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 $2.6times$ (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.
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