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 $A$. The computational cost of its single-pass version can be decreased by running it in mixed precision, where the expensive products with $A$ are computed in a lower than the working precision. We bound the extra finite precision error which is compared to the error of the Nystr\"{o}m approximation in exact arithmetic and identify when the approximation quality is not affected by the low precision computations. The mixed precision Nystr\"{o}m method can be used to inexpensively construct a limited memory preconditioner for the conjugate gradient method. We bound the condition number of the preconditioned coefficient matrix, and experimentally show that such preconditioner can be effective.

翻译：低等级矩阵近似值出现在一些科学计算应用中。 我们考虑使用 Nystr\"{o}m 方法来接近正半确定基质基体, 美元。 其单方版本的计算成本可以通过混合精度运行来降低, 以比工作精确度低的更低计算值计算出昂贵的美元产品。 我们在精确算数中绑定了与 Nystr\"{o}m 近似值错误比较的超限精确误差, 并在近似质量不受低精确度计算影响时确定。 混合精度 Nystr\\\\\\{ o}m 方法可以用来廉价地为同源梯度方法构建一个有限的记忆先决条件。 我们约束了设定参数矩阵的条件, 实验性地表明, 此类先决条件可以有效 。