The Role of the Inner Product in Stopping Criteria for Conjugate Gradient Iterations |
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| Authors: | S. F. Ashby M. J. Holst A. Manteuffel P. E. Saylor |
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| Affiliation: | (1) Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, P.O. Box 808, L-561, Livermore, CA 94551, USA;(2) Department of Mathematics, UC San Diego, La Jolla, CA 92093, USA;(3) Department of Applied Mathematics, University of Colorado at Boulder, Campus Box 526, Boulder, CO 80309-0526, USA;(4) Department of Computer Science, University of Illinois, Urbana, IL 61801, USA |
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| Abstract: | Two natural and efficient stopping criteria are derived for conjugate gradient (CG) methods, based on iteration parameters. The derivation makes use of the inner product matrix B-defining the CG method. In particular, the relationship between the eigenvalues and B-norm of a matrix is investigated, and it is shown that the ratio of largest to smallest eigenvalues defines the B-condition number of the matrix. Upper and lower bounds on various measures of the error are also given. The compound stopping criterion presented here is an obvious  default in software packages because it does not require any additional norm computations. |
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| Keywords: | Stopping criteria conjugate gradient methods B-normal matrices |
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