Improve [SD]SYEVD performance by parallelizing [SD]LAED3 #5355
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This pull request introduces a parallelized version of the [SD]LAED3 routine, a key component of the [SD]SYEVD eigensolver for symmetric matrices. OpenBLAS replaces certain LAPACK routines with custom-parallelized versions, and this PR aligns with that strategy.
The [SD]SYEVD routine consists of three main steps:
While PR #5221 improved [SD]SYTRD performance on arm64 by adding tuned [SD]SYMV kernels, this PR focuses on the second step, [SD]STEDC, by parallelizing the internal [SD]LAED3 routine.
Note that [SD]STEDC exhibits poorer scalability with increasing thread counts compared to [SD]SYTRD and [SD]ORMTR. As a result, the proportion of time spent in [SD]STEDC within [SD]SYEVD execution increases with higher thread counts as shown in the following graph.
The parallel [SD]LAED3 reduces the execution time of [SD]STEDC by approximately half in multi-threaded environments. This leads to an overall [SD]SYEVD performance improvement of 1.3x to 1.8x.
I understand that improvements at the LAPACK level are relatively rare in OpenBLAS. Therefore, the parallel [SD]LAED3 implementation has been carefully designed to minimize impact on the library’s structure and to adhere to OpenBLAS’s existing thread management. The parallelization is achieved by setting the necessary parameters in the 'blas_queue_t' structure and calling 'exec_blas(num_cpu, queue)'.