  Asynchronous methods for solving systems of linear equations have been
researched since Chazan and Miranker's pioneering 1969 paper on chaotic
relaxation. The underlying idea of asynchronous methods is to avoid processor
idle time by allowing the processors to continue to make progress even if not
all progress made by other processors has been communicated to them.
  Historically, the applicability of asynchronous methods for solving linear
equations was limited to certain restricted classes of matrices, such as
diagonally dominant matrices. Furthermore, analysis of these methods focused on
proving convergence in the limit. Comparison of the asynchronous convergence
rate with its synchronous counterpart and its scaling with the number of
processors were seldom studied, and are still not well understood.
  In this paper, we propose a randomized shared-memory asynchronous method for
general symmetric positive definite matrices. We rigorously analyze the
convergence rate and prove that it is linear, and is close to that of the
method's synchronous counterpart if the processor count is not excessive
relative to the size and sparsity of the matrix. We also present an algorithm
for unsymmetric systems and overdetermined least-squares. Our work presents a
significant improvement in the applicability of asynchronous linear solvers as
well as in their convergence analysis, and suggests randomization as a key
paradigm to serve as a foundation for asynchronous methods.
