) {

        nc = (j!=nb-1 || _nc==0) ? NC : _nc;

        for (l=0; l<kb; ++l) {

            kc    = (l!=kb-1 || _kc==0) ? KC   : _kc;

            _beta = (l==0) ? beta : 1.0;

            pack_B(kc, nc,

                   &B[l*KC*incRowB+j*NC*incColB], incRowB, incColB,

                   _B);

            for (i=0; i<mb; ++i) {

                mc = (i!=mb-1 || _mc==0) ? MC : _mc;

                pack_A(mc, kc,

                       &A[i*MC*incRowA+l*KC*incColA], incRowA, incColA,

                       _A);

                dgemm_macro_kernel(mc, nc, kc, alpha, _beta,

                                   &C[i*MC*incRowC+j*NC*incColC],

                                   incRowC, incColC);

            }

        }

    }

}

Benchmark Results

In bench/ we have extracted the benchmarks suite from the ATLAS project:

$shell> cd bench                                                         
$shell> ./xdl3blastst                                                    
./xdl3blastst 
--------------------------------- GEMM ----------------------------------
TST# A B    M    N    K ALPHA  LDA  LDB  BETA  LDC  TIME MFLOP SpUp  TEST
==== = = ==== ==== ==== ===== ==== ==== ===== ==== ===== ===== ==== =====
   0 N N  100  100  100   1.0 1000 1000   1.0 1000  0.00 1315.8 1.00 -----
   0 N N  100  100  100   1.0 1000 1000   1.0 1000  0.00 2649.0 2.01 PASS 
   1 N N  200  200  200   1.0 1000 1000   1.0 1000  0.01 1961.3 1.00 -----
   1 N N  200  200  200   1.0 1000 1000   1.0 1000  0.00 3283.4 1.67 PASS 
   2 N N  300  300  300   1.0 1000 1000   1.0 1000  0.03 2019.3 1.00 -----
   2 N N  300  300  300   1.0 1000 1000   1.0 1000  0.02 3375.6 1.67 PASS 
   3 N N  400  400  400   1.0 1000 1000   1.0 1000  0.06 2066.4 1.00 -----
   3 N N  400  400  400   1.0 1000 1000   1.0 1000  0.04 3382.2 1.64 PASS 
   4 N N  500  500  500   1.0 1000 1000   1.0 1000  0.12 2064.9 1.00 -----
   4 N N  500  500  500   1.0 1000 1000   1.0 1000  0.07 3419.3 1.66 PASS 
   5 N N  600  600  600   1.0 1000 1000   1.0 1000  0.34 1270.4 1.00 -----
   5 N N  600  600  600   1.0 1000 1000   1.0 1000  0.13 3449.5 2.72 PASS 
   6 N N  700  700  700   1.0 1000 1000   1.0 1000  0.65 1060.5 1.00 -----
   6 N N  700  700  700   1.0 1000 1000   1.0 1000  0.20 3444.5 3.25 PASS 
   7 N N  800  800  800   1.0 1000 1000   1.0 1000  0.97 1050.7 1.00 -----
   7 N N  800  800  800   1.0 1000 1000   1.0 1000  0.30 3468.3 3.30 PASS 
   8 N N  900  900  900   1.0 1000 1000   1.0 1000  1.39 1052.0 1.00 -----
   8 N N  900  900  900   1.0 1000 1000   1.0 1000  0.42 3461.3 3.29 PASS 
   9 N N 1000 1000 1000   1.0 1000 1000   1.0 1000  1.72 1164.3 1.00 -----
   9 N N 1000 1000 1000   1.0 1000 1000   1.0 1000  0.57 3494.0 3.00 PASS 
10 tests run, 10 passed

Lines with PASS show results for our own implementation. Lines with ----- belong to the reference implementation.

We can visualize the benchmarks with gnuplot. First we write the results to a report file and use grep to separate results for our implementation and the reference implementation:

$shell> ./xdl3blastst > report                                           
$shell> grep PASS report > demo-pure-c                                   
$shell> grep "\ \-\-\-\-\-$" report > refBLAS

Then we can use gnuplot to create a svg of the benchmark results:

ulmBLAS/bench/bench.gps

set terminal svg size 940,480

set output "bench.svg"

set xlabel "Matrix dimensions N=M=K"

set ylabel "MFLOPS"

set yrange [0:9600]

set title "Compute C + A*B"

set key outside

plot "refBLAS" using 4:13 with linespoints lt 2 title "Netlib RefBLAS", "demo-pure-c" using 4:13 with linespoints lt 4 title "demo-pure-c"

Feeding gnuplot with this script

$shell> gnuplot bench.gps

creates

Sensitivity to Compilers

Unfortunately different compilers, flags and versions can have a big influence on the performance. Using SSE intrinsics or inline assembly code in the micro kernel solves this problem.

Here I can demonstrate a significant performance drop if I compile with clang instead of gcc-4.8. Note that in other cases my clang compiler performs much better optimizations.

