Synopsis
Excerpt from A Note on the Convergence of Alternating Direction Methods
Convergence of the Single parameter alternating direction methods of Douglas, Peaceman, and Rachford [3] 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.
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A Note on the Convergence of Alternating Direction Methods (Classic Reprint)
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Paperback. Condition: New. Print on Demand. This book delves into alternating direction methods, a crucial numerical approach to solving elliptic difference equations. These equations arise commonly in many fields. The author provides a comprehensive analysis of the convergence rates for two fundamental alternating direction methods: the Douglas-Rachford method and the Peaceman-Rachford method. Notably, these methods are applicable to a broad range of elliptic equations with variable coefficients defined in arbitrary regions. The author goes beyond existing proofs to develop novel estimates, demonstrating that the error in the solution is reduced significantly with each iteration, providing valuable insights into the effectiveness of these methods. The book's findings advance our understanding of alternating direction methods, solidifying their importance as powerful tools for solving elliptic equations. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. print-on-demand item. Seller Inventory # 9780243082995_0
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A Note on the Convergence of Alternating Direction Methods Classic Reprint
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