Linearizing the Method of Conjugate Gradients

The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse systems of equations Ax = b, where A ∈ R is symmetric positive definite. Let xk denote the k–th iterate of CG. In this paper we obtain an expression for Jk, the Jacobian matrix of xk with respect to b. We use this expression to obtain computable bounds on the spectral norm condition number of xk, and to design algorithms to compute or estimate Jkv and J T k v for a given vector v. We also discuss several applications in which these ideas may be used. Numerical experiments are performed to illustrate the theory.