Published by Pearson (October 30, 2017) © 2018
The most readable and relevant numerical analysis text is now infused with web links at point-of-use.
Numerical Analysis, 3rd Edition is written for students of engineering, science, mathematics, and computer science who have completed elementary calculus and matrix algebra. The book covers both standard topics and some of the more advanced numerical methods used by computational scientists and engineers, while maintaining a level appropriate for undergraduates.
Students learn to construct and explore algorithms for solving science and engineering problems while situating these algorithms in a landscape of some potent and far-reaching principles. Specifically, the author cultivates a grasp of the fundamental concepts that permeate numerical analysis, including convergence, complexity, conditioning, compression, orthogonality, and its competing concerns of accuracy and efficiency.
MATLAB® software is used both for exposition of algorithms and as a suggested platform for student assignments and projects. The 3rd Edition is web enhanced, with over 200 short URLs that take students beyond the book to useful digital resources created to support their use of the text.
- These Spotlights comment on the topic at hand and make informal connections to other expressions of the same concept elsewhere in the book, helping students synthesize new material with what they already know.
- Appendix B is a short MATLAB tutorial that can be used as a first introduction to students who have not used MATLAB, or as a reference to students already familiar with the software.
MATLAB Code (goo.gl/VxzXyw): Longer instances of MATLAB code are available for students in *.m format.
Solutions to Selected Exercises (goo.gl/2j5gI7): In previous editions a Student Solutions Manual was available for purchase separately. The 3rd Edition gives students access to solutions for selected exercises online at no extra charge.
Additional examples (goo.gl/lFQb0B): Each section of the 3rd Edition is enhanced with extra new examples, designed to reinforce the text exposition and to ease the reader's transition to active solution of exercises and computer problems. The worked-out solutions of these examples, more than one hundred in total, are available online. Some of the solutions are in video format (created by the author).
The homepage for all web content supporting the text is goo.gl/zQNJeP.
MATLAB Code (goo.gl/VxzXyw): Longer instances of MATLAB code are available for students in *.m format.
Solutions to Selected Exercises (goo.gl/2j5gI7): In previous editions a Student Solutions Manual was available for purchase separately. The 3rd Edition gives students access to solutions for selected exercises online at no extra charge.
Additional examples (goo.gl/lFQb0B): Each section of the 3rd Edition is enhanced with extra new examples, designed to reinforce the text exposition and to ease the reader's transition to active solution of exercises and computer problems. The worked-out solutions of these examples, more than one hundred in total, are available online. Some of the solutions are in video format (created by the author).
The homepage for all web content supporting the text is goo.gl/zQNJeP.
CHAPTER 0
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 formats0.3.3 Addition of floating point numbers
0.4 Loss of Significance
0.5 Review of Calculus
Software and Further Reading
CHAPTER 1
1.1 The Bisection Method
1.1.1 Bracketing a root1.1.2 How accurate and how fast?
1.2 Fixed-Point Iteration
1.2.1 Fixed points of a function1.2.4 Stopping criteria
1.3 Limits of Accuracy
1.3.1 Forward and backward error1.3.3 Sensitivity of root-finding
1.4 Newton’s Method
1.4.1 Quadratic convergence of Newton’s Method1.4.2 Linear convergence of Newton’s Method
1.5 Root-Finding without Derivatives
1.5.1 Secant Method and variants1.5.2 Brent’s Method
Reality Check 1: Kinematics of the Stewart platform
Software and Further Reading
CHAPTER 2
2.1 Gaussian Elimination
2.1.1 Naive Gaussian elimination2.1.2 Operation counts
2.2 The LU Factorization
2.2.1 Matrix form of Gaussian elimination2.2.3 Complexity of the LU factorization
2.3 Sources of Error
2.3.1 Error magnification and condition number2.3.2 Swamping
2.4 The PA = LU Factorization
2.4.1 Partial pivoting2.4.3 PA = LU factorization
Reality Check 2: The Euler–Bernoulli Beam
2.5 Iterative Methods
2.5.1 Jacobi Method2.5.4 Sparse matrix computations
2.6 Methods for symmetric positive-definite matrices
2.6.1 Symmetric positive-definite matrices2.6.4 Preconditioning
2.7 Nonlinear Systems of Equations
2.7.1 Multivariate Newton’s Method2.7.2 Broyden’s Method
Software and Further Reading
CHAPTER 3
3.1 Data and Interpolating Functions
3.1.1 Lagrange interpolation3.1.5 Representing functions by approximating polynomials
3.2 Interpolation Error
3.2.1 Interpolation error formula3.2.3 Runge phenomenon
