A taxonomy for conjugate gradient methods
The conjugate method of Hestenes and Stiefel is an effective method to solve large, sparse hermitian positive definite (hpd) systems of linear equations, Ax = b. Generalizations to non-hpd matrices have long been sought. The recent theory of Faber and Manteuffle gives necessary and sufficient conditions for the existence of CG method. This paper uses these conditions to develop and organize such methods. We show that any CG method for Ax = b is characterized by and hpd inner product matrix B and a left preconditioning matrix C. At each step the method minimizes the B-norm of the error over a Krylov space. This characterization is then used to classify known and new methods. Finally, it is shown how eigenvalue estimates may be obtained from the iteration parameters, generalizing the well known connection between CG and Lanczos. Such estimates allow implementation of a stopping criterion based more nearly on the true error. 29 refs., 4 tabs.
- Research Organization:
- Illinois Univ., Urbana (USA). Dept. of Computer Science
- DOE Contract Number:
- FG02-87ER25026; W-7405-ENG-48
- OSTI ID:
- 7244989
- Report Number(s):
- DOE/ER/25026-19; UIUCDCS-R-88-1414; UILU-ENG-88-1719; ON: DE88008574
- Country of Publication:
- United States
- Language:
- English
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