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Blocking Tutorial

This is a tutorial internal to the Hazy Research group to illustrate SIMD and cache blocking.

Based on the paper: Anatomy of High-Performance Many-Threaded Matrix Multiplication http://www.cs.utexas.edu/users/flame/pubs/blis3_ipdps14.pdf

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Requirements

To run the code, you will need:

• A Haswell processor or later (AVX2 and FMA support) http://en.wikipedia.org/wiki/Advanced_Vector_Extensions http://en.wikipedia.org/wiki/FMA_instruction_set
• The OpenMP library
• I used g++ 4.9.2 but any compiler supporting AVX2 and FMA will work
• OpenBLAS (for comparison vs. OpenBLAS. Disable if not interested) http://www.openblas.net/

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Instructions

To compile:

1. Change compile.sh to point to OpenBLAS or some other BLAS To run without OpenBLAS, modify the .cpp file to not call OpenBLAS (remove references to cblas.h, cblas_sgemm)

2. Compile with: bash compile.sh

3. Run with: ./m

This will run two of the matrix multiply implementations and compare:

• The outputs (to ensure they match)
• The total time and GFLOPS

4. To change which version to compare against, edit which function is called (sgemm_naive, sgemm_16x6_block_parallel, etc.)

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Experiments

For NxN single precision matrix multiplication on a Haswell with 2 cores, 4 threads, L1 I$/D$: 2 x 32 kB 8-way L2$ 2 x 256 kB 8-way L3$ 1 x 4 MB 16-way

Naive       8330.310 milliseconds GFlops=1.699310
+SIMD       1057.134 milliseconds GFlops=13.390711
+Blocking   273.5390 milliseconds GFlops=51.750485
+Threads    138.5209 milliseconds GFlops=102.192277
OpenBLAS    117.7092 milliseconds GFlops=120.259758

For various N:

N=960 (~1 million elements per matrix):
This code   101.2 GFLOPS    
OpenBLAS    98.3 GFLOPS     

N=1920 (~4 million elements per matrix):
This code   104.1 GFLOPS    
OpenBLAS    121.2 GFLOPS    

N=2880 (~8 million elements per matrix):
This code   91.7 GFLOPS
OpenBLAS    103.9 GFLOPS