## Iterative Methods for Ill-Conditioned Linear Systems From Optimization (1998)

Citations: | 5 - 1 self |

### BibTeX

@MISC{Gould98iterativemethods,

author = {Nicholas I. M. Gould},

title = {Iterative Methods for Ill-Conditioned Linear Systems From Optimization},

year = {1998}

}

### OpenURL

### Abstract

Preconditioned conjugate-gradient methods are proposed for solving the ill-conditioned linear systems which arise in penalty and barrier methods for nonlinear minimization. The preconditioners are chosen so as to isolate the dominant cause of ill conditioning. The methods are stablized using a restricted form of iterative refinement. Numerical results illustrate the approaches considered. 1 Email : n.gould@rl.ac.uk 2 Current reports available from "http://www.rl.ac.uk/departments/ccd/numerical/reports/reports.html". Department for Computation and Information Atlas Centre Rutherford Appleton Laboratory Oxfordshire OX11 0QX August 26, 1998. 1 INTRODUCTION 1 1 Introduction Let A and H be, respectively, full-rank m by n (m n) and symmetric n by n real matrices. Suppose furthermore that any nonzero coefficients in this data are modest, that is the data is O(1). (1) We consider the iterative solution of the linear system (H +A T D \Gamma1 A)x = b (1.1) where b is modest an...