数值分析

2012-6  

机械工业出版社  

（美）Timothy Sauer  

646  

无  

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。

萨奥尔（Timothy Sauer），美国乔治梅森大学数学系教授。1982年于加州大学伯克利分校获得数学专业博士学位，师从著名数学家Robin Hartshorne。他的主要研究领域为动力系统、计算数学和数学生物学。他是《SIAM Journal on Applied Dynamical Systems》、《Journal of Difference Equations and Applications》和《Physica D》等学术期刊的编委。

PREFACE
CHAPTER0 Fundamentals
0．1 Evaluating a Polynomial
0．2 Binary Numbe
0．2．1 Decimal to binary
0．2．2 Binary to decimaI
0．3 Floating Point Representation of ReaI Numbe
0．3．1 Floating point fclrmats
0．3．2 Machine reDresentatiOn
0．3．3 Addition offloating point numbe
0．4 Loss of Significance
0．5 Review of Calculus
Software and Further Reading
CHAPTER 1 Solving Equatio
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 Wilkion polynomial
1．3．3 Seitivity 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 Equatio
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 Spae matrix computatio
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 Equatio
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 Functio
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 functio 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 conditio
3．5 BEzier Curves
Reality Check3：Fonts from Bezier curves
SoftWare and Further Reading
CHAPTER 4Least Squares
4．1 Least Squares and the NormaI Equatio
4．1．1 Incoistent systems of equatio
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 reflecto
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 paramete
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 Equatio
6．1 Initial Value Problems
6．1．1 Euler's Method
6．1．2 Existence,uniqueness．and continuity for solutio
6．1．3 Fit. order Iinear equatio
6．2 Analysis of IVP Solve
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 Equatio
6．3．1 Higher 0rder equatio
6．3．2 Computer simulation：the pendulum
6．3．3 Computer simulation：orbitaI mechanics
6．4 Runge. Kutta Methods and Applicatio
6．4．1 The Runge. Kutta family
6．4．2 Computer simulation：the Hodgkin. Huxley neuron
6．4．3 Computer simulation：the Lorenz equatio
RealityCheck 6The Tacoma Narrows Bridge
6．5 Variable Step. Size Methods
6．5．1 Embedded Runge. Kutta pai
6．5．2 0rder 4／5 methods
6．6 Implicit Methods and Sti仟Equatio
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 Solutio 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 Equatio
8．1 Parabolic Equatio
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：Equatio
8．2．1 The wave equation
8．2．2 The CFL condition
8．3 Elliptic Equatio
8．3．1 Finite Difference Method for elliptic equatio
RealityCheck8：Heat distribution on a cooling fin
8．3．2 Finite Element Method for elliptic equatio
8．4 Nonlinear partial differential equatio
8．4．1 Implicit Newton solver
8．4．2 Nonlinear equatio in two space dimeio
Software and Further Reading
CHAPTER 9 Random Numbe and Applicatio
9．1 Random Numbe
9．1．1 Pseudo. random numbe
9．1．2 Exponential and normal random numbe
9．2 Monte Carlo Simulation
9．2．1 Power Iaws for Monte Carlo estimation
9．2．2 Quasi. random numbe
9．3 Discrete and Continuous Brownian Motion
9．3．1 Random walks
9．3．2 Continuous Brownian motion
9．4 Stochastic DifFerential Equatio
9．4．1 Adding noise to differential equatio
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 Trafoml
10．1．1 Complex arithmetic
10．1．2 Discrete FourierTraform
10．1．3 The Fast FourierTraform
10．2 Trigonometric Interpolation
10．2．1 The DFT Interpolation Theorem
10．2．2 E币cient evaluation of trigonometric functio
10．3 The FFT and Signal Processing
10．3．1 Orthogonality and interpolation
10．3．2 Least squares fitting with trigonometric functio
10．3．3 Sound，noise，and filtering
Relity Check10：The Wiener Filter
Software and Further Reading
CHAPTER 11 Compression
11．1 The Discrete Cosine Traform
11．1．1 One. dimeionaI DCT
11．1．2 The DCT and least squares approximation
11．2 Two. DimeionaI DCT and lmage Compression
11．2．1 Two. dimeional 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 CosineTraform
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 lnvee 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 Applicatio of the SVD
12．4．1 Properties of the SVD
12．4．2 Dimeion reduction
12．4．3 Compression
12．4．4 Calculating the SVD
Software and Further Reading
CHAPTER 13 Optimization
13．1 Uncotrained Optimization without Derivatives
13．1．1 Golden Section Search
13．1．2 Successive parabolic interpolation
13．1．3 Nelder. Mead search
13．2 Uncotrained 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 Eigenvecto
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 Functio
B．6 Matrix 0peratio
B．7 Animation and Movies
ANSWERS T0 SELECTED EXERCISES
BIBLIOGRAPHY
INDEX

版权页： 插图： We must truncate the number in some way,and in so doing we necessarily make a small error.One method,called chopping,is to simply throw away the bits that fall off the end-that is,those beyond the 52nd bit to the right of the decimal point.This protocol is simple,but it is biased in that it always moves che result toward zero. The alternative method is rounding.In base 10,numbers are customarily rounded up if the next digit is 5 or higher,and rounded down otherwise.In binary,this corresponds to rounding up if the bit is 1.Specifically,the important bit in the double precision format is the 53rd bit to the right of the radix point,the first one lying outside of the box.The default rounding technique,implemented by the IEEE standard,is to add 1 to bit 52(round up)if bit 53 is 1,and to do nothing(round down)to bit 52 if bit 53 is 0,with one exception: If the bits following bit 52 are 10000...,exactly halfway between up and down,we round up or round down according to which Choice makes the final bit 52 equal to 0.(Here we are dealing with the mantissa only,since the sign does not play a role.) Why is there the strange exceptional case? Except for this case,the rule means rounding to the normalized floating point number closest to the original number-hence its name,the Rounding to Nearest Rule.The error made in rounding will be equally likely to be up or down.Therefore,the exceptional case,the case where there are two equally distant floating point numbers to round to,should be decided in a way that doesn't prefer up or down systematically.This is to try to avoid the possibility of an unwanted slow drift in long calculations due simply to a biased rounding.The choice to make the final bit 52 equal to 0 in the case of a tie is somewhat arbitrary,but at least it does not display a preference up or down.Problem 8 sheds some light on why the arbitrary choice of 0 is made in case of a tie.

《数值分析(英文版•第2版)》内容新颖，讲解细致，实用性强，受到广泛好评，被美国多所大学采纳为教材或指定为参考书。《数值分析(英文版•第2版)》内容循序渐进，从基本概念开始，逐步深入到复杂概念。突出了数值分析的5个重要思想：收敛性、复杂性、条件作用、压缩和正交性。

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数值分析 PDF格式下载

对于数值计算方面非常好

很好的数值计算英文书籍，对提高数学专业英文写作有一定的帮助。

看过的英文版书文字都小

讲得比较全，也很清晰，入门，查阅都是可以的

相比于高教影印的那本数学分析，这本书内容不少，但更精炼，相对于后者，少了一些练习而已。因此，这本书更适合给老鸟做参考书。

很好的书，确实经典~

如题，书中介绍多为经典的数值算法，有很多新的进展没有包含进去，比方说：利用区间分析改进后的牛顿求根法，可以求出方程式所有的根，书里面就没有提到。作为07年出版的一本书来说，这未免有些让人失望。比这本书早出几年，而且同样也是第二版的，N.J. 阅读更多

内容很好，就是印刷质量有点差，原版蓝色的插图以及表格，到这儿就成灰白色的了！