APM2613
Numerical Methods I

Numerical Analysis

(10th edition)

by Richard L. Burden, J. Douglas Faires, Annette M. Burden

1. Mathematical Preliminaries and Error Analysis
1.1. Review of Calculus
1.2. Round-off Errors and Computer Arithmetic
1.3. Algorithms and Convergence
1.4. Numerical Software
2. Solutions of Equations in One Variable
2.1. The Bisection Method
2.2. Fixed-Point Iteration
2.3. Newton's Method and Its Extensions
2.4. Error Analysis for Iterative Methods
2.5. Accelerating Convergence
2.6. Zeros of Polynomials and Müller's Method
2.7. Numerical Software and Chapter Review
3. Interpolation and Polynomial Approximation
3.1. Interpolation and the Lagrange Polynomial
3.2. Data Approximation and Neville's Method
3.3. Divided Differences
3.4. Hermite Interpolation
3.5. Cubic Spline Interpolation
3.6. Parametric Curves
3.7. Numerical Software and Chapter Review
4. Numerical Differentiation and Integration
4.1. Numerical Differentiation
4.2. Richardson's Extrapolation
4.3. Elements of Numerical Integration
4.4. Composite Numerical Integration
4.5. Romberg Integration
4.6. Adaptive Quadrature Methods
4.7. Gaussian Quadrature
4.8. Multiple Integrals
4.9. Improper Integrals
4.10. Numerical Software and Chapter Review
5. Initial-Value Problems for Ordinary Differential Equations
5.1. The Elementary Theory of Initial-Value Problems
5.2. Euler's Method
5.3. Higher-Order Taylor Methods
5.4. Runge-Kutta Methods
5.5. Error Control and the Runge-Kutta-Fehlberg Method
5.6. Multistep Methods
5.7. Variable Step-Size Multistep Methods
5.8. Extrapolation Methods
5.9. Higher-Order Equations and Systems of Differential Equations
5.10. Stability
5.11. Stiff Differential Equations
5.12. Numerical Software
6. Direct Methods for Solving Linear Systems
6.1. Linear Systems of Equations
6.2. Pivoting Strategies
6.3. Linear Algebra and Matrix Inversion
6.4. The Determinant of a Matrix
6.5. Matrix Factorization
6.6. Special Types of Matrices
6.7. Numerical Software
7. Iterative Techniques in Matrix Algebra
7.1. Norms of Vectors and Matrices
7.2. Eigenvalues and Eigenvectors
7.3. The Jacobi and Gauss-Siedel Iterative Techniques
7.4. Relaxation Techniques for Solving Linear Systems
7.5. Error Bounds and Iterative Refinement
7.6. The Conjugate Gradient Method
7.7. Numerical Software
8. Approximation Theory
8.1. Discrete Least Squares Approximation
8.2. Orthogonal Polynomials and Least Squares Approximation
8.3. Chebyshev Polynomials and Economization of Power Series
8.4. Rational Function Approximation
8.5. Trigonometric Polynomial Approximation
8.6. Fast Fourier Transforms
8.7. Numerical Software
9. Approximating Eigenvalues
9.1. Linear Algebra and Eigenvalues
9.2. Orthogonal Matrices and Similarity Transformations
9.3. The Power Method
9.4. Householder's Method
9.5. The QR Algorithm
9.6. Singular Value Decomposition
9.7. Numerical Software
10. Numerical Solutions of Nonlinear Systems of Equations
10.1. Fixed Points for Functions of Several Variables
10.2. Newton's Method
10.3. Quasi-Newton Methods
10.4. Steepest Descent Techniques
10.5. Homotopy and Continuation Methods
10.6. Numerical Software
11. Boundary-Value Problems for Ordinary Differential Equations
11.1. The Linear Shooting Method
11.2. The Shooting Method for Nonlinear Problems
11.3. Finite-Difference Methods for Linear Problems
11.4. Finite-Difference Methods for Nonlinear Problems
11.5. The Rayleigh-Ritz Method
11.6. Numerical Software
12. Numerical Solutions to Partial Differential Equations
12.1. Elliptic Partial Differential Equations
12.2. Parabolic Partial Differential Equations
12.3. Hyperbolic Partial Differential Equations
12.4. An Introduction to the Finite-Element Method
12.5. Numerical Software