TerminologyΒΆ

The following notation and terminology is used in this documentation.

A sparsity pattern of a square matrix is a set of positions (i,j) where the matrix is zero. A sparsity pattern is symmetric if (j,i) is in the sparsity pattern whenever (i,j) is in the sparsity pattern. Note that the entries of the matrix outside the sparsity pattern are allowed to be zero. Those entries are called numerical zeros, as opposed to the structural zeros in the sparsity pattern.

A symmetric sparsity pattern is called chordal if every positive definite matrix A with the sparsity pattern can be factored as

P^T A P  = L L^T

where P is a permutation matrix, and L is a lower triangular matrix with the same sparsity pattern as the lower triangular part of P^T A P. In other words, there exists a symmetric reordering of the matrix that has a Cholesky factorization with zero fill-in. Such a reordering is called a perfect elimination ordering for the sparsity pattern.

For simplicity, we will refer to a symmetric matrix with a chordal sparsity pattern as a chordal symmetric matrix. This terminology is somewhat ambiguous, because there can be many sparsity patterns associated with the same matrix. When we use the term it will be clear from the context which sparsity pattern is used. A nonsymmetric matrix X is chordal if the sparsity pattern of X + X^T is chordal.

The projection of a matrix on a sparsity pattern S is defined as

\mathrm{proj}(X)_{ij} =
\left\{ \begin{array}{ll}
    0       & (i,j) \in S \\
    X_{ij}  & (i,j) \not\in S.
\end{array}\right.

A sparsity pattern R is an embedding of the sparsity pattern S if R is a subset of S.