Fields of values and iterative methods

Abstract The performance of an iterative scheme to solve x = T x + c , T ϵ C n × n , c ϵ C n , is often judged by spectral properties of T . If T is not normal, it is however well known that only conclusions about the assymptotic behavior of an iterative method can be drawn from spectral information. To anticipate the progress of the iteration after a finite number of steps, the knowledge of the eigenvalues alone is often useless. In addition, the spectrum of T may be highly sensitive to perturbations if T is not normal. An iterative method which—on the basis of some spectral information—is predicted to converge rapidly for T may well diverge if T is slightly perturbed. In practice, the convergence of the iteration x m = T x m -1 + c is therefore frequently measured by some norm [Verbar] T [Verbar], rather than by the spectral radius ϱ( T ). But apart from the fact that norms lead to error estimates which are often too pessimistic, they cannot be used to analyze more general schemes such as, e.g., the Chebyshev iterative methods. Here, we discuss another tool to analyze the behavior of an iterative method, namely the field of values W ( T ), the collection of all Rayleigh quotients of T . W ( T ) contains the eigenvalues of T , and the numerical radius μ( T ) = max z ϵ W ( T ) | z | defines a norm on C n × n . The field of values represents therefore an “intermediate concept” to judge an iterative scheme by—it is related to the spectral approach but has also certain norm properties.