Numerical analysis - Sauer, Tim. (Boston Pearson, 2012)
Contents : Machine generated contents note: ch. 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 -- ch. 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 withoutDerivatives -- 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 -- ch. 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 -- ch. 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.Bezier Curves -- Reality Check 3 Fonts from Bezier curves -- Software and Further Reading -- ch. 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, andNonlinear Least Squares -- Software and Further Reading -- ch. 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 -- ch. 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 -- ch. 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 -- ch. 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 -- ch. 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 -- ch. 10 Trigonometric Interpolation and the FFT -- 10.1.The Fourier Transform -- 10.1.1.Complex arithmetic -- 10.1.2.Discrete FourierTransform -- 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 -- ch. 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 -- ch. 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 -- ch. 13 Optimization -- 13.1.Unconstrained Optimization without Derivatives -- 13.1.1.Golden Section Search -- 13.1.2.Successive parabolicinterpolation -- 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.Block Multiplication -- A.3.Eigenvalues and Eigenvectors -- A.4.Symmetric Matrices -- A.5.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.
Includes bibliographical references and index.
Keywords: Numerical analysis