1. Adaptive Strassen and ATLAS’s DGEMM
Paolo D’Alberto (CMU)
and
Alexandru Nicolau (UCI)
12/2/2005
HPC Asia

2. The Problem: Matrix Computations
The evolution of systems is modeled by matrix computations
The prediction and evaluation of such models (of complex systems) is fundamental in scientific computing.
For example, the solution of linear equations or the solution of least square systems.
12/2/2005
HPC Asia

3. The Problem: BLAS
The Basic Linear Algebra Subroutines is an interface describing a set of (basic) matrix and vector computations
Historically, the BLAS was a set of algorithms
Library implementing the BLAS are the back-bone of nowadays high performance computations
For ScaLAPACK
ESSL, PHiPac and ATLAS
12/2/2005
HPC Asia

4. The Problem: ATLAS
Implementation of BLAS 3 are based on Matrix Multiplication
In practice, ATLAS automatically generates a custom-tailored MM:
It probes the system
It tailors a kernel of MM to a specific system
It uses the MM as a basic routine for the other BLAS-3 routines
12/2/2005
HPC Asia

5. Matrix Multiplication (basics)= * C2 C3 A2 B3 B2 B3
C0= A0B0 + A1B2
C1= A0B1 + A1B3
C3= A2B1 + A3B3
C2= A2B0 + A3B2
12/2/2005
HPC Asia

6. The Problem: MM ATLAS uses this classic matrix multiply
For square matrices of size nxn, the algorithm takes O(n3)
It achieves 80-90% of peak performance
Strassen’s algorithm for large problems.
Because it reduces the number of computations (thus shortening the execution time)
We investigate the effects on single-processor systems
12/2/2005
HPC Asia

7. The Problem: Strassen’s
Strassen’s for 2n–size matrices O(nlog 7)
For even-size matrices, one recursive step is always applicable
Otherwise
Dynamic and static padding
Peeling:
For odd-size matrices [Hauss 97 & Luo 2004]:
12/2/2005
HPC Asia

8. Odd-Size Square MatricesB 2n B0 A0 2n
2n+1 2n 2n
A0 * B0 is an even-size problem.
Strassen is applied once more
2n+1
12/2/2005
HPC Asia

9. Our Approach: balanced division
For any matrix size, we apply a balanced Strassen’s division process
This reduces the number of computations further than an odd/even size problem (or padded)
Balanced division = balanced workload
Thus, predictable performance
Balanced sized operands
Better data cache utilization
12/2/2005
HPC Asia

10. Balanced Division Matrices
Near Square: m = n+p with min|n-p| B0 A0 A1 B1 n m A3 B2 B3 A2 p n p m
The quadrants are near square matrices.
At any step of the recursion, all sub-matrices are near square matrices
12/2/2005
HPC Asia

11. Balanced Matrices (New matrix add and multiplication)
The balanced division with Strassen’s recursion needs a new MA definition
because addition of matrices of different sizes
We generalize the operations such that:
The algorithm is correct
The extra control for the irregular sizes is completely negligible and only for matrix additions
12/2/2005
HPC Asia

12. Experimental Results We considered 14 systems
We hand coded the MA for each specific system
We measure performance of ATLAS’s MM and MA
We specify an adaptive recursion point size for each system
We encode the recursion point in the algorithm
We measured the relative performance Strassen vs ATLAS
We report the details for three systems shortly
12/2/2005
HPC Asia

13. 12/2/2005
HPC Asia

14. Opteron Strassen + ATLAS ATLAS’s Performance (the higher the better)
12/2/2005
HPC Asia

15. 8600 PA-RISC
Strassen + ATLAS
ATLAS’s Performance
12/2/2005
HPC Asia

16. ALPHA
Strassen + ATLAS
ATLAS’s Performance
12/2/2005
HPC Asia

17. Conclusions Our approach uses the balanced division as Strassen’s does
We performed an exhaustive testing of performance
Some architectures do not offer practical opportunity for S’s
We use benchmarking of ATLAS’s MM and MA for specific code tuning.
In the spirit of adaptive software packages
We speed up ATLAS’s MM without introducing any overhead
Due to data layout or extra control.
12/2/2005
HPC Asia

18. Future work The algorithm extends to rectangular matrices
We will characterize its performance
Parallel formulation and performance
Power management
MM and MA compose the application however they have different architecture utilization
Hardware configurations adaptation (e.g., Xscale)
12/2/2005
HPC Asia