Cholesky downdating on a hypercube

Least squares modifications associated with the addition or deletion of data often involve updating or downdating the Cholesky factor of the observation matrix. We describe and compare parallel implementations for the hypercube of three methods for down-dating the Cholesky factor: an orthogonal scheme, a hyperbolic scheme, and a hybrid scheme combining the first two. The computational complexities of these algorithms differ significantly, but the parallel implementations of all three have communication complexity similar to solving triangular systems. In computational tests on an Intel iPSC hypercube, the algorithms performed similarly, suggesting a preference for the orthogonal method based on stability considerations. The methods we describe can be adapted to the parallel computation of general orthogonal factorizations, but our discussion is motivated by applications in signal processing using windowed recursive least squares filtering for near real-time solutions.