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 ResiduaIGMRESMethod
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
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