The role of the inner product in stopping criteria for conjugate gradient iterations

被引：23

作者：

Ashby, SF

Holst, MJ

Manteuffel, TA

Saylor, PE

机构：

[1] Univ Calif Lawrence Livermore Natl Lab, Ctr Appl Sci Comp, Livermore, CA 94551 USA

[2] Univ Calif San Diego, Dept Math, La Jolla, CA 92093 USA

[3] Univ Colorado, Dept Appl Math, Boulder, CO 80309 USA

[4] Univ Illinois, Dept Comp Sci, Urbana, IL 61801 USA

来源：

BIT | 2001年 / 41卷 / 01期

基金：

美国国家航空航天局; 美国国家科学基金会;

关键词：

stopping criteria; conjugate gradient methods; B-normal matrices;

D O I：

10.1023/A:1021961616621

中图分类号：

TP31 [计算机软件];

学科分类号：

081202 ; 0835 ;

摘要：

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.

引用

页码：26 / 52

页数：27