Functional Analysis and Numerical Mathematics

Functional Analysis and Numerical Mathematics

1st Edition - January 1, 1966

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  • Author: Lothar Collatz
  • eBook ISBN: 9781483264004

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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


    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


    Remarks on Schauder’s Fixed-Point Theorem

    26.7 Lemmas on Compact Sets

    26.8 Two Formulations of Schauder’s Fixed-Point Theorem


    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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