Beschreibung
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.
Autorenporträt
Inhaltsangabe1 Successive approximations.- §1. Existence of the fixed point of a contraction operator.- 1.1. Contraction operators.- 1.2. Use of an equivalent norm.- 1.3. Relative uniqueness of the solution.- 1.4. Spectral radius of a linear operator.- 1.5. Operators which commute with contraction operators.- 1.6. Case of a compact set.- 1.7. Estimating the Lipschitz constant.- 1.8. Equations with uniform contraction operators.- 1.9. Local implicit function theorem.- §2. Convergence of successive approximations.- 2.1. Successive approximations.- 2.2. Equations with a contraction operator.- 2.3. Linear equations.- 2.4. Fractional convergence.- 2.5. Nonlinear equations.- 2.6. Effect of the initial approximation on estimate of convergence rate.- 2.7. Acceleration of convergence.- 2.8. Distribution of errors.- 2.9. Effect of round-off errors.- §3. Equations with monotone operators.- 3.1. Statement of the problem.- 3.2. Cones in Banach spaces.- 3.3. Solvability of equations with monotone operators.- 3.4. Non-trivial positive solutions.- 3.5. Equations with concave operators.- 3.6. Use of powers of operators.- 3.7. The contracting mapping principle in metric spaces.- 3.8. On special metric.- 3.9. Equations with power nonlinearities.- 3.10. Equations with uniformly concave operators.- §4. Equations with nonexpansive operators.- 4.1. Nonexpansive operators.- 4.2. Example.- 4.3. Selfadjoint nonexpansive operators.- 2 Linear equations.- §5. Bounds for the spectral radius of a linear operator.- 5.1. Spectral radius.- 5.2. Positive linear operators.- 5.3. Indecomposable operators.- 5.4. Comparison of the spectral radii of different operators.- 5.5. Lower bounds for the spectral radius.- 5.6. Upper bounds for the spectral radius.- 5.7. Strict inequalities.- 5.8. Examples.- §6. The block method for estimating the spectral radius.- 6.1. Fundamental theorem.- 6.2. Examples.- 6.3. Conic norm.- 6.4. Generalized contracting mapping principle.- §7. Transformation of linear equations.- 7.1. General scheme.- 7.2. Chebyshev polynomials.- 7.3. Equations with selfadjoint operators in Hilbert spaces.- 7.4. Equations with compact operators.- 7.5. Seidel's method.- 7.6. Remark on convergence.- §8. Method of minimal residuals.- 8.1. Statement of the problem.- 8.2. Convergence of the method of minimal residuals.- 8.3. The moment inequality.- 8.4. ?-processes.- 8.5. Computation scheme.- 8.6. Application to equations with nonselfadjoint operators.- §9. Approximate computation of the spectral radius.- 9.1. Use of powers of an operator.- 9.2. Positive operators.- 9.3. Computation of the greatest eigenvalue.- 9.4. Approximate computation of the eigenvalues of selfadjoint operators.- 9.5. The method of normal chords.- 9.6. The method of orthogonal chords.- 9.7. Simultaneous computation of several iterations.- §10. Monotone iterative processes.- 10.1. Statement of the problem.- 10.2. Choice of initial approximations.- 10.3. Acceleration of convergence.- 10.4. Equations with nonpositive operators.- 3 Equations with smooth operators.- §11. The Newton-Kantorovich method.- 11.1. Linearization of equations.- 11.2. Convergence.- 11.3. Further investigation of convergence rate.- 11.4. Global convergence condition.- 11.5. Simple zeros.- §12. Modified Newton-Kantorovich method.- 12.1. The modified method.- 12.2. Fundamental theorem.- 12.3. Uniqueness ball.- 12.4. Modified method with perturbations.- 12.5. Equations with compact operators.- 12.6. Equations with nondifferentiable operators.- 12.7. Remark on the Newton-Kantorovich method.- §13. Approximate solution of linearized equations.- 13.1. Statement of the problem.- 13.2. Convergence theorem.- 13.3. Application of the method of minimal residuals.- 13.4. Choice of initial approximations.- §14. A posteriori error estimates.- 14.1. Error estimates and existence theorems.- 14.2. Linearization of the equation.- 14.3. Rotation of a finite-dimensional vector field.- 14.4. Rotation of a compact vector field.- 14.5. Index of a fixed point
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