Numerical Analysis

Höfundur: Timothy Sauer (Útgáfa: 3)
Numerical Analysis

Kaup valmöguleikar

This is the eBook of the printed book and may not include any media, website access codes, or print supplements that may come packaged with the bound book. For courses in Numerical Analysis.   Helps students construct and explore algorithms for solving science and engineering problems Numerical Analysis, 3rd Edition is for students of engineering, science, mathematics, and computer science who have completed elementary calculus and matrix algebra.

It covers both standard topics and some of the more advanced numerical methods used by computational scientists and engineers, while remaining readable and relevant for undergraduates.   Sauer discusses the fundamental concepts of numerical analysis: convergence, complexity, conditioning, compression, and orthogonality. Throughout, Spotlight features comment on each of these concepts as they are addressed, and make connections to other expressions of the same principle elsewhere in the book.

Nánar um bókina

Útgefandi
Pearson Education (US)
ISBN
9780134697376
Print ISBN
9780134696454
Format
Page Fidelity (PDF)
Útgáfa
3
Höfundar
Timothy Sauer
Tungumál
English
Útgefið
2018-03-01
Prent takmörkun á líftíma
100
Prent takmörkun
2
Afritunar takmörkun
2

Kaflar

  • Title Page
  • Copyright Page
  • Contents
  • Preface
  • 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 0f 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 Iterativemethods
  • 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 Generalizedminimum 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-Sizemethods
  • 6.5.1 Embedded runge–Kutta pairs
  • 6.5.2 Order 4/5 methods
  • 6.6 Implicitmethods 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 Elementmethod
  • 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
  • 8.3.2 Finite Element Method for elliptic equations
  • Reality Check 8: Heat Distribution on a Cooling Fin
  • 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 Brownianmotion
  • 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 Geometry of the SVD
  • 12.3.2 Finding the SVD in general
  • 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:Matrix Algebra
  • A.1 Matrix Fundamentals
  • A.2 Systems of linear Equations
  • A.3 Block Multiplication
  • A.4 Eigenvalues and Eigenvectors
  • A.5 Symmetricmatrices
  • A.6 Vector Calculus
  • Appendix B: Introduction toMatlab
  • B.1 Starting MATLAB
  • B.2 Graphics
  • B.3 Programming in MATLAB
  • B.4 Flowcontrol
  • B.5 Functions
  • B.6 Matrix Operations
  • B.7 Animation and Movies
  • Answers to Selected Exercises
  • Bibliography
  • Index
  • Back Cover