Reduction of an [Formula: see text] nonsymmetric matrix to Hessenberg form which takes [Formula: see text] flops and [Formula: see text] I/Os is a major performance bottleneck in the computing of its eigenvalues. Usually to improve the performance, this Hessenberg reduction is computed in two steps: the first one reduces the matrix to a banded Hessenberg form, and the second one further reduces it to Hessenberg form by incorporating more matrix-matrix operations in the computation. We analyse the two steps of the Hessenberg reduction problem on the external memory model (of Aggarwal and Vitter) for their I/O complexities. We propose and analyse a tile based algorithm for the first step of the reduction and show that it takes [Formula: see text] I/Os. For the reduction of a banded Hessenberg matrix of bandwidth [Formula: see text] to Hessenberg form, we propose an algorithm, which uses tight packing of bulges, and requires only [Formula: see text] I/Os. Combining the results of the two steps of the reduction, we show that the Hessenberg reduction can be performed in [Formula: see text] I/Os, when [Formula: see text] is sufficiently large. To the best of our knowledge, the proposed algorithm is the first I/O optimal algorithm for Hessenberg reduction; the worst case I/O complexity matches the known lower bound of the problem.

中文翻译：

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一种用于 Hessenberg 约简的输入/输出高效算法

将 [Formula: see text] 非对称矩阵简化为 Hessenberg 形式，其中 [Formula: see text] 触发器和 [Formula: see text] I/O 是计算其特征值的主要性能瓶颈。通常为了提高性能，这种 Hessenberg 约简分两步计算：第一个步骤将矩阵简化为带状 Hessenberg 形式，第二个步骤通过在计算中加入更多的矩阵矩阵运算进一步将其简化为 Hessenberg 形式。我们分析了外部存储器模型（Aggarwal 和 Vitter）上 Hessenberg 缩减问题的两个步骤的 I/O 复杂性。我们提出并分析了一种基于瓦片的算法，用于简化的第一步，并表明它需要 [公式：见文本] I/O。为了减少带宽的带状 Hessenberg 矩阵 [公式：见文本] 到 Hessenberg 形式，我们提出了一种算法，它使用紧包的凸起，并且只需要 [公式：见文本] I/O。结合归约这两个步骤的结果，我们表明当 [公式：参见文本] 足够大时，可以在 [公式：参见文本] I/O 中执行 Hessenberg 归约。据我们所知，所提出的算法是第一个用于 Hessenberg 约简的 I/O 优化算法；最坏情况下的 I/O 复杂度与问题的已知下限相匹配。提出的算法是Hessenberg减少的第一个I/O优化算法；最坏情况下的 I/O 复杂度与问题的已知下限相匹配。提出的算法是Hessenberg减少的第一个I/O优化算法；最坏情况下的 I/O 复杂度与问题的已知下限相匹配。