An Optimized Framework for Matrix Factorization on the New Sunway Many-core Platform

Matrix factorization functions are used in many areas and often play an important role in the overall performance of the applications. In the LAPACK library, matrix factorization functions are implemented with blocked factorization algorithm, shifting most of the workload to the high-performance Level-3 BLAS functions. But the non-blocked part, the panel factorization, becomes the performance bottleneck, especially for small- and medium-size matrices that are the common cases in many real applications. On the new Sunway many-core platform, the performance bottleneck of panel factorization can be alleviated by keeping the panel in the LDM for the panel factorization. Therefore, we propose a new framework for implementing matrix factorization functions on the new Sunway many-core platform, facilitating the in-LDM panel factorization. The framework provides a template class with wrapper functions, which integrates inter-CPE communication for the Level-1 and Level-2 BLAS functions with flexible interfaces and can accommodate different partitioning schemes. With the framework, writing panel factorization code with data residing in the LDM space can be done with much higher productivity. We implemented three functions (d.etr f, d.eqr f, and dpotr f) based on the framework and compared our work with aCPE_BLAS version, which uses the original LAPACK implementation linked with optimized BLAS library that runs on the CPE mesh. Using the most favorable partitioning, the panel factorization part achieves speedup of up to 26.3, 19.1, and 18.2 for the three matrix factorization functions. For the whole function, our implementation is based on a carefully tuned recursion framework, and we added specific optimization to some subroutines used in the factorization functions. Overall, we obtained average speedup of 9.76 on d.etr f, 10.12 on d.eqr f, and 4.16 on dpotr f, compared to the CPE_BLAS version. Based on the current template class, our work can be extended to support more categories of linear algebra functions.