Functional Analysis and Numerical Mathematics
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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.
Table of Contents
Translator’s Note
Preface to the German Edition
Notation
Chapter 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 10.7
Chapter 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 Solutions
Chapter 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 Problems
Appendix
Remarks on Schauder’s Fixed-Point Theorem
26.7 Lemmas on Compact Sets
26.8 Two Formulations of Schauder’s Fixed-Point Theorem
References
Author Index
Subject Index
Product details
- No. of pages: 494
- Language: English
- Copyright: © Academic Press 1966
- Published: January 1, 1966
- Imprint: Academic Press
- eBook ISBN: 9781483264004
About the Author
Lothar Collatz
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