Table of contents for Numerical analysis : mathematics of scientific computing / David Kincaid, Ward Cheney.


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Numerical Analysis: What Is It?                         1
1  Mathematical Preliminaries                              3
1.0 Introduction                                      3
1.1 Basic Concepts and Taylor's Theorem               3
1.2  Orders of Convergence and Additional Basic Concepts  15
1.3  Difference Equations                            28
2  Computer Arithmetic                                    37
2.0 Introduction                                     37
2.1 Floating-Point Numbers and Roundoff Errors       37
2.2 Absolute and Relative Errors: Loss of Significance  55
2.3 Stable and Unstable Computations: Conditioning   64
3  Solution of Nonlinear Equations                        73
3.0 Introduction                                     73
3.1 Bisection (Interval Halving) Method              74
3.2  Newton's Method                                 81
3.3 Secant Method                                    93
3.4 Fixed Points and Functional Iteration           100
3.5 Computing Roots of Polynomials                  109
3.6  Homotopy and Continuation Methods              130
4  Solving Systems of Linear Equations                  139
4.0 Introduction                                    139
4.1 Matrix Algebra                                  140
4.2 LU and Cholesky Factorizations                 149
4.3 Pivoting and Constructing an Algorithm         163
4.4  Norms and the Analysis of Errors              186
4.5 Neumann Series and Iterative Refinement        197
4.6 Solution of Equations by Iterative Methods     207
4.7 Steepest Descent and Conjugate Gradient Methods  232
4.8 Analysis of Roundoff Error in the Gaussian Algorithm  245
5  Selected Topics in Numerical Linear Algebra         254
5.0 Review of Basic Concepts                       254
5.1 Matrix Elgenvalue Problem: Power Method        257
5.2 Schur's and Gershgorin's Theorems              265
5.3 Orthogonal Factorizations and Least-Squares Problems  273
5.4 Singular-Value Decomposition and Pseudoinverses  287
5.5 QR-Algorithm of Francis for the Elgenvalue Problem  298
6  Approximating Functions                             308
6.0 Introduction                                   308
6.1 Polynomial Interpolation                       308
6.2 Divided Differences                            327
6.3 Hermlte Interpolation                          338
6.4 Spline Interpolation                           349
6.5 B-Splines: Basic Theory                        366
6.6 B-Splines: Applications                        377
6.7 Taylor Series                                  388
6.8 Best Approximation: Least-Squares Theory       392
6.9 Best Approximation: Chebyshev Theory           405
6.10 Interpolation in Higher Dimensions             420
6.11 Continued Fractions                            438
6.12 Trigonometric Interpolation                    445
6.13  Fast Fourier Transform                        451
6.14 Adaptive Approximation                         460
7  Numerical Differentiation and Integration           465
7.1 Numerical Differentiation and Richardson Extrapolation  465
7.2 Numerical Integration Based on Interpolation   478
7.3 Gaussian Quadrature                            492
7.4 Romberg Integration                            502
7.5 Adaptive Quadrature                            507
7.6 Sard's Theory of Approximating Functionals     513
77 Bernoulli Polynomials and the Euler-Maclaurin Formula  519
8 Numerical Solution of Ordinary
Differential Equations                              524
8.0 Introduction                                   524
8.1 The Existence and Uniqueness of Solutions      524
8.2 Taylor-Series Method                           530
8.3 Runge-Kutta Methods                            539
8.4  Multistep Methods                             549
8.5 Local and Global Errors: Stability             557
8.6 Systems and Higher-Order Ordinary Differential Equations  565
8.7 Boundary-Value Problems                        572
8.8 Boundary-Value Problems: Shooting Methods      581
8.9 Boundary-Value Problems: Finite-Differences    589
8.10  Boundary-Value Problems: Collocation          593
8.11 Linear Differential Equations                  597
8.12 Stiff Equations                                608
9  Numerical Solution of Partial Differential Equations 615
9.0 Introduction                                   615
9.1 Parabolic Equations: Explicit Methods          615
9.2 Parabolic Equations: Implicit Methods          623
9.3 Problems Without Time Dependence: Finite-Differences  629
9.4 Problems Without Time Dependence: Galerkin Methods  634
9.5 First-Order Partial Differential Equations: Characteristics  642
9.6 Quasillnear Second-Order Equations: Characteristics  650
9.7 Other Methods for Hyperbolic Problems          660
9.8 Multigrid Method                               667
9.9 Fast Methods for Poisson's Equation            676
10  Linear Programming and Related Topics               681
10.1 Convexity and Linear Inequalities              681
10.2 Linear Inequalities                            689
10.3  Unear Programming                               695
10.4 The Simplex Algorithm                            700
11  Optimization                                          711
11.0 Introduction                                    711
11.1  One-Variable Case                              712
11.2  Descent Methods                                 716
11.3  Analysis of Quadratic Objective Functions       719
11.4  Quadratic-Fitting Algorithms                   721
11.5  Nelder-Mead Algorithm                          722
11.6 Simulated Annealing                             723
11.7  Genetic Algorithms                              724
11.8 Convex Programming                              725
11.9 Constrained Minimization                        726
11.10  Pareto Optimization                            727
Appendix A    An Overview of Mathematical Software                  731
Bibliography                                          745



Library of Congress subject headings for this publication: Numerical analysis