The additive Peaceman–Rachford method

A new version of the parallel alternating direction implicit (ADI) method by Peaceman and Rachford for solving systems of linear algebraic equations with positive-definite coefficient matrices represented as sums of two commuting terms is suggested. The algorithms considered are suited for solving two-dimensional grid boundary-value problems with separable variables, as well as the Sylvester and Lyapunov matrix equations. The approach to parallelization speed up proposed in the paper is based on representing rational functions as sums of partial fractions. An additive version of the factorized ADI method for solving Sylvester's equation is described. Estimates of the speed up obtained by increasing the number of computational units are presented. These estimates demonstrate a potential advantage of using the additive algorithms when implemented on a supercomputer with large number of processors or cores.