Numerical Analysis

Timothy Sauer earned his Ph.D. in mathematics at the University of California–Berkeley in 1982, and is currently a professor at George Mason University. He has published articles on a wide range of topics in applied mathematics, including dynamical systems, computational mathematics, and mathematical biology.

• CHAPTER 0      Fundamentals 0.1 Evaluating a Polynomial 0.2 Binary Numbers 0.2.1 Decimal to binary 0.2.2 Binary to decimal 0.3 Floating Point Representation of Real Numbers 0.3.1 Floating point formats 0.3.2 Machine representation 0.3.3 Addition of floating point numbers 0.4 Loss of Significance 0.5 Review of Calculus Software and Further Reading  CHAPTER 1      Solving Equations 1.1 The Bisection Method 1.1.1 Bracketing a root 1.1.2 How accurate and how fast? 1.2 Fixed-Point Iteration 1.2.1 Fixed points of a function 1.2.2 Geometry of Fixed-Point Iteration 1.2.3 Linear convergence of Fixed-Point Iteration 1.2.4 Stopping criteria 1.3 Limits of Accuracy 1.3.1 Forward and backward error 1.3.2 The Wilkinson polynomial 1.3.3 Sensitivity of root-finding 1.4 Newton’s Method 1.4.1 Quadratic convergence of Newton’s Method 1.4.2 Linear convergence of Newton’s Method 1.5 Root-Finding without Derivatives 1.5.1 Secant Method and variants 1.5.2 Brent’s Method Reality Check 1: Kinematics of the Stewart platform Software and Further Reading   CHAPTER 2      Systems of Equations 2.1 Gaussian Elimination 2.1.1 Naive Gaussian elimination 2.1.2 Operation counts 2.2 The LU Factorization 2.2.1 Matrix form of Gaussian elimination 2.2.2 Back substitution with the LU factorization 2.2.3 Complexity of the LU factorization 2.3 Sources of Error 2.3.1 Error magnification and condition number 2.3.2 Swamping 2.4 The PA = LU Factorization 2.4.1 Partial pivoting 2.4.2 Permutation matrices 2.4.3 PA = LU factorization Reality Check 2: The Euler–Bernoulli Beam 2.5 Iterative Methods 2.5.1 Jacobi Method 2.5.2 Gauss–Seidel Method and SOR 2.5.3 Convergence of iterative methods 2.5.4 Sparse matrix computations 2.6 Methods for symmetric positive-definite matrices 2.6.1 Symmetric positive-definite matrices 2.6.2 Cholesky factorization 2.6.3 Conjugate Gradient Method 2.6.4 Preconditioning 2.7 Nonlinear Systems of Equations 2.7.1 Multivariate Newton’s Method 2.7.2 Broyden’s Method Software and Further Reading  CHAPTER 3      Interpolation 3.1 Data and Interpolating Functions 3.1.1 Lagrange interpolation 3.1.2 Newton’s divided differences 3.1.3 How many degree d polynomials pass through n points? 3.1.4 Code for interpolation 3.1.5 Representing functions by approximating polynomials 3.2 Interpolation Error 3.2.1 Interpolation error formula 3.2.2 Proof of Newton form and error formula 3.2.3 Runge phenomenon 3.3 Chebyshev Interpolation 3.3.1 Chebyshev’s theorem 3.3.2 Chebyshev polynomials 3.3.3 Change of interval 3.4 Cubic Splines 3.4.1 Properties of splines 3.4.2 Endpoint conditions 3.5 Bézier Curves Reality Check 3: Fonts from Bézier curves Software and Further Reading CHAPTER 4      Least Squares 4.1 Least Squares and the Normal Equations 4.1.1 Inconsistent systems of equations 4.1.2 Fitting models to data 4.1.3 Conditioning of least squares 4.2 A Survey of Models 4.2.1 Periodic data 4.2.2 Data linearization 4.3 QR Factorization 4.3.1 Gram–Schmidt orthogonalization and least squares 4.3.2 Modified Gram–Schmidt orthogonalization 4.3.3 Householder reflectors 4.4 Generalized Minimum Residual (GMRES) Method 4.4.1 Krylov methods 4.4.2 Preconditioned GMRES 4.5 Nonlinear Least Squares 4.5.1 Gauss–Newton Method 4.5.2 Models with nonlinear parameters 4.5.3 The Levenberg–Marquardt Method Reality Check 4: GPS, Conditioning, and Nonlinear Least Squares Software and Further Reading CHAPTER 5      Numerical Differentiation and Integration 5.1 Numerical Differentiation 5.1.1 Finite difference formulas 5.1.2 Rounding error 5.1.3 Extrapolation 5.1.4 Symbolic differentiation and integration 5.2 Newton–Cotes Formulas for Numerical Integration 5.2.1 Trapezoid Rule 5.2.2 Simpson’s Rule 5.2.3 Composite Newton–Cotes formulas 5.2.4 Open Newton–Cotes Methods 5.3 Romberg Integration 5.4 Adaptive Quadrature 5.5 Gaussian Quadrature Reality Check 5: Motion Control in Computer-Aided Modeling Software and Further Reading CHAPTER 6      Ordinary Differential Equations 6.1 Initial Value Problems 6.1.1 Euler’s Method 6.1.2 Existence, uniqueness, and continuity for solutions 6.1.3 First-order linear equations 6.2 Analysis of IVP Solvers 6.2.1 Local and global truncation error 6.2.2 The explicit Trapezoid Method 6.2.3 Taylor Methods 6.3 Systems of Ordinary Differential Equations 6.3.1 Higher order equations 