Concise text introduces numerical analysis as a practical, problem-solving discipline, focusing on fundamentals of functional analysis and approximation theory, the major results of theoretical numerical analysis; and specific topics that illustrate the power and usefulness of theoretical analysis. A knowledge of advanced calculus is assumed. 1979 edition.

Table of Contents for Theoretical Numerical Analysis: An Introduction to Advanced Techniques

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

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