Numerical Analysis

Table of Contents
 Preface
Numerical Analysis: What Is It?

 Mathematical Preliminaries
1.1 Basic Concepts and Taylor's Theorem
1.2 Orders of Convergence and Additional Basic Concepts
1.3 Difference Equations
 Computer Arithmetic
2.1 FloatingPoint Numbers and Roundoff Errors
2.2 Absolute and Relative Errors; Loss of Significance
2.3 Stable and Unstable Computations; Conditioning
 Solution of Nonlinear Equations
3.1 Bisection (Interval Halving) Method
3.2 Newton's Method
3.3 Secant Method
3.4 Fixed Points and Functional Iteration
3.5 Computing Zeros of Polynomials
3.6 Homotopy and Continuation Methods
 Solving Systems of Linear Equations
4.1 Matrix Algebra
4.2 The LU and Cholesky Factorizations
4.3 Pivoting and Constructing an Algorithm
4.4 Norms and the Analysis of Errors
4.5 Neumann Series and Iterative Refinement
4.6 Solution of Equations by Iterative Methods
4.7 Steepest Descent and Conjugate Gradient Methods
4.8 Analysis of Roundoff Error in the Gaussian Algorithm
 Selected Topics in Numerical Linear Algebra
5.1 Matrix Eigenvalue Problem: Power Method
5.2 Schur's and Gershgorin's Theorems
5.3 Orthogonal Factorizations and LeastSquares Problems
5.4 SingularValue Decomposition and Pseudoinverses
5.5 The QRAlgorithm of Francis for the Eigenvalue Problem
 Approximating Functions
6.1 Polynomial Interpolation
6.2 Divided Differences
6.3 Hermite Interpolation
6.4 Spline Interpolation
6.5 The BSplines: Basic Theory
6.6 The BSplines: Applications
6.7 Taylor Series
6.8 Best Approximation: LeastSquares Theory
6.9 Best Approximation: Chebyshev Theory
6.10 Interpolation in Higher Dimensions
6.11 Continued Fractions
6.12 Trigonometric Interpolation
6.13 Fast Fourier Transform
6.14 Adaptive Approximation
 Numerical Differentiation and Integration
7.1 Numerical Differentiation and Richardson Extrapolation
7.2 Numerical Integration Based on Interpolation
7.3 Gaussian Quadrature
7.4 Romberg Integration
7.5 Adaptive Quadrature
7.6 Sard's Theory of Approximating Functionals
7.7 Bernoulli Polynomials and the EulerMaclaurin Formula
 Numerical Solution of Ordinary Differential Equations
8.1 The Existence and Uniqueness of Solutions
8.2 TaylorSeries Method
8.3 RungeKutta Methods
8.4 MultiStep Methods
8.5 Local and Global Errors; Stability
8.6 Systems and HigherOrder Ordinary Differential Equations
8.7 BoundaryValue Problems
8.8 BoundaryValue Problems: Shooting Methods
8.9 BoundaryValue Problems: FiniteDifference Methods
8.10 BoundaryValue Problems: Collocation
8.11 Linear Differential Equations
8.12 Stiff Equations
 Numerical Solution of Partial Differential Equations
9.1 Parabolic Equations: Explicit Methods
9.2 Parabolic Equations: Implicit Methods
9.3 Problems Without Time Dependence: FiniteDifference Methods
9.4 Problems Without Time Dependence: Galerkin and Ritz Methods
9.5 FirstOrder Partial Differential Equations; Characteristic Curves
9.6 QuasiLinear SecondOrder Equations; Characteristics
9.7 Other Methods for Hyperbolic Problems
9.8 Multigrid Method
9.9 Fast Methods for Poisson's Equation
 Linear Programming and Related Topics
10.1 Convexity and Linear Inequalities
10.2 Linear Inequalities
10.3 Linear Programming
10.4 The Simplex Algorithm
 Appendix: An Overview of Mathematical Software
 Bibliography
 Index