$shell> clang --version                                                  
Apple LLVM version 5.1 (clang-503.0.40) (based on LLVM 3.4svn)
Target: x86_64-apple-darwin13.3.0
Thread model: posix
$shell> gcc-4.8 --version                                                
gcc-4.8 (Homebrew gcc 4.8.3_1) 4.8.3
Copyright (C) 2013 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
$shell> cd src                                                           
$shell> clang -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level3/atl_dgemm_nn.o level3/dgemm_nn.c 
$shell> make                                                             
ar cru ../libatlulmblas.a  auxiliary/atl_xerbla.o  level1/atl_dasum.o level1/atl_daxpy.o level1/atl_dcopy.o level1/atl_ddot.o level1/atl_dnrm2.o level1/atl_drot.o level1/atl_drotg.o level1/atl_drotm.o level1/atl_drotmg.o level1/atl_dscal.o level1/atl_dswap.o level1/atl_idamax.o  level3/atl_dgemm.o level3/atl_dgemm_nn.o level3/atl_dsymm.o level3/atl_stubs.o
ranlib ../libatlulmblas.a
$shell> cd ../bench                                                      
$shell> make                                                             
gfortran -o xdl1blastst l1blastst.o libtstatlas.a ../libatlulmblas.a ../librefblas.a
gfortran -o xdl3blastst l3blastst.o libtstatlas.a ../libatlulmblas.a ../librefblas.a
$shell> ./xdl3blastst > report                                           
$shell> cat report                                                       
./xdl3blastst 
--------------------------------- GEMM ----------------------------------
TST# A B    M    N    K ALPHA  LDA  LDB  BETA  LDC  TIME MFLOP SpUp  TEST
==== = = ==== ==== ==== ===== ==== ==== ===== ==== ===== ===== ==== =====
   0 N N  100  100  100   1.0 1000 1000   1.0 1000  0.00 1798.6 1.00 -----
   0 N N  100  100  100   1.0 1000 1000   1.0 1000  0.00 1430.6 0.80 PASS 
   1 N N  200  200  200   1.0 1000 1000   1.0 1000  0.01 1962.0 1.00 -----
   1 N N  200  200  200   1.0 1000 1000   1.0 1000  0.01 1501.4 0.77 PASS 
   2 N N  300  300  300   1.0 1000 1000   1.0 1000  0.03 2018.6 1.00 -----
   2 N N  300  300  300   1.0 1000 1000   1.0 1000  0.04 1522.0 0.75 PASS 
   3 N N  400  400  400   1.0 1000 1000   1.0 1000  0.06 2027.7 1.00 -----
   3 N N  400  400  400   1.0 1000 1000   1.0 1000  0.08 1537.3 0.76 PASS 
   4 N N  500  500  500   1.0 1000 1000   1.0 1000  0.12 2068.5 1.00 -----
   4 N N  500  500  500   1.0 1000 1000   1.0 1000  0.16 1542.4 0.75 PASS 
   5 N N  600  600  600   1.0 1000 1000   1.0 1000  0.32 1364.4 1.00 -----
   5 N N  600  600  600   1.0 1000 1000   1.0 1000  0.28 1546.4 1.13 PASS 
   6 N N  700  700  700   1.0 1000 1000   1.0 1000  0.65 1060.4 1.00 -----
   6 N N  700  700  700   1.0 1000 1000   1.0 1000  0.44 1547.2 1.46 PASS 
   7 N N  800  800  800   1.0 1000 1000   1.0 1000  0.99 1036.1 1.00 -----
   7 N N  800  800  800   1.0 1000 1000   1.0 1000  0.66 1554.3 1.50 PASS 
   8 N N  900  900  900   1.0 1000 1000   1.0 1000  1.38 1056.1 1.00 -----
   8 N N  900  900  900   1.0 1000 1000   1.0 1000  0.94 1553.5 1.47 PASS 
   9 N N 1000 1000 1000   1.0 1000 1000   1.0 1000  1.72 1164.4 1.00 -----
   9 N N 1000 1000 1000   1.0 1000 1000   1.0 1000  1.28 1556.4 1.34 PASS 
10 tests run, 10 passed

We can visualize this again

$shell> grep PASS report > demo-pure-c-with-clang

By adopting the gnuplot script

ulmBLAS/bench/bench2.gps

set terminal svg size 940,480

set output "bench2.svg"

set xlabel "Matrix dimensions N=M=K"

set ylabel "MFLOPS"

set yrange [0:9600]

set title "Compute C + A*B"

set key outside

plot "refBLAS" using 4:13 with linespoints lt 2 title "Netlib RefBLAS", "demo-pure-c" using 4:13 with linespoints lt 4 title "demo-pure-c", "demo-pure-c-with-clang" using 4:13 with linespoints lt 5 title "demo-pure-c (with clang)"

and feeding gnuplot with this script

$shell> gnuplot bench2.gps

Although the performance drops dramatically we see that it is stable. The is no drop depending on cache sizes like in the refBLAS case. Therefore it still beats the reference implementation if problem sizes are large enough.