Publisher's 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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