Preconditioning Reduced Matrices

We study preconditioning strategies for linear systems with positive-definite matrices of the form $Z^T GZ$, where Z is rectangular and G is symmetric but not necessarily positive definite. The preconditioning strategies are designed to be used in the context of a conjugate-gradient iteration, and are suitable within algorithms for constrained optimization problems. The techniques have other uses, however, and are applied here to a class of problems in the calculus of variations. Numerical tests are also included.