A Matrix Framework for Conjugate Gradient Methods and Some Variants of CG with Less Synchronization Overhead

We will present a matrix framework for the conjugate gradient methods, which is expressed in terms of whole vector sequences instead of single vectors or initial parts of sequences. Using this framework extremely concise derivations of the conjugate gradient method, the Lanczos algorithm, and methods such as GMRES, QMR, CGS, can be given. This framework is then used to present some equivalent forms of computing the inner products in the conjugate gradient and Lanczos algorithms. Such equivalent formulations can perform all inner product calculations of a single iteration simultaneously, thereby making the method more efficient in a parallel computing context. 1 Matrix Framework In his 1965 book, Householder [4] presented a short derivation of the conjugate gradient method using a matrix framework. By introducing matrices whose columns are the elements of a vector sequence, e.q. X = (x1, . . .), it becomes possible to express statements about sequences as matrix equations. For instance...

• Type: Conference
• Year: 1993
• Where: PPSC
• Authors: Eduardo F. D'Azevedo, Victor Eijkhout, Charles H. Romine
• Keywords: Conjugate Gradient Method, Matrix Framework, PPSC 1993, PPSC 2001, Vector Sequence