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This paper introduces HSMA-TRSM, a hierarchical shared memory-aware optimization framework designed to enhance the performance of left-side lower-triangular TRSM operations on NVIDIA A100, H800, and Hygon DCU Z100 GPUs. By implementing a pipelined compute-memory overlap mechanism and a dual thread-group seven-stage pipeline strategy, the authors effectively address shared memory limitations for double complex types, achieving significant speedups. The framework also incorporates a compile-time configuration selection that optimally adjusts block sizes based on matrix scale and hardware characteristics, resulting in peak speedups of 2.05x over cuBLAS and 2.06x over rocBLAS.
HSMA-TRSM achieves over 2x speedup on key GPU platforms by optimizing shared memory usage for complex triangular solves, revealing untapped performance potential in BLAS operations.
Triangular Solve with Multiple Right-hand Sides (TRSM) is a fundamental BLAS Level-3 operation that underpins LU/Cholesky decomposition, sparse direct solvers, and matrix inversion. In the left-side lower-triangular case studied in this paper, efficient GPU implementation remains challenging because forward substitution introduces strict row-wise dependencies, and shared memory is too scarce to hold both operand matrices for wide data types such as double complex. This paper presents HSMA-TRSM, a hierarchical shared memory-aware optimization framework for left-side lower-triangular TRSM on NVIDIA A100, NVIDIA H800, and Hygon DCU Z100 accelerators. For the small-scale regime (m,n<=64), we design a pipelined compute-memory overlap mechanism through loop unrolling and instruction reordering, and propose a dual thread-group seven-stage pipeline strategy to address shared memory constraints for double complex types. For large-scale problems, we introduce a diagonal block decoupling optimization with an O(IB)shared-memory footprint for diagonal block inversion, enabling adaptive block size selection based on matrix scale and hardware characteristics. A compile-time configuration selection framework based on offline profiling and online lookup selects the optimal block size per platform with zero runtime overhead. Evaluated on NVIDIA A100, H800, and Hygon DCU Z100, HSMA-TRSM achieves peak speedups of 2.05xover cuBLAS and 2.06xover rocBLAS. The gains are strongest in shared-memory-constrained double-complex small cases and in large real-type cases where adaptive blocking improves GEMM-dominated updates, while mature vendor kernels leave less optimization headroom in some regimes.
