An Alternating-Direction-Implicit Iteration Technique

Introduction. In recent years, considerable effort has been devoted to the analysis of alternating-direction-implicit (ADI) iteration schemes for solving large systems of linear equations [1, 2, 3, 4]. The basic formulation by Peaceman and Rachford [1] and analysis by Douglas [2] laid the groundwork for an extension by Sheldon and Wachspress [3] to a wider class of problems. In [3] and in the work of Birkhoff and Varga [4] the relationship between convergence rate and the commutation of certain matrices was described. In this paper we present some new convergence proofs and extend the analysis beyond the class of problems previously considered. We also describe a formulation for computation on a high speed computer which involves a transformation of the usual equations in order that fast memory requirements and the number of arithmetic operations be reduced. The pioneering work of Peaceman, Rachford and Douglas included application to parabolic and elliptic differential equations. Our application has been to the elliptic difference equations which arise in neutron diffusion calculations. Our approach differs in three respects: 1. The original equations are conditioned in a manner indicated by the generalized theory. 2. Iteration parameters are based on a minimax principle. 3. Computation logic has been modified to reduce computer time and memory requirements.