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Tables of Contents for Introduction to Numerical Analysis
Chapter/Section Title
Page #
Page Count
Preface to the Third Edition
vii
Preface to the Second Edition
ix
Error Analysis
1
36
Representation of Numbers
2
2
Round off Errors and Floating-Point Arithmetic
4
5
Error Propagation
9
12
Examples
21
6
Interval Arithmetic; Statistical Round off Estimation
27
10
Exercises for Chapter 1
33
3
References for Chapter 1
36
1
Interpolation
37
108
Interpolation by Polynomials
38
21
Theoretical Foundation: The Interpolation Formula of Lagrange
38
2
Neville's Algorithm
40
3
Newtons Interpolation Formula: Divided Differences
43
5
The Error in Polynomial Interpolation
48
3
Hermite Interpolation
51
8
Interpolation by Rational Functions
59
15
General Properties of Rational Interpolation
59
5
Inverse and Reciprocal Differences. Thiele's Continued Fraction
64
4
Algorithms of the Neville Type
68
5
Comparing Rational and Polynomial Interpolation
73
1
Trigonometric Interpolation
74
23
Basic Facts
74
6
Fast Fourier Transforms
80
8
The Algorithms of Goertzel and Reinsch
88
4
The Calculation of Fourier Coefficients. Attenuation Factors
92
5
Interpolation by Spline Functions
97
48
Theoretical Foundations
97
4
Determining Interpolating Cubic Spline Functions
101
6
Convergence Properties of Cubic Spline Functions
107
4
B-Splines
111
6
The Computation of B-Splines
117
4
Multi-Resolution Methods and B-Splines
121
13
Exercises for Chapter 2
134
9
References for Chapter 2
143
2
Topics in Integration
145
45
The Integration Formulas of Newton and Cotes
146
5
Peano's Error Representation
151
5
The Euler-Maclaurin Summation Formula
156
4
Integration by Extrapolation
160
5
About Extrapolation Methods
165
6
Gaussian Integration Methods
171
10
Integrals with Singularities
181
9
Exercises for Chapter 3
184
4
References for Chapter 3
188
2
Systems of Linear Equations
190
99
Gaussian Elimination. The Triangular Decomposition of a Matrix
190
10
The Gauss-Jordan Algorithm
200
4
The Choleski Decompostion
204
3
Error Bounds
207
8
Round off Error Analysis for Gaussian Elimination
215
6
Round off Errors in Solving Triangular Systems
221
2
Orthogonalization Techniques of Householder and Gram-Schmidt
223
8
Data Fitting
231
16
Linear Least Squares. The Normal Equations
232
3
The Use of Orthogonalization in Solving Linear Least-Squares Problems
235
1
The Condition of the Linear Least-Squares Problem
236
5
Nonlinear Least-Squares Problems
241
2
The Pseudoinverse of a Matrix
243
4
Modification Techniques for Matrix Decompositions
247
9
The Simplex Method
256
12
Phase One of the Simplex Method
268
4
Appendix: Elimination Methods for Sparse Matrices
272
17
Exercises for Chapter 4
280
6
References for Chapter 4
286
3
Finding Zeros and Minimum Points by Iterative Methods
289
75
The Development of Iterative Methods
290
3
General Convergence Theorems
293
5
The Convergence of Newton's Method in Several Variables
298
4
A Modified Newton Method
302
14
On the Convergence of Minimization Methods
303
5
Application of the Convergence Criteria to the Modified Newton Method
308
5
Suggestions for a Practical Implementation of the Modified Newton Method. A Rank-One Method Due to Broyden
313
3
Roots of Polynomials. Application of Newton's Method
316
12
Sturm Sequences and Bisection Methods
328
5
Bairstow's Method
333
2
The Sensitivity of Polynomial Roots
335
3
Interpolation Methods for Determining Roots
338
6
The Δ2-Method of Aitken
344
5
Minimization Problems without Constraints
349
15
Exercises for Chapter 5
358
3
References for Chapter 5
361
3
Eigenvalue Problems
364
101
Introduction
364
2
Basic Facts on Eigenvalues
366
3
The Jordan Normal Form of a Matrix
369
6
The Fiobenius Normal Form of a Matrix
375
4
The Schur Normal Form of a Matrix; Hermitian and Normal Matrices; Singular Values of Matrixes
379
7
