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 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