Applied Numerical Analysis Using MATLAB, 2/e (IE-Paperback)

Description

This text is appropriate for undergraduate courses on numerical methods and numerical analysis found in engineering, mathematics & computer science departments.

 

Each chapter uses introductory problems from specific applications. These easy-to-understand problems clarify for the reader the need for a particular mathematical technique. Numerical techniques are explained with an emphasis on why they work.

Table of Contents

Preface

 

1 Foundations 1

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

1.3.1 Error

1.3.2 Convergence

1.3.3 Getting Better Results

1.4 Using MATLAB

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 47

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

 

3 Solving Linear Systems: Direct Methods 95

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

 

4 LU and QR Factorization 135

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 Efficient Conversion to Hessenberg Form

4.4.3 Using MATLAB’s Functions

4.5 Chapter Wrap-Up

 

5 Eigenvalues and Eigenvectors 179

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

 

6 Solving Linear Systems: Iterative Methods 213

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

 

7 Nonlinear Functions of Several Variables 251

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

 

8 Interpolation 275

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

 

9 Approximation 333

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

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

9.3 Function Approximation at a Point

9.3.1 Pad´e Approximation

9.3.2 Taylor Approximation

9.4 Further Topics

9.4.1 Bezier Curves

9.4.2 Using MATLAB’s Functions

9.5 Chapter Wrap-Up

 

10 Fourier Methods 373

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

 

11 Numerical Differentiation and Integration 405

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

 

12 Ordinary Differential Equations: Fundamentals 445

12.1 Euler’s Method

12.1.1 Geometric 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, Stiffness, Stability 499

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 Stiff ODE

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

 

14 ODE: Boundary-Value Problems 561

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

 

15 Partial Differential Equations 593

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 Coefficients

15.4.4 Using MATLAB

15.5 Chapter Wrap-Up

 

Bibliography 643

 

Answers 653

 

Index 667