Overview

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® 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.

Full Product Details

Author:   Timothy Sauer
Publisher:   Pearson Education (US)
Imprint:   Pearson
Edition:   2nd edition
Dimensions:   Width: 20.30cm , Height: 2.80cm , Length: 25.40cm
Weight:   1.230kg
ISBN:  

9780321783677

ISBN 10:   0321783670
Pages:   672
Publication Date:   14 December 2011
Audience:   College/higher education ,  Tertiary & Higher Education
Replaced By:   9781292023588
Format:   Hardback
Publisher's Status:   Out of Print
Availability:   Awaiting stock  

Table of Contents

Preface 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 0.6 Software and Further Reading   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 and error magnification 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 1.6 Software and Further Reading   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 Backsolving with the LU factorization             2.2.2 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 2.8 Software and Further Reading   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: Constructing fonts from Bézier splines 3.6 Software and Further Reading   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 4.2 Linear and nonlinear models             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 Levenberg-Marquardt method REALITY CHECK 4: GPS, conditioning and nonlinear least squares 4.6 Software and Further Reading   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 modelling 5.6 Software and Further Reading   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 6.8 Software and Further Reading   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 7.4 Software and Further Reading   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 8.5 Software and Further Reading   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 9.5 Software and Further Reading   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 10.4 Software and Further Reading   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 using the MDCT 11.5 Software and Further Reading   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 QR             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 12.5 Software and Further Reading   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             13.2.4 Nonlinear least squares REALITY CHECK 13: Molecular conformation and numerical optimization 13.3 Software and Further Reading   APPENDIX Appendix A: Matrix Algebra A.1 Matrix fundamentals A.2 Block multiplication A.3 Eigenvalues and eigenvectors A.4 Symmetric matrices A.5 Vector calculus   Appendix B: Introduction to MATLAB B.1 Starting MATLAB B.2 MATLAB graphics B.3 Programming in MATLAB B.4 Flow control B.5 Functions B.6 Matrix operations B.7 Animation and movies References

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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.

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