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Fast matrix multiplication

Austin R. Benson and Grey Ballard

This software contains implementations of fast matrix multiplication algorithms for sequential and shared-memory parallel environments.

To cite this work, please use:

Austin R. Benson and Grey Ballard. "A framework for practical parallel fast matrix multiplication". In Proceedings of the 20th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming (PPoPP), 2015.

An extended version of the paper is available on arxiv.

License

Copyright 2014 Sandia Corporation. Under the terms of Contract DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government retains certain rights in this software.

This software is released under the BSD 2-Clause license. Please see the LICENSE file.

Setup

The code requires:

  • Intel MKL
  • Compiler supporting C++11 and OpenMP

The Makefile depends on an included file that specifies the compiler and the run-time mode. You must specify this file in the first line of the Makefile. For an example, see the file make.incs/make.inc.edison, which contains the information for running on NERSC's Edison machine. The MODE variable specifies sequential or parallel mode. The DEFINES variable can specify the type of parallelism if running in parallel mode. The DEFINES variable also specifies the naming convention for BLAS routines. By default, names are considered to be like "dgemm". However, by defining BLAS_POST, names are considered to be like "dgemm_", i.e., the routines have a trailing underscore. The MKL_ROOT variable must be set for your machine.

We did most testing using the Intel compiler (icpc). Depending on the version of g++, the OpenMP task constructs can be different and the hybrid shared-memory parallel code may crash. Sequential mode, DFS parallel, and BFS parallel should work with g++.

Building examples

First, use the code generator to generate the algorithms:

  cd codegen
  bash gen_all_algorithms.sh 0

Some simple codes that use the fast algorithms are in the examples directory. For example, you can build and run the (4, 3, 3) algorithm:

make fast433
./build/fast433

Building tests

We now assume that all of the algorithms have been gernated with the code generator (see above). The tests are built and run with:

make matmul_tests
./build/matmul_tests -all 1

The tests are just for correctness of the algorithms, not for performance. You should see output like:

STRASSEN_1: 257, 500, 55
Maximum relative difference: 3.87253e-15

This test runs one step of Strassen's algorithm, multiplying a 257 x 500 matrix with a 500 x 55 matrix. The maximum relative difference is an error measure:

max_{ij} |C_{ij} - D_{ij}| / |C_{ij}|,

where C is the result computed with the fast algorithm and D is the result computed with the classical algorithm. For all of the exact fast algorithms, the error should be around 1e-14 or 1e-15. The approximate algorithms (e.g., Bini's) have larger error. Typically, additional recursive steps leads to a larger error.

Building with different parallel methods

The BFS, DFS, and HYBRID parallel algorithms are compile-time options. In your make include file in the make.incs directory, to use DFS:

MODE=openmp
DEFINES += -D_PARALLEL_=1

Switch the _PARALLEL_ define to 2 for BFS or 3 for HYBRID. For an example, run

make fast424
./build/fast424

DGEMM curve benchmarks

Build and run the benchmark for the dgemm curves:

  make dgemm_curves
  ./build/dgemm_curves 1  # N x N x N
  ./build/dgemm_curves 2  # N x 800 x N
  ./build/dgemm_curves 3  # N x 800 x 800

The output is a semi-colon separated list, where each item loooks like:

M K N num_trials time;

The M, K, and N terms specify the matrix dimensions: M x K multiplied by K x N. The time is in milliseconds and is the total time to run num_trials multiplies. For example,

1200 800 1200 5 104.87;

means that it took 104.87 milliseconds to multiply a 1200 x 800 matrix by a 800 x 1200 matrix five times. To build with parallelism enabled, you need to define the _PARALLEL_ (see make.incs/make.inc.linux). To run without dynamic threads (i.e., mkl_set_dynamic(0)), append a second argument, e.g.:

./build/dgemm_curves 1 1 # Square timings without dynamic thread allocation

Fast algorithms benchmarks

Build benchmarking code for all of the fast algorithms:

make matmul_benchmarks

The build takes a while because we are compiling all of the algorithms. To run a small test to benchmark MKL against Strassen with one, two, and three levels of recursion:

./build/matmul_benchmarks -square_test 1

The output format is specified in data/README.md.

To run all of the benchmarks for the tall-and-skinny matrix multiplied by a small square matrix (N x k x k for fixed k):

./build/matmul_benchmarks -ts_square_like 1