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Numerical Analysis, Second Edition, is a modern and readable text. This book covers not only the standard topics but also some more advanced numerical methods being used by computational scientists and engineers-topics such as compression, forward and backward error analysis, and iterative methods of solving equations-all while maintaining a level of discussion appropriate for undergraduates. Each chapter contains a Reality Check, which is an extended exploration of relevant application areas that can launch individual or team projects. MATLAB (R) is used throughout to demonstrate and implement numerical methods. The Second Edition features many noteworthy improvements based on feedback from users, such as new coverage of Cholesky factorization, GMRES methods, and nonlinear PDEs.
Contents
Preface0. Fundamentals0.1 Evaluating a polynomial0.2 Binary numbers0.2.1 Decimal to binary0.2.2 Binary to decimal0.3 Floating point representation of real numbers0.3.1 Floating point formats0.3.2 Machine representation0.3.3 Addition of floating point numbers0.4 Loss of significance0.5 Review of calculus0.6 Software and Further Reading1. Solving Equations1.1 The Bisection Method1.1.1 Bracketing a root1.1.2 How accurate and how fast?1.2 Fixed point iteration1.2.1 Fixed points of a function1.2.2 Geometry of Fixed Point Iteration1.2.3 Linear Convergence of Fixed Point Iteration1.2.4 Stopping criteria1.3 Limits of accuracy1.3.1 Forward and backward error1.3.2 The Wilkinson polynomial1.3.3 Sensitivity and error magnification1.4 Newton's Method1.4.1 Quadratic convergence of Newton's method1.4.2 Linear convergence of Newton's method1.5 Root-finding without derivatives1.5.1 Secant method and variants1.5.2 Brent's MethodREALITY CHECK 1: Kinematics of the Stewart platform1.6 Software and Further Reading2. Systems of Equations2.1 Gaussian elimination2.1.1 Naive Gaussian elimination2.1.2 Operation counts2.2 The LU factorization2.2.1 Backsolving with the LU factorization2.2.2 Complexity of the LU factorization2.3 Sources of error2.3.1 Error magnification and condition number2.3.2 Swamping2.4 The PA=LU factorization2.4.1 Partial pivoting2.4.2 Permutation matrices2.4.3 PA = LU factorizationREALITY CHECK 2: The Euler-Bernoulli Beam2.5 Iterative methods2.5.1 Jacobi Method2.5.2 Gauss-Seidel Method and SOR2.5.3 Convergence of iterative methods2.5.4 Sparse matrix computations2.6 Methods for symmetric positive-definite matrices2.6.1 Symmetric positive-definite matrices2.6.2 Cholesky factorization2.6.3 Conjugate Gradient Method2.6.4 Preconditioning2.7 Nonlinear systems of equations2.7.1 Multivariate Newton's method2.7.2 Broyden's method2.8 Software and Further Reading3. Interpolation3.1 Data and interpolating functions3.1.1 Lagrange interpolation3.1.2 Newton's divided differences3.1.3 How many degree d polynomials pass through n points?3.1.4 Code for interpolation3.1.5 Representing functions by approximating polynomials3.2 Interpolation error3.2.1 Interpolation error formula3.2.2 Proof of Newton form and error formula3.2.3 Runge phenomenon3.3 Chebyshev interpolation3.3.1 Chebyshev's Theorem3.3.2 Chebyshev polynomials3.3.3 Change of interval3.4 Cubic splines3.4.1 Properties of splines3.4.2 Endpoint conditions3.5 Bezier curvesREALITY CHECK 3: Constructing fonts from Bezier splines3.6 Software and Further Reading4. Least Squares4.1 Least squares and the normal equations4.1.1 Inconsistent systems of equations4.1.2 Fitting models to data4.2 Linear and nonlinear models4.1.3 Conditioning of least squares4.2 A survey of models4.2.1 Periodic data4.2.2 Data linearization4.3 QR factorization4.3.1 Gram-Schmidt orthogonalization and least squares4.3.2 Modified Gram-Schmidt orthogonalization4.3.3 Householder reflectors4.4 Generalized Minimum Residual (GMRES) Method4.4.1 Krylov methods4.4.2 Preconditioned GMRES4.5 Nonlinear least squares4.5.1 Gauss-Newton method4.5.2 Models with nonlinear parameters4.5.3 Levenberg-Marquardt methodREALITY CHECK 4: GPS, conditioning and nonlinear least squares4.6 Software and Further Reading5. Numerical Differentiation and Integration5.1 Numerical differentiation5.1.1 Finite difference formulas5.1.2 Rounding error5.1.3 Extrapolation5.1.4 Symbolic differentiation and integration5.2 Newton-Cotes formulas for numerical integration5.2.1 Trapezoid rule5.2.2 Simpson's Rule5.2.3 Composite Newton-Cotes Formulas5.2.4 Open Newton-Cotes methods5.3 Romberg integration5.4 Adaptive quadrature5.5 Gaussian quadratureREALITY CHECK 5: Motion control in computer-aided modelling5.6 Software and Further Reading6. Ordinary Differential Equations6.1 Initial value problems6.1.1 Euler's method6.1.2 Existence, uniqueness, and