Alternating-direction implicit method

The alternating-direction implicit method, usually shortened to ADI, is an iterative technique from numerical linear algebra. In its matrix form it solves Sylvester matrix equations, and it is popular for the large matrix equations that arise in systems theory and control because it can be arranged to build solutions in a memory-efficient factored form. Each iteration is a two step process that alternately updates the column and row spaces of the approximate solution, and its convergence depends strongly on the choice of shift parameters, for which near-optimal values are known in some cases and heuristic strategies are used in others. The same idea also solves parabolic and elliptic partial differential equations, where it became a classic way to model heat conduction and the diffusion equation in two or more dimensions: each time step is split so that one direction is treated implicitly and then the other, leaving simple tridiagonal systems that are cheap to solve. It is an example of an operator splitting method. The method was developed at Humble Oil in the mid-1950s by Jim Douglas Jr, Henry Rachford and Don Peaceman.