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1 Mathematical Preliminaries 1

1.1 Review of Calculus 2

1.2 Roundoff Errors and Computer Arithmetic 18

1.3 Algorithms and Convergence 31

1.4 Numerical Software 40

2 Solutions of Equations in One Variable 47

2.1 The Bisection Method 48

2.2 Fixed-Point Iteration 55

2.3 Newton’s Method 66

2.4 Error Analysis for Iterative Methods 78

2.5 Accelerating Convergence 86

2.6 Zeros of Polynomials and Müller’s Method 91

2.7 Survey of Methods and Software 101

3 Interpolation and Polynomial Approximation 104

3.1 Interpolation and the Lagrange Polynomial 107

3.2 Divided Differences 122

3.3 Hermite Interpolation 133

3.4 Cubic Spline Interpolation 141

3.5 Parametric Curves 156

3.6 Survey of Methods and Software 163

4 Numerical Differentiation and Integration 166

4.1 Numerical Differentiation 167

4.2 Richardson’s Extrapolation 178

4.3 Elements of Numerical Integration 186

4.4 Composite Numerical Integration 196

4.5 Romberg Integration 207

4.6 Adaptive Quadrature Methods 213

4.7 Gaussian Quadrature 220

4.8 Multiple Integrals 227

4.9 Improper Integrals 241

4.10 Survey of Methods and Software 247

5 Initial-Value Problems for Ordinary Differential Equations 249

5.1 The Elementary Theory of Initial-Value Problems 251

5.2 Euler’s Method 256

5.3 Higher-Order Taylor Methods 266

5.4 Runge-Kutta Methods 272

5.5 Error Control and the Runge-Kutta-Fehlberg Method 282

5.6 Multistep Methods 289

5.7 Variable Step-Size Multistep Methods 301

5.8 Extrapolation Methods 307

5.9 Higher-Order Equations and Systems of Differential Equations 313

5.10 Stability 324

5.11 Stiff Differential Equations 334

5.12 Survey of Methods and Software 342

6 Direct Methods for Solving Linear Systems 344

6.1 Linear Systems of Equations 345

6.2 Pivoting Strategies 359

6.3 Linear Algebra and Matrix Inversion 370

6.4 The Daterminant of a Matrix 383

6.5 Matrix Factorization 388

6.6 Special Types of Matrices 398

6.7 Survey of Methods and Software 413

7 Iterative Techniques in Matrix Algebra 417

7.1 Norms of Vectors and Matrices 418

7.2 Eigenvalues and Eigenvectors 430

7.3 Iterative Techniques for Solving Linear Systems 437

7.4 Error Bounds and Iterative Refinement 454

7.5 The Conjugate Gradient Method 465

7.6 Survey of Methods and Software 481

8 Approximation Theory 483

8.1 Discrete Least Squares Approximation 484

8.2 Orthogonal Polynomials and Least Squares Approximation 498

8.3 Chebyshev Polynomials and Economization of Power Series 507

8.4 Rational Function Approximation 517

8.5 Trigonometric Polynomial Approximation 529

8.6 Fast Fourier Transforms 537

8.7 Survey of Methods and Software 548

9 Approximating Eigenvalues 550

9.1 Linear Algebra and Eigenvalues 551

9.2 The Power Method 560

9.3 Householder’s Method 577

9.4 The QR Algorithm 585

9.5 Survey of Methods and Software 597

10 Numerical Solutions of Nonlinear Systems of Equations 600

10.1 Fixed Points for Functions of Several Variables 602

10.2 Newton’s Method 611

10.3 Quasi-Newton Methods 620

10.4 Steepest Descent Techniques 628

10.5 Homotopy and Continuation Methods 635

10.6 Survey of Methods and Software 643

11 Boundary-Value Problems for Ordinary Differential Equations 645

11.1 The Linear Shooting Method 646

11.2 The Shooting Method for Nonlinear Prblems 653

11.3 Finite-Difference Methods for Linear Problems 660

11.4 Finite-Difference Methods for Nonlinear Problems 667

11.5 The Rayleigh-Ritz Method 672

11.6 Survey of Methods and Software 688

12 Numerical Solutions to Partial Differential Equations 691

12.1 Elliptic Partial Differential Equations 694

12.2 Parabolic Partial Differential Equations 704

12.3 Hyperbolic Partial Differential Equations 718

12.4 An Introduction to the Finite-Element Method 726

12.5 Survey of Methods and Software 741

Bibliography 743

Answers to Selected Exercises 753

Index 831