计算大型病态Hankel矩阵的最小特征值
Computing the smallest eigenvalue of large ill-conditioned Hankel matrices
AI总结：
针对病态Hankel矩阵的最小特征值问题，提出一种需极高精度算术的并行算法，发现直接方法比常用高效方法更有效，并通过消息传递与共享内存结合实现近完美可扩展性。
摘要：
本文提出了一种并行算法，用于求解特定形式的病态Hankel矩阵的最小特征值，该算法需要使用极高精度的算术运算。令人惊讶的是，我们发现，在这些条件下，通常设计用于高效的方法实际上不如直接方法高效。然后，我们开发了该算法的并行实现，考虑了个别算术运算异常高的成本。我们的方法结合了消息传递和共享内存，实现了近完美的可扩展性和对网络延迟的高容忍度。因此，我们能够求解比以前可能更大的矩阵，并有可能将这项工作扩展到具有更高并行度的系统。
英文摘要：
This paper presents a parallel algorithm for finding the smallest eigenvalue of a particular form of ill-conditioned Hankel matrix, which requires the use of extremely high precision arithmetic. Surprisingly, we find that commonly-used approaches that are designed for high efficiency are actually less efficient than a direct approach under these conditions. We then develop a parallel implementation of the algorithm that takes into account the unusually high cost of individual arithmetic operations. Our approach combines message passing and shared memory, achieving near-perfect scalability and high tolerance for network latency. We are thus able to find solutions for much larger matrices than has been previously possible, with the potential for extending this work to systems with greater levels of parallelism.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
