Functional Analysis and Numerical Mathematics

Functional Analysis and Numerical Mathematics focuses on the structural changes which numerical analysis has undergone, including iterative methods, vectors, integral equations, matrices, and boundary value problems. The publication first examines the foundations of functional analysis and applications, including various types of spaces, convergence and completeness, operators in Hilbert spaces, vector and matrix norms, eigenvalue problems, and operators in pseudometric and other special spaces. The text then elaborates on iterative methods. Topics include the fixed-point theorem for a general iterative method in pseudometric spaces; special cases of the fixed-point theorem and change of operator; iterative methods for differential and integral equations; and systems of equations and difference methods. The manuscript takes a look at monotonicity, inequalities, and other topics, including monotone operators, applications of Schauder's theorem, matrices and boundary value problems of monotone kind, discrete Chebyshev approximation and exchange methods, and approximation of functions. The publication is a valuable source of data for mathematicians and researchers interested in functional analysis and numerical mathematics.

Translator’s NotePreface to the German EditionNotationChapter I Foundations of Functional Analysis and Applications 1. Typical Problems in Numerical Mathematics 1.1 Some General Concepts 1.2 Solutions of Equations 1.3 Properties of the Solutions of Equations 1.4 Extremum Problems with and without Constraints 1.5 Expansions (Determination of Coefficients) 1.6 Evaluations of Expressions 2. Various Types of Spaces 2.1 Hölder’s and Minkowski’s Inequalities 2.2 The Topological Space 2.3 Quasimetric and Metric Spaces 2.4 Linear Spaces 2.5 Normed Spaces 2.6 Unitary Spaces and Schwarz Inequality 2.7 The Parallelogram Equation 2.8 Orthogonality in Unitary Spaces, Bessel’s Inequality 3. Orderings 3.1 Partial Ordering and Complete Ordering 3.2 Lattices 3.3 Pseudometric Spaces 4. Convergence and Completeness 4.1 Convergence in a Pseudometric Space 4.2 Cauchy Sequences 4.3 Completeness, Hilbert Spaces, and Banach Spaces 4.4 Continuity Properties 4.5 Direct Consequences for Hilbert Spaces, Subspaces 4.6 Complete Orthonormal Systems in Hilbert Spaces 4.7 Examples 4.8 Weak Convergence 5. Compactness 5.1 Relative Compactness and Compactness 5.2 Examples of Compactness 5.3 Arzelà’s Theorem 5.4 Compact Sets of Functions Generated by Integral Operators 6. Operators in Pseudometric and Other Special Spaces 6.1 Linear and Bounded Operators 6.2 Composition of Operators 6.3 The Inverse Operator 6.4 Examples of Operators 6.5 Inverse Operators of Neighboring Operators 6.6 Condition Number of a Linear, Bounded Operator 6.7 Error Estimates for an Iteration Process 6.8 Riesz’s Theorem and Theorem of Choice 6.9 A Theorem by Banach on Sequences of Operators 6.10 Application to Quadrature Formulas 7. Operators in Hilbert Spaces 7.1 The Adjoint Operator 7.2 Examples 7.3 Differential Operators for Functions of a Single Variable 7.4 Differential Operators for Functions of Several Variables 7.5 Completely Continuous Operators 7.6 Completely Continuous Integral Operators 7.7 Estimates for the Remainder Term for Holomorphic Functions 7.8 A Bound for the Truncation Error of Quadrature Formulas 7.9 A Fundamental Principle of Variational Calculus 8. Eigenvalue Problems 8.1 General Eigenvalue Problems 8.2 Spectrum of Operators in a Metric Space 8.3 Inclusion Theorem for Eigenvalues 8.4 Projections 8.5 Extremum Properties of the Eigenvalues 8.6 Two Minimum Principles for Differential Equations 8.7 Ritz’s Method 9. Vector and Matrix Norms 9.1 Vector Norms 9.2 Comparison of Different Vector Norms 9.3 Matrix Norms 9.4 From Matrix Theory 9.5 Euclidean Vector Norm and Consistent Matrix Norms 9.6 Other Vector Norms and Subordinate Matrix Norms 9.7 Transformed Norms 10. Further Theorems on Vector and Matrix Norms 10.1 Dual Vector Norms 10.2 Determination of Some Dual Norms 10.3 Powers of Matrices 10.4 A Minimum Property of the Spectral Norm 10.5 Deviation of a Matrix from Normality 10.6 Spectral Variation of Two Matrices 10.7 Selected Problems to Chapter I 10.8 Hints to Selected Problems of Section 10.7Chapter II Iterative Methods 11. The Fixed-Point Theorem for a General Iterative Method in Pseudometric Spaces 11.1 Iterative Methods and Simple Examples 11.2 Iterative Methods for Differential Equations 11.3 The General Fixed-Point Theorem 11.4 Proof of the General Fixed-Point Theorem 11.5 Uniqueness Theorem 12. Special Cases of the Fixed-Point Theorem and Change of Operator 12.1 Special Case of a Linear Auxiliary Operator P 12.2 Special Case of a Metric Space with P a Scalar Factor 12.3 Special Case of a Metric Space with P a Nonlinear, Real-Valued Function 12.4 Iteration with a Perturbed Operator and Questions Concerning the Accuracy 12.5 Error Estimates for the Perturbed Operator 13. Iterative