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 …

The computational solution of problems can be restricted by the availability of solution
methods for linear (ized) systems of equations. In conjunction with iterative methods …

We introduce a new notion of graph sparsification based on spectral similarity of graph
Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier …

Spectral graph theory is the study and exploration of graphs through the eigenvalues and
eigenvectors of matrices naturally associated with those graphs. It is intuitively related to …

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 …

We show how to perform sparse approximate Gaussian elimination for Laplacian matrices.
We present a simple, nearly linear time algorithm that approximates a Laplacian by the …

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 …

We present an algorithm that on input of an n-vertex m-edge weighted graph G and a value
k produces an incremental sparsifier ̂G with n-1+m/k edges, such that the relative condition …

Krylov subspace methods (KSMs) are iterative algorithms for solving large, sparse linear
systems and eigenvalue problems. Current KSMs rely on sparse matrix-vector multiply …

Although there are many advanced and specialized texts and handbooks on algorithms,
until now there was no book that focused exclusively on the wide variety of data structures …