美国萨奥尔编著的《数值分析》是一本优秀的数值分析教材，书中不仅全面论述了数值分析的基本方法，还深入浅出地介绍了计算机和工程领域使用的一些高级数值方法，如压缩、前向和后向误差分析、求解方程组的迭代方法等。每章的“实例检验”部分结合数值分析在各领域的具体应用实例，进一步探究如何更好地应用数值分析方法解决实际问题。此外，书中含有一些算法的matlab实现代码，并且每章都配有大量难度适宜的习题和计算机问题，便于读者学习、巩固和提高。

 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