Direct and Incomplete Cholesky Factorizations with Static Supernodes

Introduction Incomplete factorizations of sparse symmetric positive definite (SSPD) matrices have been used to generate preconditioners for various iterative solvers. These solvers generally use preconditioners derived from the matrix system, , in order to reduce the total number of iterations until convergence. In this report, we investigate the findings of ref. [1] on their method for computing preconditioners from SSPD matrix. In particular, we focus on their first supernodal Cholesky factorization algorithm designed for matrices with naturally occurring block structures. The supernodal incomplete Cholesky algorithm for preconditioner generation is motivated by how the Cholesky factorization accesses column nodes, the overhead from indirect addressing of SSPD matrix , and the memory advantages obtained from level 3 BLAS routines with dense blocking. We introduce this motivation and explain some priors such as supernodal elimination trees [2] in the background section. In Matlab, we implement the above algorithm along with several comparable to illustrate a proof of correctness and to support the motivating claims. Partial results are shown in the methods section. Last, we experiment with the dropping strategies used in the incomplete factorization for both randomized and structured matrices. Our findings and the analysis are in the experiments section.