Numerical Analysis: Mathematics of Scientific Computing

Cover......Page 1
Title......Page 2
Copyright......Page 3
Contents......Page 6
Preface......Page 10
Numerical Analysis: What Is It?......Page 16
1.1 Basic Concepts and Taylor's Theorem......Page 18
1.2 Orders of Convergence and Additional Basic Concepts......Page 30
1.3 Difference Equations......Page 43
2.1 Floating-Point Numbers and Roundoff Errors......Page 52
2.2 Absolute and Relative Errors: Loss of Significance......Page 70
2.3 Stable and Unstable Computations: Conditioning......Page 79
3.0 Introduction......Page 88
3.1 Bisection (Interval Halving) Method......Page 89
3.2 Newton's Method......Page 96
3.3 Secant Method......Page 108
3.4 Fixed Points and Functional Iteration......Page 115
3.5 Computing Roots of Polynomials......Page 124
3.6 Homotopy and Continuation Methods......Page 145
4.0 Introduction......Page 154
4.1 Matrix Algebra......Page 155
4.2 LU and Cholesky Factorizations......Page 164
4.3 Pivoting and Constructing an Algorithm......Page 178
4.4 Norms and the Analysis of Errors......Page 201
4.5 Neumann Series and Iterative Refinement......Page 212
4.6 Solution of Equations by Iterative Methods......Page 222
4.7 Steepest Descent and Conjugate Gradient Methods......Page 247
4.8 Analysis of Roundoff Error in the Gaussian Algorithm......Page 260
5.0 Review of Basic Concepts......Page 269
5.1 Martix Eigenvalue Problem: Power Method......Page 272
5.2 Schur's and Gershgorin's Theorems......Page 280
5.3 Orthogonal Factorizations and Least-Squares Problems......Page 288
5.4 Singular-Value Decomposition and Pseudo inverses......Page 302
5.5 QR-Algorithm of Francis for the Eigenvalue Problem......Page 313
6.1 Polynomial Interpolation......Page 323
6.2 Divided Differences......Page 342
6.3 Hermite Interpolation......Page 353
6.4 Spline Interpolation......Page 364
6.5 B-Splines: Basic Theory......Page 381
6.6 B-Splines: Applications......Page 392
6.7 Taylor Series......Page 403
6.8 Best Approximation: Least-Squares Theory......Page 407
6.9 Best Approximation: Chebyshev Theory......Page 420
6.10 Interpolation in Higher Dimensions......Page 435
6.11 Continued Fractions......Page 453
6.12 Trigonometric Interpolation......Page 460
6.13 Fast Fourier Transform......Page 466
6.14 Adaptive Approximation......Page 475
7.1 Numerical Differentiation and Richardson Extrapolation......Page 480
7.2 Numerical Integration Based on Interpolation......Page 493
7.3 Gaussian Quadrature......Page 507
7.4 Romberg Integration......Page 517
7.5 Adaptive Quadrature......Page 522
7.6 Sard's Theory of Approximating Functionals......Page 528
7.7 Bernoulli Polynomials and the Euler-Maclaurin Formula......Page 534
8.1 The Existence and Uniqueness of Solutions......Page 539
8.2 Taylor-Series Method......Page 545
8.3 Runge-Kutta Methods......Page 554
8.4 Multistep Methods......Page 564
8.5 Local and Global Errors: Stability......Page 572
8.6 Systems and Higher-Order Ordinary Differential Equations......Page 580
8.7 Boundary-Value Problems......Page 587
8.8 Boundary-Value Problems: Shooting Methods......Page 596
8.9 Boundary-Value Problems: Finite-Differences......Page 604
8.10 Boundary-Value Problems: Collocation......Page 608
8.11 Linear Differential Equations......Page 612
8.12 Stiff Equations......Page 623
9.1 Parabolic Equations: Explicit Methods......Page 630
9.2 Parabolic Equations: Implicit Methods......Page 638
9.3 Problems Without Time Dependence: Finite-Differences......Page 644
9.4 Problems Without Time Dependence: Galerkin Methods......Page 649
9.5 First-Order Partial Differential Equations: Characteristics......Page 657
9.6 Quasilinear Second-Order Equations: Characteristics......Page 665
9.7 Other Methods for Hyperbolic Problems......Page 675
9.8 Multigrid Method......Page 682
9.9 Fast Method s for Poisson's Equation......Page 691
10.1 Convexity and Linear Inequalities......Page 696
10.2 Linear Inequalities......Page 704
10.3 Linear Programming......Page 710
10.4 The Simplex Algorithm......Page 715
11.0 Introduction......Page 726
11.1 One-Variable Case......Page 727
11.2 Descent Methods......Page 731
11.3 Analysis of Quadratic Objective Functions......Page 734
11.4 Quadratic-Fitting Algorithms......Page 736
11.5 Nelder-Mead Algorithm......Page 737
11.6 Simulated Annealing......Page 738
11.7 Genetic Algorithms......Page 739
11.8 Convex Programming......Page 740
11.9 Constrained Minimization......Page 741
11.10 Pareto Optimization......Page 742
Appendix A: An Overview of Mathematical Software......Page 746
Bibliography......Page 760
B......Page 786
C......Page 787
D......Page 788
E......Page 789
G......Page 790
I......Page 791
L......Page 792
N......Page 794
O......Page 795
P......Page 796
R......Page 797
S......Page 798
T......Page 800
U......Page 802
Z......Page 803
Back Cover......Page 804