  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.
