Low rank matrix approximations appear in a number of scientific computing applications. We consider the Nystr\"{o}m method for approximating a positive semidefinite matrix $A$. In the case that $A$ is very large or its entries can only be accessed once, a single-pass version may be necessary. In this work, we perform a complete rounding error analysis of the single-pass Nystr\"{o}m method in two precisions, where the computation of the expensive matrix product with $A$ is assumed to be performed in the lower of the two precisions. Our analysis gives insight into how the sketching matrix and shift should be chosen to ensure stability, implementation aspects which have been commented on in the literature but not yet rigorously justified. We further develop a heuristic to determine how to pick the lower precision, which confirms the general intuition that the lower the desired rank of the approximation, the lower the precision we can use without detriment. We also demonstrate that our mixed precision Nystr\"{o}m method can be used to inexpensively construct limited memory preconditioners for the conjugate gradient method and derive a bound the condition number of the resulting preconditioned coefficient matrix. We present numerical experiments on a set of matrices with various spectral decays and demonstrate the utility of our mixed precision approach.

翻译：低秩矩阵近似出现在众多科学计算应用中。本文考虑使用Nyström方法近似正半定矩阵$A$。当$A$规模极大或只能对其元素进行单次访问时，可能需要采用单次版本。本研究对两种精度下的单次Nyström方法进行了完整的舍入误差分析，其中与$A$的昂贵矩阵乘积运算假设以两种精度中较低者执行。我们的分析揭示了如何选择草绘矩阵和偏移量以确保稳定性——这些实现细节虽在文献中有所讨论，但此前尚未得到严格论证。我们进一步开发了一种确定低精度选取的启发式方法，证实了直观认知：近似目标秩越低，可安全使用的精度越低。此外，我们展示了混合精度Nyström方法可低成本构建共轭梯度法的有限记忆预条件子，并推导了所得预条件系数矩阵的条件数上界。我们针对一组具有不同谱衰减特性的矩阵进行了数值实验，验证了所提混合精度方法的有效性。