# Linearizing the Method of Conjugate Gradients

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

Résultats de recherche: Contribution à un journal/une revueArticle

### Résumé

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 originale Anglais 110-126 SIAM Journal on Matrix Analysis and Applications 35 1 14 févr. 2014 https://doi.org/10.1137/120889848 Publié - 2014

### Empreinte digitale

Jacobian matrices
Spectral Norm
Algorithm Design
Jacobian matrix
Iterative Solution
Condition number
Iterate
Positive definite
System of equations
Experiments
Numerical Experiment
Denote
Estimate

### Citer ceci

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abstract = "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.",
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author = "Serge Gratton and David Titley-Peloquin and Philippe Toint and {Tshimanga Ilunga}, Jean",
year = "2014",
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journal = "SIAM Journal on Matrix Analysis and Applications",
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Linearizing the Method of Conjugate Gradients. / Gratton, Serge; Titley-Peloquin, David; Toint, Philippe; Tshimanga Ilunga, Jean.

Dans: SIAM Journal on Matrix Analysis and Applications, Vol 35, Numéro 1, 2014, p. 110-126.

Résultats de recherche: Contribution à un journal/une revueArticle

TY - JOUR

T1 - Linearizing the Method of Conjugate Gradients

AU - Gratton, Serge

AU - Titley-Peloquin, David

AU - Toint, Philippe

AU - Tshimanga Ilunga, Jean

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - linear algebra

KW - sesnistivity analysis

U2 - 10.1137/120889848

DO - 10.1137/120889848

M3 - Article

VL - 35

SP - 110

EP - 126

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

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