Stochastic Preconditioning for Diagonally Dominant Matrices

Doubly Diagonally Dominant Matrices

We consider the class of doubly diagonally dominant matrices (A = [ ajj] E C”, ‘, la,,1 l”jjl > Ck+ i laiklCk+ jlajkl. i #j) and its subclasses. We give necessary and sufficient conditions in terms of the directed graph for an irreducibly doubly diagonally dominant matrix to be a singular matrix or to be an H-matrix. As in the case of diagonal dominance, we show that the Schur complements of do...

Relative Perturbation Theory for Diagonally Dominant Matrices

OF DISSERTATION RELATIVE PERTURBATION THEORY FOR DIAGONALLY DOMINANT MATRICES Diagonally dominant matrices arise in many applications. In this work, we exploit the structure of diagonally dominant matrices to provide sharp entrywise relative perturbation bounds. We first generalize the results of Dopico and Koev to provide relative perturbation bounds for the LDU factorization with a well condi...

Solving Symmetric Diagonally-Dominant Systems by Preconditioning

In this paper we design support-tree preconditioners for n × n matrices with m nonzeros that aresymmetric and diagonally-dominant with a nonnegative diagonal (SDD matrix). This reduces to de-signing such a preconditioner for a Laplacian matrix, A, which can be interpreted as an undirectednonnegatively-weighted graph, G with n vertices and m edges. Preconditioners accelerate the ...

A note on nonnegative diagonally dominant matrices

Subdirect sums of doubly diagonally dominant matrices

The problem of when the k-subdirect sum of a doubly diagonally dominant matrix (DDD matrix) is also a DDD matrix is studied. Some sufficient conditions are given. The same situation is analyzed for diagonally dominant matrices and strictly diagonally dominant matrices. Additionally, some conditions are also derived when card(S)>card(S1) which was not studied by Bru, Pedroche and Szyld [Electron...