Iterative method

Iterative methods solve a linear system by forming subsequent approximations to the solution. The two main classes are the Stationary Iterative methods, and the more general Krylov Subspace methods.

Stationary iterative methods solve a system with an operator approximating the original one, and based on a measurement of the error (the residual) form a correction equation for which this process is repeated. While these methods are simple to derive, implement, and analyse, convergence is only guaranteed for a limited class of matrices.

Krylov subspace methods form an orthogonal basis of the sequence of subsequent matrix powers times the initial residual (the Krylov sequence). The approximations to the solution are then formed by minimizing the residual over the subspace formed. The prototypical method in this class is the Conjugate Gradient Method.

Since these methods form a basis, the method trivially converges in N iterations, where N is the system size. However, in the presence of rounding errors this statement does not hold; moreover, in practice N can be very large, and the iterative process reaches sufficient accuracy already far earlier. The analysis of these methods is hard, being a complicated function of the spectrum of the operator.

Preconditioners: the approximating operator that appears in Stationary Iterative methods can also be incorporated in Krylov subspace methods (alternatively, preconditioned Krylov methods can be considered as accelerations of Stationary iterative methods), where they become transformations of the original operator to a presumably better conditioned one. The construction of preconditioners is a large research area.

History: probably the first iterative method appeared in a letter of Gauss to a student of his. He proposed solving a 4-by-4 system of equations by repeatedly solving the component in which the residual was largest. The theory of Stationary Iterative methods was solidly established with the work of D.M. Young starting the 1950s. The Conjugate Gradient method was also invented in the 1950s, with independent developments by Lanczos and Hestenes and Stiefel, but its nature and applicability was minunderstood at the time. Only in the 1970s was it realized that conjugacy based methods work very well for partial differential equations, especially of elliptic type.