Linearizing the Method of Conjugate Gradients

Serge Gratton, David Titley-Peloquin, Philippe Toint, Jean Tshimanga Ilunga

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The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse systems of equations Ax=b, where A 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 Jk.v and JkT.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.
langue originaleAnglais
Pages (de - à)110-126
journalSIAM Journal on Matrix Analysis and Applications
Numéro de publication1
Date de mise en ligne précoce14 févr. 2014
Les DOIs
Etat de la publicationPublié - 2014

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