Applied Numerical Analysis
Using MATLAB®
Second Edition
Laurene V. Fausett
Texas A&M
University-Commerce
PEARSON
Prentice
Hall
Upper Saddle River, NJ 07458
Contents
Preface
1
Foundations
1.1 Introductory Examples
1.1.1 Nonlinear Equations
1.1.2 Linear Systems
1.1.3 Numerical Integration
1.2 Useful Background
1.2.1 Results from Calculus
1.2.2 Results from Linear Algebra
1.2.3 A Little Information about Computers
1.3 Some Basic Issues
1.3.1 Error
1.3.2 Convergence
1.3.3 Getting Better Results
1.4 UsingMATLAB
1.4.1 Command Window Computations
1.4.2 M-Files
1.4.3 Programming in MATLAB
1.4.4 Matrix Multiplication
1.5 Chapter Wrap-Up
2 Functions of One Variable
2.1 Bisection Method
2.2 Secant-Type Methods
2.2.1 Regula Falsi
2.2.2 Secant Method
2.2.3 Analysis
2.3 Newton's Method
2.4 Muller's Method
2.5 Minimization
2.5.1 Golden-Section Search
2.5.2 Brent's Method
2.6 Beyond the Basics
2.6.1 Using MATLAB's Functions
2.6.2 Laguerre's Method
2.6.3 Zeros of a Nonlinear Function
2.7 Chapter Wrap-Up
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3
Solving Linear Systems: Direct Methods
3.1 Gaussian Elimination
3.1.1 Basic Method
3.1.2 Row Pivoting
3.2 Gauss-Jordan
3.2.1 Inverse of a Matrix
3.3 Tridiagonal Systems
3.4 Further Topics
3.4.1 MATLAB's Methods
3.4.2 Condition of a Matrix
3.4.3 Iterative Refinement
3.5 Chapter Wrap-Up
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4
LU and QR Factorization
4.1 LU Factorization
4.1.1 Using Gaussian Elimination
4.1.2 Direct LU Factorization
4.1.3 Applications
4.2 Matrix Transformations
4.2.1 Householder Transformation
4.2.2 Givens Rotations
4.3 QR Factorization
4.3.1 Using Householder Transformations
4.3.2 Using Givens Rotations
4.4 Beyond the Basics
4.4.1 LU Factorization with Implicit Row Pivoting
4.4.2 EfRcient Conversion to Hessenberg Form
4.4.3 Using MATLAB's Functions
4.5 Chapter Wrap-Up
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5
Eigenvalues and Eigenvectors
5.1 Power Method
5.1.1 Basic Power Method
5.1.2 Rayleigh Quotient
5.1.3 Shifted Power Method
5.1.4 Accelerating Convergence
5.2 Inverse Power Method
5.2.1 General Inverse Power Method
5.2.2 Convergence
5.3 QR Method
5.3.1 Basic QR Method
5.3.2 Better QR Method
5.3.3 Finding Eigenvectors
5.3.4 Accelerating Convergence
5.4 Further Topics
5.4.1 Singular Value Decomposition
5.4.2 MATLAB's Methods
5.5 Chapter Wrap-Up
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VII
6
Solving Linear Systems: Iterative Methods
6.1 Jacobi Method
6.2 Gauss-Seidel Method
6.3 Successive Over-Relaxation
6.4 Beyond the Basics
6.4.1 MATLAB's Built-In Functions
6.4.2 Conjugate Gradient Methods
6.4.3 GMRES
6.4.4 Simplex Method
6.5 Chapter Wrap-Up
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7
Nonlinear Functions of Several Variables
7.1 Nonlinear Systems
7.1.1 Newton's Method
7.1.2 Secant Methods
7.1.3 Fixed-Point Iteration
7.2 Minimization
7.2.1 Descent Methods
7.2.2 Quasi-Newton Methods
7.3 Further Topics
7.3.1 Levenberg-Marquardt Method
7.3.2 Nelder-Mead Simplex Search
7.4 Chapter Wrap-Up
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8
Interpolation
8.1 Polynomial Interpolation
8.1.1 Lagrange Form
8.1.2 Newton Form
8.1.3 Difficulties
8.2 Hermite Interpolation
8.3 Piecewise Polynomial Interpolation
8.3.1 Piecewise Linear Interpolation
8.3.2 Piecewise Quadratic Interpolation
8.3.3 Piecewise Cubic Hermite Interpolation
8.3.4 Cubic Spline Interpolation
8.4 Beyond the Basics
8.4.1 Rational-Function Interpolation
8.4.2 Using MATLAB's Functions
8.5 Chapter Wrap-Up
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9 Approximation
9.1 Least-Squares Approximation
9.1.1 Approximation by a Straight Line
9.1.2 Approximation by a Parabola
