面向多核架构的对称正定矩阵高效分块求逆
Towards an Efficient Tile Matrix Inversion of Symmetric Positive Definite Matrices on Multicore Architectures
AI总结：
针对多核架构上对称正定矩阵求逆问题，提出基于动态任务调度和编译器优化（数组重命名、循环反转和流水线）的分块算法，实现LAPACK代码到高效并行分块算法的转换。
摘要：
当前顺序数值线性代数库（如LAPACK）中的算法在多核架构上并行化效果不佳。近年来，一种新的算法家族——分块算法被引入。先前的研究表明，可以编写高效且可扩展的分块算法来执行Cholesky分解、（伪）LU分解和QR分解。在这篇扩展摘要中，我们研究了对称正定矩阵求逆的计算问题。我们观察到，使用动态任务调度器，将现有的LAPACK代码转换为可直接执行的分块算法相对容易。然而，我们证明需要应用非平凡的编译器技术（数组重命名、循环反转和流水线）来进一步提高应用程序的并行性。我们给出了初步的实验结果。
英文摘要：
The algorithms in the current sequential numerical linear algebra libraries (e.g. LAPACK) do not parallelize well on multicore architectures. A new family of algorithms, the tile algorithms, has recently been introduced. Previous research has shown that it is possible to write efficient and scalable tile algorithms for performing a Cholesky factorization, a (pseudo) LU factorization, and a QR factorization. In this extended abstract, we attack the problem of the computation of the inverse of a symmetric positive definite matrix. We observe that, using a dynamic task scheduler, it is relatively painless to translate existing LAPACK code to obtain a ready-to-be-executed tile algorithm. However we demonstrate that non trivial compiler techniques (array renaming, loop reversal and pipelining) need then to be applied to further increase the parallelism of our application. We present preliminary experimental results.
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