Combinatorial preconditioning — where it appears
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
Also named here as star mesh transform, stretch — the same set of essays touches all of them, so they are one junction rather than several.
Eliminating a vertex is a graph operation
Gaussian elimination on a Laplacian deletes a vertex and joins its neighbours into a clique with conductances wᵢwⱼ over Σw. The matrix that remains is still a graph — symmetric, zero row sums, nonpositive off the diagonal — and the ordering decides whether the fill is thirty-one edges or four hundred and sixty-five.
A preconditioner that is a tree
Every eigenvalue of a tree-preconditioned Laplacian is at least one and at most the total stretch — a combinatorial integer with no arithmetic in it. Measured, the bound is two to four times loose, and on a grid the preconditioner makes the conditioning worse by a factor of 1.85 at every size.
Named alongside it
The objects these essays reach for when they reach for this one.
Graph laplacianStar mesh transformStretchConditioningEffective resistanceFillGeneralised eigenvalueGraph eliminationMinimum degreeOrderingPreconditioningSchur complement