數(shù)值分析

內(nèi)容概要

美國薩奧爾編著的《數(shù)值分析》是一本優(yōu)秀的數(shù)值分析教材，書中不僅全面論述了數(shù)值分析的基本方法，還深入淺出地介紹了計算機和工程領域使用的一些高級數(shù)值方法，如壓縮、前向和后向誤差分析、求解方程組的迭代方法等。每章的“實例檢驗”部分結合數(shù)值分析在各領域的具體應用實例，進一步探究如何更好地應用數(shù)值分析方法解決實際問題。此外，書中含有一些算法的matlab實現(xiàn)代碼，并且每章都配有大量難度適宜的習題和計算機問題，便于讀者學習、鞏固和提高。

作者簡介

薩奧爾（Timothy Sauer），美國喬治梅森大學數(shù)學系教授。1982年于加州大學伯克利分校獲得數(shù)學專業(yè)博士學位，師從著名數(shù)學家Robin Hartshorne。他的主要研究領域為動力系統(tǒng)、計算數(shù)學和數(shù)學生物學。他是《SIAM Journal on Applied Dynamical Systems》、《Journal of Difference Equations and Applications》和《Physica D》等學術期刊的編委。

書籍目錄

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．a(chǎn)nd 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幣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

章節(jié)摘錄

版權頁：   插圖：   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.

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《數(shù)值分析(英文版?第2版)》內(nèi)容新穎，講解細致，實用性強，受到廣泛好評，被美國多所大學采納為教材或指定為參考書?！稊?shù)值分析(英文版?第2版)》內(nèi)容循序漸進，從基本概念開始，逐步深入到復雜概念。突出了數(shù)值分析的5個重要思想：收斂性、復雜性、條件作用、壓縮和正交性。

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