A black box at the end of the rainbow: searching for the perfect preconditioner

Solution of large systems of linear algebraic equations is required in many areas of science and technology, and when several million unknown variables are involved, this apparently simple problem can assume gargantuan proportions. During the last half century many methods have been developed with the aim of providing reliable and cost-effective solutions to a wide range of system types. Many of these use 'pre-conditioners' to improve efficiency. The end-user, perhaps with limited knowledge of the mathematical basis, is presented with the task of choosing between a bewildering array of competing linear solvers and preconditioners. Here we survey currently available techniques and consider the feasibility of producing 'black-box' software capable of solving any linear system without further user input.