SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page C313-C334, January 2020.
In this work, we deal with the $QR$ factorization of block-tridiagonal matrices, where the blocks are dense and rectangular. This work is motivated by a novel method for computing geodesics over Riemannian manifolds. If blocks are reduced sequentially along the diagonal, only limited parallelism is available. We propose a matrix permutation approach based on the nested dissection method which improves parallelism at the cost of additional computations and storage. We show how operations can be arranged to keep this extra cost as low as possible. We provide a detailed analysis of the approach showing that this extra cost is bounded. Finally, we present an implementation for shared memory systems which relies on task parallelism and makes use of a runtime system. Experimental results support the conclusions of our analysis and show that the proposed approach leads to good performance and scalability.

中文翻译：

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块三对角矩阵的并行QR分解

SIAM科学计算杂志，第42卷，第6期，第C313-C334页，2020年1月。
在这项工作中，我们处理块三对角矩阵的$ QR $因式分解，其中块是密集和矩形的。这项工作是由一种新颖的方法来计算黎曼流形上的测地线。如果沿着对角线顺序减少块，则只能使用有限的并行度。我们提出了一种基于嵌套分解方法的矩阵置换方法，该方法以额外的计算和存储为代价提高了并行度。我们展示了如何安排运营以将这笔额外费用保持在尽可能低的水平。我们对该方法进行了详细的分析，显示出这种额外费用是有限的。最后，我们提出了一种基于任务并行性并利用运行时系统的共享内存系统的实现。