论文标题
引导块兰斯佐斯（Lanczos）用于大维度特征值问题
Bootstrapped Block Lanczos for large-dimension eigenvalue problems
论文作者
论文摘要
兰开斯算法已证明自己是一个有价值的矩阵本质固定器，用于大幅度的问题，多达数亿甚至数千亿美元。使用任何Lanczos算法的计算成本由稀疏基质矢量乘法的数量支配到达到合适的收敛。 Block Lanczos用稀疏的矩阵矩阵乘法代替了稀疏矩阵矢量乘法，这更有效，但是对于随机选择的起始块（或枢轴），需要更多的乘法才能达到收敛。我们发现，一个自举枢轴块，即是由在截短的空间中计算出的近似特征向量构成的初始块，导致乘法数量急剧减少，并以随机枢轴的形式胜过标准矢量兰斯佐斯和块lanczos。加快加速的关键条件是，枢轴块与最终融合向量具有非平凡的重叠。我们在核结构的配置相互作用代码中实施了这种方法，并将其缩短的时间减少了两倍或多个，最高为十倍。
The Lanczos algorithm has proven itself to be a valuable matrix eigensolver for problems with large dimensions, up to hundreds of millions or even tens of billions. The computational cost of using any Lanczos algorithm is dominated by the number of sparse matrix-vector multiplications until suitable convergence is reached. Block Lanczos replaces sparse matrix-vector multiplication with sparse matrix-matrix multiplication, which is more efficient, but for a randomly chosen starting block (or pivot), more multiplications are required to reach convergence. We find that a bootstrapped pivot block, that is, an initial block constructed from approximate eigenvectors computed in a truncated space, leads to a dramatically reduced number of multiplications, significantly outperforming both standard vector Lanczos and block Lanczos with a random pivot. A key condition for speed-up is that the pivot block have a non-trivial overlap with the final converged vectors. We implement this approach in a configuration-interaction code for nuclear structure, and find a reduction in time-to-solution by a factor of two or more, up to a factor of ten.
