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 |
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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 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 -