Numerical Analysis, 3rd edition

CHAPTER 0     Fundamentals

0.1 Evaluating a Polynomial

0.2 Binary Numbers

    0.2.1 Decimal to binary

    0.2.2 Binary to decimal

0.3 Floating Point Representation of Real Numbers

    0.3.1 Floating point formats
    0.3.2 Machine representation

    0.3.3 Addition of floating 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 How accurate and how fast?

1.2 Fixed-Point Iteration

    1.2.1 Fixed points of a function
    1.2.2 Geometry of Fixed-Point Iteration
    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 Brent’s Method

Reality Check 1: Kinematics of the 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 Factorization

    2.2.1 Matrix form of Gaussian elimination
    2.2.2 Back substitution with the LU factorization

    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 Partial 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 Preconditioning

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 interval

3.4 Cubic Splines

    3.4.1 Properties of splines

    3.4.2 Endpoint conditions

3.5 Bézier Curves

Reality Check 3: Fonts from Bézier curves

Software and Further Reading

CHAPTER 4     Least Squares

4.1 Least Squares and the Normal Equations

    4.1.1 Inconsistent systems of equations
    4.1.2 Fitting models to data

    4.1.3 Conditioning of least 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 orthogonalization and least squares
    4.3.2 Modified Gram–Schmidt orthogonalization

    4.3.3 Householder reflectors

4.4 Generalized Minimum Residual (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

Reality Check 4: GPS, Conditioning, and Nonlinear Least Squares

Software and Further Reading

CHAPTER 5     Numerical Differentiation and Integration

5.1 Numerical 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 Numerical Integration

    5.2.1 Trapezoid Rule
    5.2.2 Simpson’s Rule
    5.2.3 Composite Newton–Cotes formulas

    5.2.4 Open Newton–Cotes Methods

5.3 Romberg Integration

5.4 Adaptive Quadrature

5.5 Gaussian Quadrature

Reality Check 5: Motion Control in Computer-Aided Modeling

Software and Further Reading

CHAPTER 6     Ordinary Differential 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 linear 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 Differential Equations

    6.3.1 Higher order equations
    6.3.2 Computer simulation: the pendulum

    6.3.3 Computer simulation: orbital 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

Reality Check 6: The Tacoma Narrows Bridge

6.5 Variable Step-Size Methods

    6.5.1 Embedded Runge–Kutta pairs

    6.5.2 Order 4/5 methods

6.6 Implicit Methods and Stiff 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 Check 7: 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 8     Partial Differential Equations

8.1 Parabolic Equations

    8.1.1 Forward Difference Method
    8.1.2 Stability analysis of Forward Difference Method
    8.1.3 Backward Difference Method

    8.1.4 Crank–Nicolson Method

8.2 Hyperbolic 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

Reality Check 8: 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 laws 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 Numerical methods for SDEs

Reality Check 9: The Black–Scholes Formula

Software and Further Reading

CHAPTER 10     Trigonometric Interpolation and the FFT

10.1 The Fourier Transform

    10.1.1 Complex arithmetic
    10.1.2 Discrete Fourier Transform

    10.1.3 The Fast Fourier Transform

10.2 Trigonometric Interpolation

    10.2.1 The DFT Interpolation Theorem

    10.2.2 Efficient 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

Reality Check 10: The Wiener Filter

Software and Further Reading

 

CHAPTER 11     Compression

11.1 The Discrete Cosine Transform

    11.1.1 One-dimensional DCT

    11.1.2 The DCT and least squares approximation

11.2 Two-Dimensional DCT and Image Compression

    11.2.1 Two-dimensional DCT
    11.2.2 Image compression

    11.2.3 Quantization

11.3 Huffman Coding

    11.3.1 Information theory and coding

    11.3.2 Huffman coding for the JPEG format

11.4 Modified DCT and Audio Compression

    11.4.1 Modified Discrete Cosine Transform

    11.4.2 Bit quantization

Reality Check 11: A Simple Audio Codec

Software and Further Reading

CHAPTER 12     Eigenvalues and Singular Values

12.1 Power Iteration Methods

    12.1.1 Power Iteration
    12.1.2 Convergence of Power Iteration
    12.1.3 Inverse Power Iteration

    12.1.4 Rayleigh Quotient Iteration

12.2 QR Algorithm

    12.2.1 Simultaneous iteration
    12.2.2 Real Schur form and the QR algorithm

    12.2.3 Upper Hessenberg form

Reality Check 12: How Search Engines Rate Page Quality

12.3 Singular Value Decomposition

    12.3.1 Finding the SVD in general

    12.3.2 Special 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 Optimization

Software and Further Reading

Appendix A

A.1 Matrix Fundamentals

A.2 Systems of linear equations

A.3 Block Multiplication

A.4 Eigenvalues and Eigenvectors

A.5 Symmetric Matrices

A.6 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 Operations

B.7 Animation and Movies

 

ANSWERS TO SELECTED EXERCISES

BIBLIOGRAPHY

INDEX