Practical Extrapolation Methods by Avram Sidi

In addition, as | z| approaches ρ, the rate of convergence of this series deteriorates. If f( z) is analytic at z = 0 but has singularities on the circle | z| = ρ for some finite ρ > 0, then its Maclaurin series converges only for | z| ρ. Let f( z) be a vector-valued function of the complex variable z, namely, f : ℂ → ℂ N, and let its Maclaurin series expansion be given. As we will see later in this chapter, these methods are closely related (actually, mathematically equivalent) to the vector extrapolation methods we have discussed in the previous chapters when the latter are applied to vector sequences that results from a fixed-point iterative scheme without actually having to know what the scheme is. In this chapter, we are concerned with the solution, via so-called Krylov subspace methods, of nonsingular linear (as opposed to nonlinear) systems of equations of the form As an interesting application, the author shows how these methods give rise to rational approximation procedures for vector-valued functions in the complex plane, a subject of importance in model reduction problems among others.ĩ.1 ▪ Projection methods for linear systems It also provides complete proofs in most places. It covers all aspects of the subject, including development of the methods, their convergence study, numerically stable algorithms for their implementation, and their various applications. It is a self-contained, up-to-date, and state-of-the-art reference on the theory and practice of the most useful methods. Vector Extrapolation Methods with Applications is the first book fully dedicated to the subject of vector extrapolation methods.

The convergence of these sequences, which is very slow in many cases, can be accelerated successfully by using suitable vector extrapolation methods. Such sequences arise, for example, in the numerical solution of systems of linear and nonlinear equations by fixed-point iterative methods, and their limits are simply the required solutions to these systems. An important problem that arises in different disciplines of science and engineering is that of computing limits of sequences of vectors of very large dimension.

Practical Extrapolation Methods by Avram Sidi