A Note on the Convergence of Alternating Direction Methods (Classic Reprint)
Excerpt from A Note on the Convergence of Alternating Direction Methods Convergence of the Single parameter alternating direction methods of Douglas, Peaceman, and Rachford  has been proved for a wide class of elliptic difference equations; see, for example, Birkhoff and varga The proof consists in showing that a certain matrix, similar to the defining matrix, has spectral radius less than one. Since the defining matrices for these methods are symmetric only when they are induced by a proper discretization of Iaplace's equation in a rectangular region, an estimate for their spectral radii does not imply a corresponding norm estimate for their rate of convergence. The purpose of this note is to present another proof of the convergence of the two basic alternating direction methods which, at the same time, provides a norm estimate for their rate of convergence. First, for simplicity, we shall consider Laplaceis equation in an arbitrary, bounded lattice region. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
- Paperback | 26 pages
- 152 x 229 x 1mm | 50g
- 11 Feb 2018
- Forgotten Books
- 7 Illustrations; Illustrations, black and white