Automatic preconditioning by limited memory quasi-Newton updating

This paper proposes a preconditioner for the conjugate gradient method ( CG ) that is designed for solving systems of equations Ax = b(i) with differen t right-hand-side vectors or for solving a sequence of slowly varying syste ms A(k)x = b(k). The preconditioner has the form of a limited memory quasi- Newton matrix and is generated using information from the CG iteration. The automatic preconditioner does not require explicit knowledge of the coeffi cient matrix A and is therefore suitable for problems where only products o f A times a vector can be computed. Numerical experiments indicate that the preconditioner has most to offer when these matrix-vector products are exp ensive to compute and when low accuracy in the solution is required. The ef fectiveness of the preconditioner is tested within a Hessian-free Newton me thod for optimization and by solving certain linear systems arising infinit e element models.