ARC Proposal: Preconditioning in non-Laplacian case

Challenges in Robotics motivate the problem of solving a system of the form AAx = A b, where the matrix A has at most two nonzero arbitrary entries (reals or block matrices) in each row. There is a lot of interest in solving these kind of systems efficiently, which correspond to the errors in ”measurements” of the position of the robot while it moves in the 3-D space, so this problem has lots of practical applications. To do this, a standard approach is to find a preconditioner P , namely a non-singular matrix and solve the system P−1(ATAx − A b) = 0. The iterative methods that are commonly used (such as preconditioned conjugate gradient) have cost that is equal to a single iteration (involving the operation of the matrix and of the preconditioner on a vector) multiplied by the number of iterations. The number of iterations in preconditioned conjugate method is bounded by c √ k(ATA,P ) where c is constant and k(AA,P ) is the condition number of the system and is equal to λmax(P −1/2ATAP−1/2) λmin(P−1/2ATAP−1/2) . So what we are interested in is to find a preconditioner P such that the condition number k(AA,P ) is as small as possible.