3.3 Chebyshev Interpolation
3.3.1 Chebyshev’s theorem3.3.3 Change of interval
3.4 Cubic Splines
3.4.1 Properties of splines3.4.2 Endpoint conditions
3.5 Bézier Curves
Reality Check 3: Fonts from Bézier curves
Software and Further Reading
CHAPTER 4
4.1 Least Squares and the Normal Equations
4.1.1 Inconsistent systems of equations4.1.3 Conditioning of least squares
4.2 A Survey of Models
4.2.1 Periodic data4.2.2 Data linearization
4.3 QR Factorization
4.3.1 Gram–Schmidt orthogonalization and least squares4.3.3 Householder reflectors
4.4 Generalized Minimum Residual (GMRES) Method
4.4.1 Krylov methods4.4.2 Preconditioned GMRES
4.5 Nonlinear Least Squares
4.5.1 Gauss–Newton Method4.5.3 The Levenberg–Marquardt Method
Reality Check 4: GPS, Conditioning, and Nonlinear Least Squares
Software and Further Reading
CHAPTER 5
5.1 Numerical Differentiation
5.1.1 Finite difference formulas5.1.4 Symbolic differentiation and integration
5.2 Newton–Cotes Formulas for Numerical Integration
5.2.1 Trapezoid Rule5.2.4 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
6.1 Initial Value Problems
6.1.1 Euler’s Method6.1.3 First-order linear equations
6.2 Analysis of IVP Solvers
6.2.1 Local and global truncation error6.2.3 Taylor Methods
6.3 Systems of Ordinary Differential Equations
6.3.1 Higher order equations6.3.3 Computer simulation: orbital mechanics
6.4 Runge–Kutta Methods and Applications
6.4.1 The Runge–Kutta family6.4.3 Computer simulation: the Lorenz equations
Reality Check 6: The Tacoma Narrows Bridge
6.5 Variable Step-Size Methods
6.5.1 Embedded Runge–Kutta pairs6.5.2 Order 4/5 methods
6.6 Implicit Methods and Stiff Equations
6.7 Multistep Methods
6.7.1 Generating multistep methods6.7.3 Implicit multistep methods
Software and Further Reading
CHAPTER 7
7.1 Shooting Method
7.1.1 Solutions of boundary value problems7.1.2 Shooting Method implementation
Reality Check 7: Buckling of a Circular Ring
7.2 Finite Difference Methods
7.2.1 Linear boundary value problems7.2.2 Nonlinear boundary value problems
7.3 Collocation and the Finite Element Method
7.3.1 Collocation7.3.2 Finite elements and the Galerkin Method
Software and Further Reading
CHAPTER 8
8.1 Parabolic Equations
8.1.1 Forward Difference Method8.1.4 Crank–Nicolson Method
8.2 Hyperbolic Equations
8.2.1 The wave equation8.2.2 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 equations8.4 Nonlinear partial differential equations
8.4.1 Implicit Newton solver8.4.2 Nonlinear equations in two space dimensions
Software and Further Reading
CHAPTER 9
9.1 Random Numbers
9.1.1 Pseudo-random numbers9.1.2 Exponential and normal random numbers
9.2 Monte Carlo Simulation
9.2.1 Power laws for Monte Carlo estimation9.2.2 Quasi-random numbers
9.3 Discrete and Continuous Brownian Motion
9.3.1 Random walks9.3.2 Continuous Brownian motion
9.4 Stochastic Differential Equations
9.4.1 Adding noise to differential equations9.4.2 Numerical methods for SDEs
Reality Check 9: The Black–Scholes Formula
Software and Further Reading
CHAPTER 10
10.1 The Fourier Transform
10.1.1 Complex arithmetic10.1.3 The Fast Fourier Transform
10.2 Trigonometric Interpolation
10.2.1 The DFT Interpolation Theorem10.2.2 Efficient evaluation of trigonometric functions
10.3 The FFT and Signal Processing
10.3.1 Orthogonality and interpolation10.3.3 Sound, noise, and filtering
Reality Check 10: The Wiener Filter
Software and Further Reading
CHAPTER 11
11.1 The Discrete Cosine Transform
11.1.1 One-dimensional DCT11.1.2 The DCT and least squares approximation
11.2 Two-Dimensional DCT and Image Compression
11.2.1 Two-dimensional DCT11.2.3 Quantization
11.3 Huffman Coding
11.3.1 Information theory and coding11.3.2 Huffman coding for the JPEG format
11.4 Modified DCT and Audio Compression
11.4.1 Modified Discrete Cosine Transform11.4.2 Bit quantization
Reality Check 11: A Simple Audio Codec
Software and Further Reading
CHAPTER 12
12.1 Power Iteration Methods
12.1.1 Power Iteration12.1.4 Rayleigh Quotient Iteration
12.2 QR Algorithm
12.2.1 Simultaneous iteration12.2.3 Upper Hessenberg form
Reality Check 12: How Search Engines Rate Page Quality
12.3 Singular Value Decomposition
12.3.1 Finding the SVD in general12.3.2 Special case: symmetric matrices
12.4 Applications of the SVD
12.4.1 Properties of the SVD12.4.4 Calculating the SVD
Software and Further Reading
CHAPTER 13
13.1 Unconstrained Optimization without Derivatives
13.1.1 Golden Section Search13.1.3 Nelder–Mead search
13.2 Unconstrained Optimization with Derivatives
13.2.1 Newton’s Method13.2.3 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
Timothy Sauer earned his Ph.D. in mathematics at the University of California–Berkeley in 1982, and is currently a professor at George Mason University. He has published articles on a wide range of topics in applied mathematics, including dynamical systems, computational mathematics, and mathematical biology.
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