Sparse Cholesky Elimination Tree

Here I derive the elimination tree for the (right-looking) sparse Cholesky algorithm for computing A = LL^T for lower triangular L and sparse matrices A. This tree forms the foundation for most sparse factorization software, even when the underlying assumptions of Cholesky (symmetric and definite) do not apply. Ultimately this tree tells us two things: 1. where nonzeros appear in the matrix L even if not present in the original A (i.e. “fill-in”) and 2. the task dependency graph of our resulting factorization. Traditionally this concept is usually presented in the context sparse triangular solves which is then used as a building-block to a Cholesky factorization. I wanted to instead work directly from a Cholesky factorization, which is what I do below.