continuity for solutions6.1.3 First-order linear equations6.2 Analysis of IVP solvers6.2.1 Local and global truncation error6.2.2 The explicit trapezoid method6.2.3 Taylor methods6.3 Systems of ordinary differential equations6.3.1 Higher order equations6.3.2 Computer simulation: The pendulum6.3.3 Computer simulation: Orbital mechanics6.4 Runge-Kutta methods and applications6.4.1 The Runge-Kutta family6.4.2 Computer simulation: The Hodgkin-Huxley neuron6.4.3 Computer simulation: The Lorenz equationsREALITY CHECK 6: The Tacoma Narrows Bridge6.5 Variable step-size methods6.5.1 Embedded Runge-Kutta pairs6.5.2 Order 4/5 methods6.6 Implicit methods and stiff equations6.7 Multistep methods6.7.1 Generating multistep methods6.7.2 Explicit multistep methods6.7.3 Implicit multistep methods6.8 Software and Further Reading7. Boundary Value Problems7.1 Shooting method7.1.1 Solutions of boundary value problems7.1.2 Shooting method implementationREALITY CHECK 7: Buckling of a circular ring7.2 Finite difference methods7.2.1 Linear boundary value problems7.2.2 Nonlinear boundary value problems7.3 Collocation and the Finite Element Method7.3.1 Collocation7.3.2 Finite elements and the Galerkin method7.4 Software and Further Reading8. Partial Differential Equations8.1 Parabolic equations8.1.1 Forward difference method8.1.2 Stability analysis of forward difference method8.1.3 Backward difference method8.1.4 Crank-Nicolson method8.2 Hyperbolic equations8.2.1 The wave equation8.2.2 The CFL condition8.3 Elliptic equations8.3.1 Finite difference method for elliptic equationsREALITY CHECK 8: Heat distribution on a cooling fin8.3.2 Finite element method for elliptic equations8.4 Nonlinear partial differential equations8.4.1 Implicit Newton solver8.4.2 Nonlinear equations in two space dimensions8.5 Software and Further Reading9. Random Numbers and Applications9.1 Random numbers9.1.1 Pseudo-random numbers9.1.2 Exponential and normal random numbers9.2 Monte-Carlo simulation9.2.1 Power laws for Monte Carlo estimation9.2.2 Quasi-random numbers9.3 Discrete and continuous Brownian motion9.3.1 Random walks9.3.2 Continuous Brownian motion9.4 Stochastic differential equations9.4.1 Adding noise to differential equations9.4.2 Numerical methods for SDEsREALITY CHECK 9: The Black-Scholes formula9.5 Software and Further Reading10. Trigonometric Interpolation and the FFT10.1 The Fourier Transform10.1.1 Complex arithmetic10.1.2 Discrete Fourier Transform10.1.3 The Fast Fourier Transform10.2 Trigonometric interpolation10.2.1 The DFT Interpolation Theorem10.2.2 Efficient evaluation of trigonometric functions10.3 The FFT and signal processing10.3.1 Orthogonality and interpolation10.3.2 Least squares fitting with trigonometric functions10.3.3 Sound, noise, and filteringREALITY CHECK 10: The Wiener filter10.4 Software and Further Reading11. Compression11.1 The Discrete Cosine Transform11.1.1 One-dimensional DCT11.1.2 The DCT and least squares approximation11.2 Two-dimensional DCT and image compression11.2.1 Two-dimensional DCT11.2.2 Image compression11.2.3 Quantization11.3 Huffman coding11.3.1 Information theory and coding11.3.2 Huffman coding for the JPEG format11.4 Modified DCT and audio compression11.4.1 Modified Discrete Cosine Transform11.4.2 Bit quantizationREALITY CHECK 11: A simple audio codec using the MDCT11.5 Software and Further Reading12. Eigenvalues and Singular Values12.1 Power iteration methods12.1.1 Power iteration12.1.2 Convergence of power iteration12.1.3 Inverse power iteration12.1.4 Rayleigh quotient iteration12.2 QR algorithm12.2.1 Simultaneous iteration12.2.2 Real Schur form and QR12.2.3 Upper Hessenberg formREALITY CHECK 12: How search engines rate page quality12.3 Singular value decomposition12.3.1 Finding the SVD in general12.3.2 Special case: symmetric matrices12.4 Applications of the SVD12.4.1 Properties of the SVD12.4.2 Dimension reduction12.4.3 Compression12.4.4 Calculating the SVD12.5 Software and Further Reading13. Optimization13.1 Unconstrained optimization without derivatives13.1.1 Golden section search13.1.2 Successive parabolic interpolation13.1.3 Nelder-Mead search13.2 Unconstrained optimization with derivatives13.2.1 Newton's method13.2.2 Steepest descent13.2.3 Conjugate gradient search13.2.4 Nonlinear least squaresREALITY CHECK 13: Molecular conformation and numerical optimization13.3 Software and Further ReadingAPPENDIXAppendix A: Matrix AlgebraA.1 Matrix fundamentalsA.2 Block multiplicationA.3 Eigenvalues and eigenvectorsA.4 Symmetric matricesA.5 Vector calculusAppendix B: Introduction to MATLABB.1 Starting MATLABB.2 MATLAB graphicsB.3 Programming in MATLABB.4 Flow controlB.5 FunctionsB.6 Matrix operationsB.7 Animation and moviesReferences
