Symbolic Factorization of Sparse Matrix
Table of Contents
The goal of symbolic factorization is to define the sparsity pattern of the \(LU\) decomposition of a sparse matrix \(A\). Recall that
\begin{equation*} LU = PAQ \end{equation*}where \(P\) and \(Q\) are row and column permutations that reflect the pivot strategy associated with the factorization process. [Source]
1. Symbolic Factorization
- generate ordering
- estimate non zero count of \(L\) matrix using symbolic analysis
- apply cholesky with the best ordering to find the actual \(LU\) factorization
This problem is normally done in two steps: (1) generating and applying the ordering to the sparse matrix and (2) applying a symbolic analysis via the graph representation to calculate the nonzero counts (i.e., nnz(L)). These steps together are normally referred to as symbolic factorization by sparse linear solver packages.
