Theoretical Numerical Analysis An Introduction to Advanced Techniques

This concise text introduces numerical analysis as a practical, problem-solving discipline. The three-part presentation begins with the fundamentals of functional analysis and approximation theory. Part II outlines the major results of theoretical numerical analysis, reviewing product integration, approximate expansion methods, the minimization of functions, and related topics. Part III considers specific subjects that illustrate the power and usefulness of theoretical analysis.
Ideal as a text for a one-year graduate course, the book also offers engineers and scientists experienced in numerical computing a simple introduction to the major ideas of modern numerical analysis. Some practical experience with computational mathematics and the ability to relate this experience to new concepts is assumed. Otherwise, no background beyond advanced calculus is presupposed. Moreover, the ideas of functional analysis used throughout the text are introduced and developed only to the extent they are needed.

Part I. Basic Concepts
  Chapter 1. Review of Functional Analysis
    1.1 Linear Spaces
    1.2 Norms
    1.3 Banach and Hilbert Spaces
    1.4 Mappings and Operators
    1.5 Classification of Problems in Computational Mathematics
  Chapter 2. Approximation Theory
    2.1 The General Interpolation Problem
    2.2 Polynomial Interpolation
    2.3 Piecewise Polynomial Approximations and Splines
    2.4 Best Approximations in Inner Product Spaces
    2.5 Best Approximations in the Maximum Norm
    2.6 Approximations in Several Variables
Part II. Theoretical Aspects of Computational Mathematics
  Chapter 3. Numerical Integration
    3.1 Interpolatory Quadrature Rules
    3.2 Product Integration
    3.3 Another Approach to Numerical Integration
  Chapter 4. The Approximate Solution of Linear Operator Equations
    4.1 Some Theorems on Linear Operators
    4.2 Approximate Expansion Methods
    4.3 Stability and Convergence
    4.4 Error Estimates and Extrapolation
    4.5 Approximate Computation of Eigenvalues
  Chapter 5. The Nonlinear Inverse Problem
    5.1 The Method of Successive Substitutions
    5.2 Some Elementary Results in the Calculus of Operators
    5.3 The Generalized Newton's Method
    5.4 The Minimization of Functionals
Part III. Special Topics
  Chapter 6. The Approximate Solution of Linear Operator Equations of the Second Kind
    6.1 Compact Operators and Equations of the Second Kind
    6.2 The Method of Degenerate Kernels for Fredholm Equations
    6.3 The Nyström Method
    6.4 Analysis of Methods for Integral Equations via Restrictions and Prolongations
    6.5 General Operator Equations of the Second Kind
  Chapter 7. The Finite Element Method
    7.1 Variational Formulation of Operator Equations and Galerkin's Method
    7.2 The Construction of the Finite Elements
    7.3 Convergence Rates for the Finite Element Method
    7.4 Stability of the Finite Element Method
  Chapter 8. The Solution of Nonlinear Operator Equations by Discretization
    8.1 Well-posedness and Condition Numbers
    8.2 Stability and Convergence
    8.3 Asymptotic Expansions of the Discretization Error
  Chapter 9. The Solution of Improperly Posed Problems
    9.1 Regularization Techniques
    9.2 Expansion Methods
  References; Index