Publication: Differentiating the Method of Conjugate Gradients
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Differentiating the Method of Conjugate Gradients

- Article in a journal -
 

Area
Perturbation Analysis

Author(s)
S. Gratton , D. Titley-Peloquin , P. Toint , J. Ilunga

Published in
SIAM Journal on Matrix Analysis and Applications

Year
2014

Abstract
The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse systems of equations Ax=b, where AinRe^n×n is symmetric positive definite. Let x_k denote the kth iterate of CG. This is a nonlinear differentiable function of b. In this paper we obtain expressions for J_k, the Jacobian matrix of x_k with respect to b. We use these expressions to obtain bounds on \|J_k\|_2, the spectral norm condition number of x_k, and discuss algorithms to compute or estimate J_kv and J_k^Tv for a given vector v.

BibTeX
@ARTICLE{
         Gratton2014DtM,
       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. This is a nonlinear differentiable
         function of $b$. In this paper we obtain expressions for $J_k$, the Jacobian matrix of $x_k$ with
         respect to $b$. We use these expressions to obtain bounds on $\|J_k\|_2$, the spectral
         norm condition number of $x_k$, and discuss algorithms to compute or estimate $J_kv$ and $J_k^Tv$
         for a given vector $v$.",
       author = "Gratton, S. and Titley-Peloquin, D. and Toint, P. and Ilunga, J.",
       title = "Differentiating the Method of Conjugate Gradients",
       journal = "SIAM Journal on Matrix Analysis and Applications",
       volume = "35",
       number = "1",
       pages = "110--126",
       year = "2014",
       doi = "10.1137/120889848",
       url = "http://dx.doi.org/10.1137/120889848",
       eprint = "http://dx.doi.org/10.1137/120889848",
       ad_area = "Perturbation Analysis"
}


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