Preconditioned conjugate gradient algorithms and software for solving large sparse linear systems
David M. Young, Kang C. Jea, Tsun‐Zee Mai · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · 1987
The classical form of the conjugate gradient method (CG method), developed by Hestenes and Stiefel, for solving the linear system Au = b is applicable when the coefficient matrix A is symmetric and positive definite (SPD). In this paper we consider various alternative forms of the CG method as well as generalizations to cases where A is not necessarily SPD. This analysis includes the ''preconditioned conjugate gradient method'' which is equivalent to conjugate gradient acceleration of a basic iterative method corresponding to a preconditioned system. Both the symmetrizable case and the nonsymmetrizable case are considered. For the nonsymmetrizable case there are very few useful theoretical results available. A package of programs, known as ITPACK, has been developed as a tool for carrying out experimental studies on various algorithms. Preliminary conclusions based on experimental results are given. 42 refs.