One of the most important chapters in modern functional analysis is the theory of approximate methods for solution of various mathematical problems. Besides providing considerably simplified approaches to numerical methods, the ideas of functional analysis have also given rise to essentially new computation schemes in problems of linear algebra, differential and integral equations, nonlinear analysis, and so on. The general theory of approximate methods includes many known fundamental results. We refer to the classical work of Kantorovich; the investigations of projection methods by Bogolyubov, Krylov, Keldysh and Petrov, much furthered by Mikhlin and Pol'skii; Tikho nov's methods for approximate solution of ill-posed problems; the general theory of difference schemes; and so on. During the past decade, the Voronezh seminar on functional analysis has systematically discussed various questions related to numerical methods; several advanced courses have been held at Voronezh Uni versity on the application of functional analysis to numerical mathe matics. Some of this research is summarized in the present monograph. The authors' aim has not been to give an exhaustive account, even of the principal known results. The book consists of five chapters.
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ISBN-13
978-94-010-2715-1 (9789401027151)
DOI
10.1007/978-94-010-2715-1
Schweitzer Klassifikation
1 Successive approximations.- §1. Existence of the fixed point of a contraction operator.- §2. Convergence of successive approximations.- §3. Equations with monotone operators.- §4. Equations with nonexpansive operators.- 2 Linear equations.- §5. Bounds for the spectral radius of a linear operator.- §6. The block method for estimating the spectral radius..- §7. Transformation of linear equations.- §8. Method of minimal residuals.- §9. Approximate computation of the spectral radius.- §10. Monotone iterative processes.- 3 Equations with smooth operators.- §11. The Newton-Kantorovich method.- §12. Modified Newton-Kantorovich method.- §13. Approximate solution of linearized equations.- §14. A posteriori error estimates.- 4 Projection methods.- § 15. General theorems on convergence of projection methods..- § 16. The Bubnov-Galerkin and Galerkin-Petrov methods.- §17. The Galerkin method with perturbations and the general theory of approximate methods.- §18. Projection methods in the eigenvalue problem.- §19. Projection methods for solution of nonlinear equations.- 5 Small solutions of operator equations.- §20. Approximation of implicit functions.- §21. Finite systems of equations.- §22. Branching of solutions of operator equations.- § 23. Simple solutions and the method of undetermined coefficients.- §24. The problem of bifurcation points.