Methods for Systems of Equations 13.1 One Single Equation 13.2 Various Iterative Methods for Systems of Equations 13.3 Convergence Criteria for Linear Systems of Equations 13.4 Row-Sum and Column-Sum Criteria 14. Systems of Equations and Difference Methods 14.1 Difference Methods for Elliptic Differential Equations 14.2 Error Estimates for Jacobi’s and Gauss-Seidel’s Iterative Methods 14.3 Group Iteration 14.4 Infinite Systems of Linear Equations 14.5 Overrelaxation and Error Estimates 14.6 Determination of the Optimal Overrelaxation Factor 14.7 Alternating-Direction Implicit Methods 15. Iterative Methods for Differential and Integral Equations 15.1 Nonlinear Boundary Value Problems 15.2 Nonlinear Ordinary Differential Equations 15.3 Integral Equations 15.4 Systems of Hyperbolic Differential Equations 15.5 Error Estimates for Hyperbolic Systems 16. Derivative of Operators in Supermetric Spaces 16.1 The Fréchet Derivative 16.2 Higher Derivatives 16.3 The Chain Rule of Differential Calculus 16.4 Some Basic Examples for the Determination of Derivatives 16.5 L-Metric Spaces 16.6 Mean Value Theorem and Taylor’s Theorem 17. Some Special Iterative Methods 17.1 Standard and Simplified Newton's Method 17.2 Error Estimate for the Simplified Newton Method 17.3 Simplified Newton Method for Nonlinear Boundary Value Problems 17.4 The Order of Iterative Methods 17.5 Iterative Methods for Equations with Holomorphic Functions, also for Multiple Zeros 17.6 General Iterative Procedure of Order k for the Solution of the Operator Equation Tu = θ 17.7 Remark on the Computational Effort Associated with Procedures of Higher Order 18. The Method of False Position (Regula Falsi) 18.1 Standard and Abbreviated Method of False Position 18.2 Abbreviated Method of False Position for Real Functions of a Single Variable 18.3 The Method of False Position for Operator Equations 18.4 Extensions to the Method of False Position 18.5 Divided Differences of an Operator and Newton’s Interpolation Polynomial 18.6 Convergence of the Method of False Position for Real Functions of One Variable 18.7 More General Methods and Examples 19. Newton’s Method with Improvements 19.1 Improved Newton’s Method and Fundamental Estimating Functions 19.2 General Convergence Theorem for the Improved Newton Methods 19.3 General Remarks Concerning the Application of Newton’s Method 19.4 Newton’s Method for Eigenvalue Problems 19.5 Newton’s Method Applied to Approximation Problems 20. Monotonicity and Extremum Principles for Newton’s Method 20.1 Class of Problems, Convex and Concave Operators 20.2 Monotonicity in Newton’s Method 20.3 Extremum Principle and Inclusion Theorem for the Roots 20.4 Examples of Nonlinear Boundary Value Problems 20.5 Investigation of Convergence 20.6 Mixed Problems for Chapter II 20.7 Hints to the SolutionsChapter III Monotonicity, Inequalities, and Other Topics 21. Monotone Operators 21.1 Definition and Examples 21.2 Monotonically Decomposable Operators 21.3 Application of Schauder’s Fixed-Point Theorem 21.4 Application of Schauder’s Theorem of Nonlinear Differential Equations 21.5 Application to Real Systems of Linear Equations 22. Further Applications of Schauder’s Theorem 22.1 Extrapolation and Error Estimates for a Monotone Sequence of Iterations 22.2 Applications to Systems of Linear Equations 22.3 Application to Linear Differential Equations 22.4 An Additional Theorem on Monotonicity 22.5 Applications to Nonlinear Integral Equations 23. Matrices and Boundary Value Problems of Monotone Kind 23.1 Matrices of Monotone Kind 23.2 Linear Boundary Value Problems of Monotone Kind for Ordinary Differential Equations 23.3 The Maximum Principle for Nonlinear Elliptic Differential Equations 23.4 Nonlinear Elliptic Differential Equations of Monotone Kind 23.5 The Special Case of Linear Elliptic Differential Equations 24. Initial Value Problems and Additional Theorems on Monotonicity 24.1 Strict Monotonicity with Parabolic Equations 24.2 The General Monotonicity Theorem 24.3 Nonlinear Hyperbolic Differential Equations 24.4 Majorization of Green’s Function and Nonlinear Boundary Value Problems 25. Approximation of Functions 25.1 Some Questions Arising with the Approximation Problem 25.2 Linear Approximation 25.3 The Set of Minimal Solutions for Rational Approximation 25.4 Existence Theorem for Rational Chebyshev Approximation 25.5 General Inclusion Theorem for the Minimal Deviation 25.6 A System of Inequalities 25.7 Applications 25.8 Rational T Approximation and Eigenvalue Problems 26. Discrete Chebyshev Approximation and Exchange Methods 26.1 Discrete T Approximation 26.2 Reference and Reference Deviation 26.3 The Center of a Reference 26.4 Exchange Methods 26.5 Mixed Problems for Chapter III 26.6 Hints to the Solutions of the ProblemsAppendix Remarks on Schauder’s Fixed-Point Theorem 26.7 Lemmas on Compact Sets 26.8 Two Formulations of Schauder’s Fixed-Point TheoremReferencesAuthor IndexSubject Index

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