A preprint on GPU performance reports that its HSMA-TRSM kernels reached peak speedups of 2.05× over cuBLAS and 2.06× over rocBLAS. The comparison covered three accelerators and four data types, but the figures describe the best reported cases across the benchmark rather than one uniform gain.
TRSM is the paper’s shorthand for a matrix-solving routine. The tested operation was the left-side lower-triangular case, run with square matrices and random-valued inputs.
One framework, several paths
The framework changes strategy according to problem size and data type. For small problems, it selects shared-memory direct solving, a dedicated partitioned pipeline for double-complex cases, and a block-GEMM path for large problems. The small-scale path combines loop unrolling and instruction reordering for compute-memory overlap with a dual thread-group, seven-stage pipeline for double-complex cases. One diagonal-block inversion scheme uses double buffering. That raises shared-memory requirements from 2×I_B to 4×I_B, while retaining O(I_B) space complexity relative to the original O(I_B²) scheme.
For large problems, a compile-time configuration framework uses offline profiling and online lookup to choose a platform-specific block size with zero runtime overhead. The evaluation used square random-input matrices in small and large regimes on the NVIDIA A100-SXM4-40GB, NVIDIA H800 PCIe and Hygon DCU Z100. Each test point ran 10 times, with the first run used as warmup and the average of the last nine reported. cuBLAS v13.2.1 was the NVIDIA baseline and rocBLAS v5.1 the DCU baseline.
Small complex cases showed the widest gaps
At matrix dimension 64, the largest reported gains were in double-complex tests. HSMA-TRSM reached 3.53 GFLOPS on DCU, 15.88 GFLOPS on A100 and 12.47 GFLOPS on H800. The reported improvements over vendor libraries were 96.1%, 14.5% and 105.4%, respectively. Peak speedup reached 2.05× on H800 and 1.96× on DCU, while float and double generally stayed in the 1.1× to 1.3× range.
Float-complex performance at the same dimension reached 11.61 GFLOPS on DCU, an 86.3% improvement over rocBLAS. For float, results were 4.64 GFLOPS on DCU and 5.75 GFLOPS on A100, corresponding to reported improvements of 18.4% over rocBLAS and 29.2% over cuBLAS.
On A100, the comparison with MAGMA at m = 64 also favored HSMA-TRSM across all four data types. Improvements were 28.9% for float, 24.6% for double, 5.1% for float complex and 41.4% for double complex.
The advantage changed with matrix size
On DCU, adaptive blocking selected a block size of 256 at m = 1024 and 512 at m = 8192. Compared with rocBLAS’s fixed block size of 128, the reported improvements were 39.3% and 23.8%, respectively. At m = 16384, the paper’s conclusion reports speedups from 1.63× to 2.06× for real types, along with up to 51.9% improvement for double precision on DCU.
Peak throughput numbers varied by platform. In large-scale testing, HSMA-TRSM reached 6,010 GFLOPS for float and 3,654 GFLOPS for double on DCU. A100 peaks were 17,802 GFLOPS for float, 16,564 for double, 18,205 for float complex and 17,111 for double complex. H800 reached 36,088 GFLOPS for float and 38,654 for float complex, while double-complex gains there were near cuBLAS parity.
The timing breakdown shows how the work was divided in A100 double-complex TRSM with block size 256. At m = 2048, diagonal inversion accounted for 20.7% of runtime and GEMM updates, the matrix-multiplication update phase, accounted for 54.8%. At m = 8192, those shares were 3.7% and 88.0%, respectively.
A result tied to a narrow benchmark
The evidence has a clear boundary. It comes from three listed accelerators, square random-input matrices, four data types and the tested small and large regimes. Performance on other architectures, nonsquare shapes and other TRSM variants remains unresolved.
The numerical check was also limited in scope. On A100, the diagonal-inversion path was compared with cuBLAS using relative Frobenius-norm error, with reported errors on the order of 10−7 for single precision and 10−16 to 10−15 for double precision.
The paper is identified as arXiv:2608.25469v1 and dated 26 Aug 2026. The work acknowledges support from the National Key Research and Development Program of China under Grant No. 2023YFB3001700.
Paper data and sources
Original title: Hierarchical Shared Memory-Aware Optimization for TRSM on GPU Platforms
Authors: Xinzhe Chen, Haowei Li, Lijuan Hu et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: 10.1145/3832810.3832817
Original paper · Full text