Linearizing the Method of Conjugate Gradients

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

Research output: Contribution to journalArticle

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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.
Original languageEnglish
Pages (from-to)110-126
JournalSIAM Journal on Matrix Analysis and Applications
Volume35
Issue number1
Early online date14 Feb 2014
DOIs
Publication statusPublished - 2014

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

Keywords

  • linear algebra
  • conjugate gradients
  • sesnistivity analysis

Cite this

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Linearizing the Method of Conjugate Gradients. / Gratton, Serge; Titley-Peloquin, David; Toint, Philippe; Tshimanga Ilunga, Jean.

In: SIAM Journal on Matrix Analysis and Applications, Vol. 35, No. 1, 2014, p. 110-126.

Research output: Contribution to journalArticle

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T1 - Linearizing the Method of Conjugate Gradients

AU - Gratton, Serge

AU - Titley-Peloquin, David

AU - Toint, Philippe

AU - Tshimanga Ilunga, Jean

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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.

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