- Linearizing the Method of Conjugate Gradients Serge Gratton(grattoncerfacs.fr) David Titley-Peloquin(dtitleypenseeiht.fr) Philippe Toint(philippe.tointfundp.ac.be) Jean Tshimanga Ilunga(jtshimangaserviware.com) Abstract: The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse systems of equations $Ax=b$, where $A\in\Re^{n\times n}$ is symmetric positive definite. Let $x_k$ denote the $k$--th iterate of CG. In this paper we obtain an expression for $J_k$, the Jacobian matrix of $x_k$ with respect to $b$. We use this expression to obtain computable bounds on the spectral norm condition number of $x_k$, and to design algorithms to compute or estimate $J_kv$ and $J_k^Tv$ 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. Keywords: Conjugate Gradients Algorithm, Lanczos Algorithm, Perturbation Analysis, Linearization, Automatic Differentiation Category 1: Applications -- Science and Engineering (Basic Sciences Applications ) Category 2: Nonlinear Optimization (Unconstrained Optimization ) Citation: Download: [PDF]Entry Submitted: 09/09/2012Entry Accepted: 09/09/2012Entry Last Modified: 09/09/2012Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.