  Starlight subtraction algorithms based on the method of Karhunen-Lo\`eve
eigenimages have proved invaluable to exoplanet direct imaging. However, they
scale poorly in runtime when paired with differential imaging techniques. In
such observations, reference frames and frames to be starlight-subtracted are
drawn from the same set of data, requiring a new subset of references (and
eigenimages) for each frame processed to avoid self-subtraction of the signal
of interest. The data rates of extreme adaptive optics instruments are such
that the only way to make this computationally feasible has been to downsample
the data. We develop a technique that updates a pre-computed singular value
decomposition of the full dataset to remove frames (i.e. a "downdate") without
a full recomputation, yielding the modified eigenimages. This not only enables
analysis of much larger data volumes in the same amount of time, but also
exhibits near-linear scaling in runtime as the number of observations
increases. We apply this technique to archival data and investigate its scaling
behavior for very large numbers of frames $N$. The resulting algorithm provides
speed improvements of $2.6\times$ (for 200 eigenimages at $N = 300$) to $140
\times$ (at $N = 10^4$) with the advantage only increasing as $N$ grows. This
algorithm has allowed us to substantially accelerate KLIP even for modest $N$,
and will let us quickly explore how KLIP parameters affect exoplanet
characterization in large $N$ datasets.
