Vector extrapolation methods with applications

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. 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. The convergence of these sequences, which is very slow in many cases, can be accelerated successfully by using suitable vector extrapolation methods. Vector Extrapolation Methods with Applications is the first book fully dedicated to the subject of vector extrapolation methods. It is a self-contained, up-to-date, and state-of-the-art reference on the theory and practice of the most useful methods. 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 also provides complete proofs in most places. 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.

Preface -- 0. Introduction and review of linear algebra -- part I. Vector extrapolation methods -- 1. Development of polynomial extrapolation methods -- 2. Unified algorithms for MPE and RRE -- 3. MPE and RRE are related -- 4. Algorithms for MMPE and SVD-MPE -- 5. Epsilon algorithms -- 6. Convergence study of extrapolation methods : part I -- 7. Convergence study of extrapolation methods : part II -- 8. Recursion relations for vector extrapolation methods -- part II. Krylov sybspace methods -- 9. Krylov subspace methods for linear systems -- 10. Krylov subspace methods for eigenvalue problems -- part III. Applications and generalizations -- 11. Miscellaneous applications of vector extrapolation methods -- 12. Rational approximations from vector-valued power series : part I -- 13. Rational approximations from vector-valued power series : part II -- 14. Applications of SMPE, SMMPE, and STEA -- 15. Vector generalizations of scalar extrapolation methods -- 16. Vector-valued rational interpolation methods -- part IV. Appendices -- Appendix A. QR factorization -- Appendix B. singular value decompositions (SVD) -- Appendix C. Moore-Penrose generalized inverse -- Appendix D. Basics of ort…