### Abstract

Original language | English |
---|---|

Pages (from-to) | 110-126 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 35 |

Issue number | 1 |

Early online date | 14 Feb 2014 |

DOIs | |

Publication status | Published - 2014 |

### Fingerprint

### Keywords

- linear algebra
- conjugate gradients
- sesnistivity analysis

### Cite this

*SIAM Journal on Matrix Analysis and Applications*,

*35*(1), 110-126. https://doi.org/10.1137/120889848

}

*SIAM Journal on Matrix Analysis and Applications*, vol. 35, no. 1, pp. 110-126. https://doi.org/10.1137/120889848

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

Research output: Contribution to journal › Article

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 - conjugate gradients

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

IS - 1

ER -