We note that many variations and hybrid implementations that lie somewhere between pure column-Cholesky and pure submatrix-Cholesky are possible. Perhaps the most important of these are the multi-frontal methods (see, e.g., ), in which updating operations are accumulated in and propagated through a series of front matrices until finally being incorporated into the ultimate target columns. Multifrontal methods have a number of attractive advantages, most of which accrue from the localization of memory references in the front matrices, thereby facilitating the effective use of memory hierarchies, including cache, virtual memory with paging, or explicit out-of-core solutions (the latter was the original motivation for these methods . In addition, since the front matrices are essentially dense, the operations on them can be done using optimized kernels, such as the BLAS, to take advantage of vectorization or any other available architectural features. For example, such techniques have been used to attain very high performance for sparse factorization on conventional vector supercomputers  and on RISC workstations .