Conjugate gradient methods for indefinite systems

Abstract

Conjugate gradient methods have often been used to solve a ~lde variety of numerical problems, including linear and nonlinear algebraic equations, eigenvalue problems and minimization problems. These applications have been s~m~lar in that they involve large numbers of variables or dimensions. In these circumstances any method of solution which involves storing a full matrix of this large order, becomes inapplicable. Thus recourse to the conjugate gradient method may be the only alternative. For problems in linear equations, the conjugate gradient method requires that the e~fficient m~trix A , is not onl~ symmetric but also positive definite. This restriction is also implicit in applications to unconstrained minimization. Yet there are many problems in which non-trivial equations have to be added to v~hat would otherwise be an elliptic system, One example is the restriction to divergence-free vectors in fluid dynamics (linear or nonlinear), and there are various other examples from partial differential equations. Also minimization problems in gersral, when involving linear or nonlinear equality constraints, provide further examples. In all these cases the coefficient matrix of the linear model has the farm