The term "iterative method" refers to a wide range of techniques that use successive approximations to obtain more accurate solutions to a linear system at each step. There are two major groups of iterative methods: stationary and nonstationary methods. Stationary methods are older, simpler to understand and implement, but usually not as effective. Nonstationary methods are a relatively recent development; their analysis is usually harder to understand, but they can be highly effective. The nonstationary methods are based on the idea of sequences of orthogonal vectors. 

The rate at which an iterative method converges depends greatly on the spectrum of the coefficient matrix. Hence, iterative methods usually involve a second matrix that transforms the coefficient matrix into one with a more favorable spectrum. The transformation matrix is called a pre conditioner. A good precondition improves the convergence of the iterative method, sufficiently to overcome the extra cost of constructing and applying the pre conditioner. Indeed, without a pre conditioner the iterative method may even fail to converge.

SolveIterative

published