  This paper provides a unified and detailed presentation of root-node style
algebraic multigrid (AMG). Algebraic multigrid is a popular and effective
iterative method for solving large, sparse linear systems that arise from
discretizing partial differential equations. However, while AMG is designed for
symmetric positive definite matrices (SPD), certain SPD problems, such as
anisotropic diffusion, are still not adequately addressed by existing methods.
Non-SPD problems pose an even greater challenge, and in practice AMG is often
not considered as a solver for such problems.
  The focus of this paper is on so-called root-node AMG, which can be viewed as
a combination of classical and aggregation-based multigrid. An algorithm for
root-node is outlined and a filtering strategy is developed, which is able to
control the cost of using root-node AMG, particularly on difficult problems.
New theoretical motivation is provided for root-node and energy-minimization as
applied to symmetric as well non-symmetric systems. Numerical results are then
presented demonstrating the robust ability of root-node to solve non-symmetric
problems, systems-based problems, and difficult SPD problems, including
strongly anisotropic diffusion, convection-diffusion, and upwind steady-state
transport, in a scalable manner. New, detailed estimates of the computational
cost of the setup and solve phase are given for each example, providing
additional support for root-node AMG over alternative methods.
