Original edition, entitled NUMERICAL ANALYSIS, 2E, 9780321783677 by SAUER, TIMOTHY, published by Pearson Education, Inc, publishing as Pearson, Copyright 2012. All rights reserved. No part of this book may be reproduced or transmitted in any form or
目录
PREFACE CHAPTER0 Fundamentals 0．1 Evaluating a Polynomial 0．2 Binary Numbers 0．2．1 Decimal to binary 0．2．2 Binary to decimaI 0．3 Floating Point Representation of ReaI Numbers 0．3．1 Floating point fclrmats 0．3．2 Machine reDresentatiOn 0．3．3 Addition offloating 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 Howaccurate and howfast? 1．2 Fixed. Point Iteration 1．2．1 Fixed points of a function 1．2．2 Geometry of Fixed. Point lteration 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 Brent3 Method Reality Check1：Kinematics ofthe 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 FactO rizatiOn 2．2．1 Matrix form of Gaussian elimination 2．2．2 Back substitution with the LU f2Ictorization 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 PartiaI 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 PrecOnditioninq 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 intervaI 3．4 Cubic Splines 3．4．1 Properties of splines 3．4．2 Endpoint conditions 3．5 BEzier Curves Reality Check3：Fonts from Bezier curves SoftWare and Further Reading CHAPTER 4Least Squares 4．1 Least Squares and the NormaI Equations 4．1．1 Inconsistent systems of equations 4．1．2 Fitting models to data 4．1．3 Conditioning of Ieast 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 OrthoaonaIizatiOn and Ieast squares 4．3．2 Modified Gram. Schmidt orthogonalization 4．3．3 Householder reflectors 4．4 Generalized Minimum ResiduaI(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． Reatity Check4：GPS，Conditioning，and Nonlinear Least Squares Software and Further Reading CHAPTER 5 NumericalDifferentiation and Inteqration 5．1 NumericaI 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 NumericaI Integration 5．2．1 Trapezoid Rule 5．2．2 Simpson's Rule 5．2．3 Composite Newton. Cotes formulas 5．2．4 0pen Newton. Cotes Methods 5．3 Romberg Integration 5．4 Adaptive Quadrature 5．5 Gaussian Quadrature Reality Check5：Motion Control in Computer. Aided Modeling SOftware and Further Reading CHAPTER 6 Ordinary Differentiai 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 Iinear 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 Difl．erential Equations 6．3．1 Higher 0rder equations 6．3．2 Computer simulation：the pendulum 6．3．3 Computer simulation：orbitaI 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 RealityCheck 6The Tacoma Narrows Bridge 6．5 Variable Step. Size Methods 6．5．1 Embedded Runge. Kutta pairs 6．5．2 0rder 4／5 methods 6．6 Implicit Methods and Sti仟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 Check7: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 8Partial Differential Equations 8．1 Parabolic Equations 8．1．1 Forward Difference Method 8．1．2 Stability analysis of Forward Difierence Method 8．1．3 Backward Di仟lerence Method 8．1．4 Crank. Nicolson Method 8．2 Hyperbolk：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 RealityCheck8：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 Iaws 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 NumericaI methods for SDEs Reality Check 9：The Black. Scholes FormulaSoftware and Further Reading CHAPTER 10 Trigonometric Interpolation andthe FFT 10．1 The Fourier Transfoml 10．1．1 Complex arithmetic 10．1．2 Discrete FourierTransform 10．1．3 The Fast FourierTransform 10．2 Trigonometric Interpolation 10．2．1 The DFT Interpolation Theorem 10．2．2 E币cient 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 Relity Check10：The Wiener Filter Software and Further Reading CHAPTER 11 Compression 11．1 The Discrete Cosine Transform 11．1．1 One. dimensionaI DCT 11．1．2 The DCT and least squares approximation 11．2 Two. DimensionaI DCT and lmage Compression 11．2．1 Two. dimensional DCT 11．2．2 lmage compression 11．23 Quantization 11．3 HufFman Coding 11．3．1 Information theory and coding 11．3．2 Huffman coding for the JPEG format 11. 14 Modified DCT and Audio Compression 11．4．1 Modified Discrete CosineTransform 11．4．2 Bit quantization Reality Check11：A Simple Audio Codec Software and Further Reading CHAPTER12 Eigenvalues and Singular Values 12．I Power Iteration Methods 12．1．1 Power Iteration 12．1．2 Convergence of Power Iteration 12．1．3 lnverse Power Iteration 12．1．4 Rayleigh Quotient Iteration 12．2 QR Algorithm 12．2．1 Simultaneous iteration 12．2．2 ReaI Schur form and the QR algorithm 12．2．3 Upper Hessenberg form Reality Check 12：How Sea~h Engines Rate Page Quality 12．3 Singular Value Decomposition 12．3．1 Finding the SVD in general 12．3．2 SpeciaI 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 0ptimization 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 0perations B．7 Animation and Movies ANSWERS T0 SELECTED EXERCISES BIBLIOGRAPHY INDEX
