Preface

Numerical Analysis: What Is It?

1. Mathematical Preliminaries
1.1 Basic Concepts and Taylor's Theorem
1.2 Orders of Convergence and Additional Basic Concepts
1.3 Difference Equations

2. Computer Arithmetic
2.1 Floating-Point Numbers and Roundoff Errors
2.2 Absolute and Relative Errors; Loss of Significance
2.3 Stable and Unstable Computations; Conditioning

3. 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

4. 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

5. 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 Least-Squares Problems
5.4 Singular-Value Decomposition and Pseudoinverses
5.5 The QR-Algorithm of Francis for the Eigenvalue Problem

6. Approximating Functions
6.1 Polynomial Interpolation
6.2 Divided Differences
6.3 Hermite Interpolation
6.4 Spline Interpolation
6.5 The B-Splines: Basic Theory
6.6 The B-Splines: Applications
6.7 Taylor Series
6.8 Best Approximation: Least-Squares 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

7. Numerical Differentiation and Integration
7.1 Numerical Differentiation and Richardson Extrapolation
7.2 Numerical Integration Based on Interpolation
7.4 Romberg Integration
7.6 Sard's Theory of Approximating Functionals
7.7 Bernoulli Polynomials and the Euler-Maclaurin Formula

8. Numerical Solution of Ordinary Differential Equations
8.1 The Existence and Uniqueness of Solutions
8.2 Taylor-Series Method
8.3 Runge-Kutta Methods
8.4 Multi-Step Methods
8.5 Local and Global Errors; Stability
8.6 Systems and Higher-Order Ordinary Differential Equations
8.7 Boundary-Value Problems
8.8 Boundary-Value Problems: Shooting Methods
8.9 Boundary-Value Problems: Finite-Difference Methods
8.10 Boundary-Value Problems: Collocation
8.11 Linear Differential Equations
8.12 Stiff Equations

9. Numerical Solution of Partial Differential Equations
9.1 Parabolic Equations: Explicit Methods
9.2 Parabolic Equations: Implicit Methods
9.3 Problems Without Time Dependence: Finite-Difference Methods
9.4 Problems Without Time Dependence: Galerkin and Ritz Methods
9.5 First-Order Partial Differential Equations; Characteristic Curves
9.6 Quasi-Linear Second-Order Equations; Characteristics
9.7 Other Methods for Hyperbolic Problems
9.8 Multigrid Method
9.9 Fast Methods for Poisson's Equation

10. 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

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 9/12/1998