1. Elements of the Laboratory
1.1 Getting Started
1.2 Scalars, Vectors and Matrices
1.3 Matrix Operations
1.4 Built-in Functions
1.5 Programming with MATLAB
1.6 Graphics and Data Files
1.7 Floating-Point Arithmetic
1.8 Error Analysis
1.9 Summary and Notes
1.10 Exercises
2. Linear Systems
2.1 Vector Spaces
2.2 Linear Maps
2.3 Systems and Linear Equations
2.4 Vector and Matrix Norms
2.5 Direct Methods
2.6 Iterative Methods
2.7 Cholesky Factorization
2.8 Determinants
2.9 Summary and Notes
2.10 Exercises
3. Orthogonality
3.1 Inner Product Spaces
3.2 Orthogonal Projections
3.3 QR Factorizations
3.4 The Least-Squares Method
3.5 Summary and Notes
3.6 Exercises
4. Eigenvalues and Eigenvectors
4.1 Matrix Eigenvalue Problems
4.2 Properties of Eigenvalues
4.3 Properties of Eigenvectors
4.4 Normal Matrices
4.5 Sensitivity of Eigenvalues
4.6 Power Iterations
4.7 Simultaneous Iterations
4.8 Singular Value Decomposition
4.9 Summary and Notes
4.10 Exercises
5. Polynomial Functions
5.1 Properties of Polynomials
5.2 Vandermonde Interpolation
5.3 Lagrange Interpolation
5.4 Newton Interpolation
5.5 Errors of Polynomial Interpolation
5.6 Polynomial Approximation
5.7 Approximation with Orthogonal Polynomials
5.8 Summary and Notes
5.9 Exercises
6. Differential and Integral Calculus
6.1 Derivatives and Finite Differences
6.2 Higher-Order Numerical Derivatives
6.3 Multi-Point First-Order Numerical Derivatives
6.4 Richardson Extrapolation
6.5 Integrals and Finite Sums
6.6 Newton-Cotes Integration Rules
6.7 Romberg Integration
6.8 Gaussian Quadrature Rules
6.9 Summary and Notes
6.10 Exercises
7. Vector Calculus
7.1 Scalar Functions of Several Variables
7.2 Partial Derivatives and Differentiability
7.3 The Gradient Vector
7.4 Paths
7.5 Vector Fields
7.6 Line Integrals
7.7 Surface Integrals
7.8 Integral Theorems
7.9 Summary and Notes
7.10 Exercises
8. Zeros and Extrema of Functions
8.1 One-dimensional root finding
8.2 Multidimensional root finding
8.3 One-dimensional minimization
8.4 Multidimensional Minimization
8.5 Summary and Notes
8.6 Exercises
9. Initial-Value Problems for ODEs
9.1 Approximations of Solutions
9.2 Single-Step Runge-Kutta Solvers
9.3 Adaptive Single-Step Solvers
9.4 Multi-step Adams Solvers
9.5 Implicit Methods for Stiff Differential Equations
9.6 Summary and Notes
9.7 Exercises
10. Boundary-Value Problems for ODEs and PDEs
10.1 Finite-Difference Methods for ODEs
10.2 Shooting Methods for ODEs
10.3 Finite-Difference Methods for Parabolic PDEs
10.4 Finite-Difference Methods for Hyperbolic PDEs
10.5 Finite-Difference Methods for Elliptic PDEs
10.6 Summary and Notes
10.7 Exercises
11. Spectral Methods
11.1 Trigonometric Approximation and Interpolation
11.2 Errors of Trigonometric Interpolation
11.3 Trigonometric Methods for Differential Equations
11.4 Summary and Notes
11.5 Exercises
12. Finite Element Methods
12.1 Spline Interpolation
12.2 Hermite Interpolation
12.3 Finite Elements for Differential Equations
12.4 Summary and Notes
12.5 Exercises
Subject Index
MATLAB Functions and Commands
Mathematical Symbols
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