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Tables of Contents for Numerical Analysis in Modern Scientific Computing
Chapter/Section Title
Page #
Page Count
Preface
vii
Outline
xi
Linear Systems
1
20
Solution of Triangular Systems
3
1
Gaussian Elimination
4
3
Pivoting Strategies and Iterative Refinement
7
7
Cholesky Decomposition for Symmetric Positive Definite Matrices
14
7
Exercises
16
5
Error Analysis
21
36
Sources of Errors
22
2
Condition of Problems
24
10
Normwise Condition Analysis
26
5
Componentwise Condition Analysis
31
3
Stability of Algorithms
34
10
Stability Concepts
35
2
Forward Analysis
37
5
Backward Analysis
42
2
Application to Linear Systems
44
13
A Zoom into Solvability
44
2
Backward Analysis of Gaussian Elimination
46
3
Assessment of Approximate Solutions
49
3
Exercises
52
5
Linear Least-Squares Problems
57
24
Least-Squares Method of Gauss
57
9
Formulation of the Problem
57
3
Normal Equations
60
2
Condition
62
3
Solution of Normal Equations
65
1
Orthogonalization Methods
66
8
Givens Rotations
68
2
Householder Reflections
70
4
Generalized Inverses
74
7
Exercises
78
3
Nonlinear Systems and Least-Squares Problems
81
38
Fixed-Point Iterations
81
5
Newton Methods for Nonlinear Systems
86
6
Gauss-Newton Method for Nonlinear Least-Squares Problems
92
7
Nonlinear Systems Depending on Parameters
99
20
Solution Structure
100
2
Continuation Methods
102
11
Exercises
113
6
Linear Eigenvalue Problems
119
32
Condition of General Eigenvalue Problems
120
3
Power Method
123
3
QR-Algorithm for Symmetric Eigenvalue Problems
126
6
Singular Value Decomposition
132
5
Stochastic Eigenvalue Problems
137
14
Exercises
148
3
Three-Term Recurrence Relations
151
28
Theoretical Background
153
5
Orthogonality and Three-Term Recurrence Relations
153
3
Homogeneous and Inhomogeneous Recurrence Relations
156
2
Numerical Aspects
158
10
Condition Number
160
6
Idea of the Miller Algorithm
166
2
Adjoint Summation
168
11
Summation of Dominant Solutions
169
3
Summation of Minimal Solutions
172
4
Exercises
176
3
Interpolation and Approximation
179
58
Classical Polynomial Interpolation
180
17
Uniqueness and Condition Number
180
4
Hermite Interpolation and Divided Differences
184
8
Approximation Error
192
1
Min-Max Property of Chebyshev Polynomials
193
4
Trigonometric Interpolation
197
7
Bezier Techniques
204
14
Bernstein Polynomials and Bezier Representation
205
6
De Casteljau Algorithm
211
7
Splines
218
19
Spline Spaces and B-Splines
219
7
Spline Interpolation
226
4
Computation of Cubic Splines
230
3
Exercises
233
4
Large Symmetric Systems of Equations and Eigenvalue Problems
237
32
Classical Iteration Methods
239
5
Chebyshev Acceleration
244
5
Method of Conjugate Gradients
249
7
Preconditioning
256
5
Lanczos Methods
261
8
Exercises
266
3
Definite Integrals
269
56
Quadrature Formulas
270
3
Newton-Cotes Formulas
273
6
Gauss-Christoffel Quadrature
279
8
Construction of the Quadrature Formula
280
5
Computation of Nodes and Weights
285
2
Classical Romberg Quadrature
287
11
Asymptotic Expansion of the Trapezoidal Sum
288
2
Idea of Extrapolation
290
5
Details of the Algorithm
295
3
Adaptive Romberg Quadrature
298
12
Principle of Adaptivity
299
2
Estimation of the Approximation Error
301
3
Derivation of the Algorithm
304
6
Hard Integration Problems
310
3
Adaptive Multigrid Quadrature
313
12
Local Error Estimation and Refinement Rules
314
4
Global Error Estimation and Details of the Algorithm
318
3
Exercises
321
4
References
325
6
Software
331
2
Index
333