Iteratively constructing preconditioners via the conjugate gradient method

John Dunagan, Nicholas J. A. Harvey

Abstract: We consider the problem of solving a symmetric, positive definite system of linear equations.The most well-known and widely-used method for solving such systemsis the preconditioned Conjugate Gradient method.The performance of this method depends crucially on knowing a good preconditioner matrix.We show that the Conjugate Gradient method itself canproduce good preconditioners as a by-product. These preconditioners allow us to derive new asymptotic bounds on the timeto solve multiple related linear systems.