6.3.2 Computer simulation: the pendulum 6.3.3 Computer simulation: orbital mechanics 6.4 Runge–Kutta Methods and Applications 6.4.1 The Runge–Kutta family 6.4.2 Computer simulation: the Hodgkin–Huxley neuron 6.4.3 Computer simulation: the Lorenz equations Reality Check 6: The Tacoma Narrows Bridge 6.5 Variable Step-Size Methods 6.5.1 Embedded Runge–Kutta pairs 6.5.2 Order 4/5 methods 6.6 Implicit Methods and Stiff Equations 6.7 Multistep Methods 6.7.1 Generating multistep methods 6.7.2 Explicit multistep methods 6.7.3 Implicit multistep methods Software and Further Reading  CHAPTER 7      Boundary Value Problems 7.1 Shooting Method 7.1.1 Solutions of boundary value problems 7.1.2 Shooting Method implementation Reality Check 7: Buckling of a Circular Ring 7.2 Finite Difference Methods 7.2.1 Linear boundary value problems 7.2.2 Nonlinear boundary value problems 7.3 Collocation and the Finite Element Method 7.3.1 Collocation 7.3.2 Finite elements and the Galerkin Method Software and Further Reading  CHAPTER 8      Partial Differential Equations 8.1 Parabolic Equations 8.1.1 Forward Difference Method 8.1.2 Stability analysis of Forward Difference Method 8.1.3 Backward Difference Method 8.1.4 Crank–Nicolson Method 8.2 Hyperbolic Equations 8.2.1 The wave equation 8.2.2 The CFL condition 8.3 Elliptic Equations 8.3.1 Finite Difference Method for elliptic equations Reality Check 8: Heat distribution on a cooling fin 8.3.2 Finite Element Method for elliptic equations 8.4 Nonlinear partial differential equations 8.4.1 Implicit Newton solver 8.4.2 Nonlinear equations in two space dimensions Software and Further Reading  CHAPTER 9      Random Numbers and Applications 9.1 Random Numbers 9.1.1 Pseudo-random numbers 9.1.2 Exponential and normal random numbers 9.2 Monte Carlo Simulation 9.2.1 Power laws for Monte Carlo estimation 9.2.2 Quasi-random numbers 9.3 Discrete and Continuous Brownian Motion 9.3.1 Random walks 9.3.2 Continuous Brownian motion 9.4 Stochastic Differential Equations 9.4.1 Adding noise to differential equations 9.4.2 Numerical methods for SDEs Reality Check 9: The Black–Scholes Formula Software and Further Reading CHAPTER 10      Trigonometric Interpolation and the FFT 10.1 The Fourier Transform 10.1.1 Complex arithmetic 10.1.2 Discrete Fourier Transform 10.1.3 The Fast Fourier Transform 10.2 Trigonometric Interpolation 10.2.1 The DFT Interpolation Theorem 10.2.2 Efficient evaluation of trigonometric functions 10.3 The FFT and Signal Processing 10.3.1 Orthogonality and interpolation 10.3.2 Least squares fitting with trigonometric functions 10.3.3 Sound, noise, and filtering Reality Check 10: The Wiener Filter Software and Further Reading  CHAPTER 11      Compression 11.1 The Discrete Cosine Transform 11.1.1 One-dimensional DCT 11.1.2 The DCT and least squares approximation 11.2 Two-Dimensional DCT and Image Compression 11.2.1 Two-dimensional DCT 11.2.2 Image compression 11.2.3 Quantization 11.3 Huffman Coding 11.3.1 Information theory and coding 11.3.2 Huffman coding for the JPEG format 11.4 Modified DCT and Audio Compression 11.4.1 Modified Discrete Cosine Transform 11.4.2 Bit quantization Reality Check 11: A Simple Audio Codec Software and Further Reading CHAPTER 12      Eigenvalues and Singular Values 12.1 Power Iteration Methods 12.1.1 Power Iteration 12.1.2 Convergence of Power Iteration 12.1.3 Inverse Power Iteration 12.1.4 Rayleigh Quotient Iteration 12.2 QR Algorithm 12.2.1 Simultaneous iteration 12.2.2 Real Schur form and the QR algorithm 12.2.3 Upper Hessenberg form Reality Check 12: How Search Engines Rate Page Quality 12.3 Singular Value Decomposition 12.3.1 Finding the SVD in general 12.3.2 Special case: symmetric matrices 12.4 Applications of the SVD 12.4.1 Properties of the SVD 12.4.2 Dimension reduction 12.4.3 Compression 12.4.4 Calculating the SVD Software and Further Reading  CHAPTER 13      Optimization 13.1 Unconstrained Optimization without Derivatives 13.1.1 Golden Section Search 13.1.2 Successive parabolic interpolation 13.1.3 Nelder–Mead search 13.2 Unconstrained Optimization with Derivatives 13.2.1 Newton’s Method 13.2.2 Steepest Descent 13.2.3 Conjugate Gradient Search Reality Check 13: Molecular Conformation and Numerical Optimization Software and Further Reading Appendix A A.1 Matrix Fundamentals A.2 Systems of linear equations A.3 Block Multiplication A.4 Eigenvalues and Eigenvectors A.5 Symmetric Matrices A.6 Vector Calculus   Appendix B B.1 Starting MATLAB B.2 Graphics B.3 Programming in MATLAB B.4 Flow Control B.5 Functions B.6 Matrix Operations B.7 Animation and Movies  ANSWERS TO SELECTED EXERCISES BIBLIOGRAPHY INDEX