This article surveys preconditioning techniques for the iterative solution of large linear
systems, with a focus on algebraic methods suitable for general sparse matrices. Covered …

We present algorithms for solving symmetric, diagonally-dominant linear systems to
accuracy ε in time linear in their number of non-zeros and log (κf (A) ε), where κf (A) is the …

We present a randomized algorithm that on input a symmetric, weakly diagonally dominant n-
by-n matrix A with m nonzero entries and an n-vector b produces an ̃x such that ‖̃x …

In this paper, we present a simple combinatorial algorithm that solves symmetric diagonally
dominant (SDD) linear systems in nearly-linear time. It uses little of the machinery that …

Laplacian matrices of graphs arise in large-scale computational applications such as
semisupervised machine learning; spectral clustering of images, genetic data, and web …

We present a preconditioning technique, called support-graph preconditioning, and use it to
analyze two classes of preconditioners. The technique was first described in a talk by Pravin …

We present support theory, a set of techniques for bounding extreme eigenvalues and
condition numbers for matrix pencils. Our intended application of support theory is to enable …

Combinatorial techniques have become essential tools across the landscape of
computational science, and some of the combinatorial ideas undergirding these tools are …

Graphical models provide a powerful general framework for encoding the structure of large-
scale estimation problems. However, the graphs describing typical real-world phenomena …

We present a linear-system solver that, given an n-by-n symmetric positive semi-definite,
diagonally dominant matrix A with m non-zero entries and an n-vector b, produces a vector …