Preface
Notation
1 Scientific Computing
1.1 Introduction
1.2 Approximations in Scientific Computation
1.3 Computer Arithmetic
1.4 Mathematical Software
1.5 Historical Notes and Further Reading
2 Systems of Linear Equations
2.1 Linear Systems
2.2 Existence and Uniqueness
2.3 Sensitivity and Conditioning
2.4 Solving Linear Systems
2.5 Special Types of Linear Systems
2.6 Iterative Methods for Linear Systems
2.7 Software for Linear Systems
2.8 Historical Notes and Further Reading
3 Linear Least Squares
3.1 Linear Least Squares Problems
3.2 Existence and Uniqueness
3.3 Sensitivity and Conditioning
3.4 Problem Transformations
3.5 Orthogonalization Methods
3.6 Singular Value Decomposition
3.7 Comparison of Methods
3.8 Software for Linear Least Squares
3.9 Historical Notes and Further Reading
4 Eigenvalue Problems
4.1 Eigenvalues and Eigenvectors
4.2 Existence and Uniqueness
4.3 Sensitivity and Conditioning
4.4 Problem Transformations
4.5 Computing Eigenvalues and Eigenvectors
4.6 Generalized Eigenvalue Problems
4.7 Computing the Singular Value Decomposition
4.8 Software for Eigenvalue Problems
4.9 Historical Notes and Further Reading
5 Nonlinear Equations
5.1 Nonlinear Equations
5.2 Existence and Uniqueness
5.3 Sensitivity and Conditioning
5.4 Convergence Rates and Stopping Criteria
5.5 Nonlinear Equations in One Dimension
5.6 Systems of Nonlinear Equations
5.7 Software for Nonlinear Equations
5.8 Historical Notes and Farther Reading
6 Optimization
6.1 Optimization Problems
6.2 Existence and Uniqueness
6.3 Sensitivity and Conditioning
6.4 Optimization in One Dimension
6.5 Unconstrained Optimization
6.6 Nonlinear Least Squares
6.7 Constrained Optimization
6.8 Software for Optimization
6.9 Historical Notes and Further Reading
7 Interpolation
7.1 Interpolation
7.2 Existence, Uniqueness, and Conditioning
7.3 Polynomial Interpolation
7.4 Piecewise Polynomial Interpolation
7.5 Software for Interpolation
7.6 Historical Notes and Further Reading
8 Numerical Integration and Differentiation
8.1 Integration
8.2 Existence, Uniqueness, and Conditioning
8.3 Numerical Quadrature
8.4 Other Integration Problems
8.5 Integral Equations
8.6 Numerical Differentiation
8.7 Richardson Extrapolation
8.8 Software for Integration and Differentiation
8.9 Historical Notes and Further Reading
9 Initial Value Problems for ODEs
9.1 Ordinary Differential Equations
9.2 Existence, Uniqueness, and Conditioning
9.3 Numerical Solution of ODES
9.4 Software for ODE Initial Value Problems
9.8 Historical Notes and Further Reading
10 Boundary Value Problems for ODEs
10.1 Boundary Value Problems
10.2 Existence, Uniqueness, and Conditioning
10.3 Shooting Method
10.4 Finite Difference Method
10.5 Collocation Method
10.6 Galerkin Method
10.7 Eigenvalue Problems
10.8 Software for ODE Boundary Value Problems
10.9 Historical Notes and Further Reading
11 Partial Differential Equations
11.1 Partial Differential Equations
11.2 Time-Dependent Problems
11.3 Time-Independent Problems
11.4 Direct Methods for Sparse Linear Systems
11.5 Iterative Methods for Linear Systems
11.6 Comparison of Methods
11.7 Software for Partial Differential Equations
11.8 Historical Notes and Further Reading
12 Fast Fourier Transform
12.1 Trigonometric Interpolation
12.2 FFT Algorithm
12.3 Applications of DFT
12.4 Wavelets
12.5 Software for FFT
12.6 Historical Notes and Further Reading
13 Random Numbers and Simulation
13.1 Stochastic Simulation
13.2 Randomness and Random Numbers
13.3 Random Number Generators
13.4 Quasi-Random Sequences
13.5 Software for Generating Random Numbers
13.6 Historical Notes and Further Reading
Bibliography
Index
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