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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. In this book we will cover two types
of iterative 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 we present are
based on the idea of sequences of orthogonal vectors. (An exception
is the Chebyshev iteration method, which is based on
orthogonal polynomials.)
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 preconditioner. The use of a good preconditioner improves the convergence of the iterative method, sufficiently to overcome the extra cost of constructing and applying the preconditioner. Indeed, without a preconditioner the iterative method may even fail to converge.