You (plus optional teammate) are tasked with the job of making the fastest matrix
multiplication program as possible for all machines. That means you cannot
specifically target a machine. But you are free to research and find all usual
architectures specification for personal and server machines. You may assume
that everything is Intel architecture (x86_64) to make life easier.
The matrix is column major. Naïve implementation is given in dgemm-naive.c and
you can run the bench-naive to see the output.
void dgemm( int m, int n, float *A, float *C )
for( int i = 0; i < m; i++ )
for( int k = 0; k < n; k++ )
for( int j = 0; j < m; j++ )
C[i+j*m] += A[i+k*m] * A[j+k*m];
C is where the result is stored and we are doing all the calculations from just one
matrix, A. You are required to do all the calculations and no optimization is
allowed on this front to make benchmarking easier. Zip contains the following files :
Makefile: to make and benchmark
benchmark.c: do not modify. It check results and produce performance numbers
dgemm-naive.c: naïve implementation as shown above
dgemm-optimize.c: your optimization
Choose at most 3 of the following common optimizations (1 per function, DO NOT
combine). The project worths 100 points, any extra points will go into your final
exam as an extra credit (if submitted on time. Having multiple projects due or
exams is not an excuse!).
• [20 points] Reordering of instructions (compiler peep-hole optimization)
• [20 points] Register blocking (reusing the same registers for multiple
• Cache optimizations (each sub-bullet counts as one optiomazation)
o [40 points] Blocking/Tiling (trying to keep the data in the cache for large
o [40 points] Pre-fetching (Copying small matrix blocks into contiguous
chunks of memory)
o [20 points] Pre-compute transpose for spatial locality
• Loop optimizations (each sub-bullet counts as one optiomazation)
o [40 points] SSE instructions
o [20 points] Reordering
o [40 points] Unrolling (at least 3 iterations)
• [40 points] Padding Matrices (odd sizes can hurt pipeline performance)
You should not use any libraries for parallel computing, such as openMP. You may
assume multiple cores. Anything else you can find or can think of is fine to increase
performance. Just remember to calculate all the results (copying from one part of
the resulting matrix to another is not allowed).
Your solution will be ran across different machines and the results aggregated.
Note that you should not optimize just for your computer or one particular matrix
size. Use your knowledge of computer architecture with all the modern features that
tries to accelerate execution. Caches will play a big role but it is not safe to assume a
particular architecture. But in general, optimizing memory accesses will lead to big
gains. Matrix size and corner cases will also matter, as the same optimization will
not work across the board. Have fun with this project.