Reduction of Matrices to Simpler Form
386
19
Reduction of a Hermitian Matrix to Tridiagonal Form: The Method of Householder
388
6
Reduction of a Hermitian Matrix to Tridiagonal or Diagonal Form: The Methods of Givens and Jacobi
394
4
Reduction of a Hermitian Matrix to Tridiagonal Form: The Method of Lanczos
398
4
Reduction to Hessenberg Form
402
3
Methods for Determining the Eigenvalues and Eigenvectors
405
31
Computation of the Eigenvalues of a Hermitian Tridiagonal Matrix
405
2
Computation of the Eigenvalues of a Hessenberg Matrix. The Method of Hyman
407
1
Simple Vector Iteration and Inverse Iteration of Wielandt
408
7
The LR and QR Methods
415
10
The Practical Implementation of the QR Method
425
11
Computation of the Singular Values of a Matrix
436
4
Generalized Eigenvalue Problems
440
1
Estimation of Eigenvalues
441
24
Exercises for Chapter 6
455
7
References for Chapter 6
462
3
Ordinary Differential Equations
465
154
Introduction
465
2
Some Theorems from tile Theory of Ordinary Differential Equations
467
4
Initial-Value Problems
471
68
One-Step Methods: Basic Concepts
471
6
Convergence of One-Step Methods
477
3
Asymptotic Expansions for the Global Discretization Error of One-Step Methods
480
3
The Influence of Rounding Errors in One-Step Methods
483
2
Practical Implementation of One-Step Methods
485
7
Multistep Methods: Examples
492
3
General Multistep Methods
495
3
An Example of Divergence
498
3
Linear Difference Equations
501
3
Convergence of Multistep Methods
504
4
Linear Multistep Methods
508
5
Asymptotic Expansions of the Global Discretization Error for Linear Multistep Methods
513
4
Practical Implementation of Multistep Methods
517
4
Extrapolation Methods for the Solution of the Initial-Value Problem
521
3
Comparison of Methods for Solving Initial-Value Problems
524
1
Stiff Differential Equations
525
6
Implicit Differential Equations. Differential-Algebraic Equations
531
5
Handling Discontinuities in Differential Equations
536
2
Sensitivity Analysis of Initial-Value Problems
538
1
Boundary-Value Problems
539
43
Introduction
539
3
The Simple Shooting Method
542
6
The Simple Shooting Method for Linear Boundary-Value Problems
548
2
An Existence and Uniqueness Theorem for the Solution of Boundary-Value Problems
550
2
Difficulties in the Execution of the Simple Shooting Method
552
5
The Multiple Shooting Method
557
4
Hints for the Practical Implementation of the Multiple Shooting Method
561
4
An Example: Optimal Control Program for a Lifting Reentry Space Vehicle
565
7
Advanced Techniques in Multiple Shooting
572
5
The Limiting Case m →∞ of the Multiple Shooting Method (General Newton's Method, Quasilinearization)
577
5
Difference Methods
582
4
Variational Methods
586
10
Comparison of the Methods for Solving Boundary-Value Problems for Ordinary Differential Equations
596
4
Variational Methods for Partial Differential Equations The Finite-Element Method
600
19
Exercises for Chapter 7
607
6
References for Chapter 7
613
6
Iterative Methods for the Solution of Large Systems of Linear Equations. Additional Methods
619
111
Introduction
619
2
General Procedures for the Construction of Iterative Methods
621
2
Convergence Theorems
623
6
Relaxation Methods
629
10
Applications to Difference Methods--An Example
639
6
Block Iterative Methods
645
2
The ADI-Method of Peaceman and Rachford
647
10
Krylov Space Methods for Solving Linear Equations
657
34
The Conjugate-Gradient Method of Hestenes and Stiefel
658
9
The GMRES Algorithm
667
13
The Biorthogonalization Method of Lanczos and the QMR algorithm
680
6
The Bi-CG and BI-CGSTAB Algorithms
686
5
Buneman's Algorithm and Fourier Methods for Solving the Discretized Poisson Equation
691
11
Multigrid Methods
702
10
Comparison of Iterative Methods
712
18
Exercises for Chapter 8
719
8
References for Chapter 8
727
3
General Literature on Numerical Methods
730
2
Index
732