9.1.3 General Least-Squares Approximation
9.1.4 Approximation for Other Functional Forms
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9.2
Continuous Least-Squares Approximation
9.2.1 Approximation Using Powers of x
9.2.2 Orthogonal Polynomials
9.2.3 Legendre Polynomials
9.2.4 Chebyshev Polynomials
Function Approximation at a Point
9.3.1 Pade Approximation
9.3.2 Taylor Approximation
Further Topics
9.4.1 Bezier Curves
9.4.2 Using MATLAB's Functions
Chapter Wrap-Up
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10 Fourier Methods
10.1 Fourier Approximation and Interpolation
10.1.1 Derivation
10.1.2 Data on Other Intervals
10.2 Radix-2 Fourier Transforms
10.2.1 Discrete Fourier Transform
10.2.2 Fast Fourier Transform
10.2.3 Matrix Form of FFT
10.2.4 Algebraic Form of FFT
10.3 Mixed-Radix FFT
10.4 Using MATLAB's Functions
10.5 Chapter Wrap-Up
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11 Numerical Differentiation and Integration
11.1 Differentiation
11.1.1 First Derivatives
11.1.2 Higher Derivatives
11.1.3 Partial Derivatives
11.1.4 Richardson Extrapolation
11.2 Numerical Integration
11.2.1 Trapezoid Rule
11.2.2 Simpson's Rule
11.2.3 Newton-Cotes Open Formulas . . .
11.2.4 Extrapolation Methods
11.3 Quadrature
11.3.1 Gaussian Quadrature
11.3.2 Other Gauss-Type Quadratures
11.4 MATLAB's Methods
11.4.1 Differentiation
11.4.2 Integration
11.5 Chapter Wrap-Up
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9.3
9.4
9.5
IX
12 Ordinary Differential Equations: Fundamentals
12.1 Euler's Method
12.1.1 Geometrie Introduction
12.1.2 Approximating the Derivative
12.1.3 Approximating the Integral
12.1.4 Using Taylor Series
12.2 Runge-Kutta Methods
12.2.1 Second-Order Runge-Kutta Methods
12.2.2 Third-Order Runge-Kutta Methods
12.2.3 Classic Runge-Kutta Method
12.2.4 Fourth-Order Runge-Kutta Methods
12.2.5 Fifth-Order Runge-Kutta Methods
12.2.6 Runge-Kutta-Fehlberg Methods
12.3 Multistep Methods
12.3.1 Adams-Bashforth Methods
12.3.2 Adams-Moulton Methods
12.3.3 Adams Predictor-Corrector Methods
12.3.4 Other Predictor-Corrector Methods
12.4 Further Topics
12.4.1 MATLAB's Methods
12.4.2 Consistency and Convergence
12.5 Chapter Wrap-Up
13 ODE: Systems, Stiffhess, Stability
13.1 Systems
13.1.1 Systems of Two ODE
13.1.2 Euler's Method for Systems
13.1.3 Runge-Kutta Methods for Systems
13.1.4 Multistep Methods for Systems
13.1.5 Second-Order ODE
13.2 StifFODE
13.2.1 BDF Methods
13.2.2 Implicit Runge-Kutta Methods
13.3 Stability
13.3.1 A-Stable and Stiffly Stable Methods
13.3.2 Stability in the Limit
13.4 Further Topics
13.4.1 MATLAB's Methods for Stiff ODE
13.4.2 Extrapolation Methods
13.4.3 Rosenbrock Methods
13.4.4 Multivalue Methods
13.5 Chapter Wrap-Up
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14 ODE: Boundary-Value Problems
14.1 Shooting Method
14.1.1 Linear ODE
14.1.2 Nonlinear ODE
14.2 Finite-Difference Method
14.2.1 Linear ODE
14.2.2 Nonlinear ODE
14.3 Function Space Methods
14.3.1 Collocation
14.3.2 Rayleigh-Ritz
14.4 Chapter Wrap-Up
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15 Partial Differential Equations
15.1 Heat Equation: Parabolic PDE
15.1.1 Explicit Method
15.1.2 Implicit Method
15.1.3 Crank-Nicolson Method
15.1.4 Insulated Boundary
15.2 Wave Equation: Hyperbolic PDE
15.2.1 Explicit Method
15.2.2 Implicit Method
15.3 Poisson Equation: Elliptic PDE
15.4 Finite-Element Method for Elliptic PDE
15.4.1 Defining the Subregions
15.4.2 Defining the Basis Functions
15.4.3 Computing the Coefncients
15.4.4 UsingMATLAB
15.5 Chapter Wrap-Up
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Bibliography
643
Ans wer s
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Index
667