Fundamentals of Numerical Mathematics for Physicists and Engineers (Hardcover)

· 제목 : Fundamentals of Numerical Mathematics for Physicists and Engineers (Hardcover) 
· 분류 : 외국도서 > 과학/수학/생태 > 수학 > 응용수학
· ISBN : 9781119425670
· 쪽수 : 400쪽
· 출판일 : 2020-06-16

Preface 1

Acknowledgments 5

Glossary of mathematical symbols 7

Part I 8

1 Solution methods for scalar nonlinear equations 9

1.1 Nonlinear equations in physics 9

1.2 Approximate roots: tolerance 12

The Bisection Method 12

1.3 Newton’s Method 17

1.4 Order of a root-finding method 20

1.5 Chord and Secant methods 24

1.6 Conditioning 27

1.7 Local and global convergence 30

Problems and exercises 35

2 Polynomial interpolation 39

2.1 Function approximation 39

2.2 Polynomial interpolation 40

2.3 Lagrange’s Interpolation 43

Equispaced grids 48

2.4 Barycentric Interpolation 51

2.5 Convergence of the interpolation method 56

Runge’s counterexample 59

2.6 Conditioning of an interpolation 62

2.7 Chebyshev’s interpolation 69

Problems and exercises 77

3 Numerical differentiation 81

3.1 Introduction 81

3.2 Differentiation matrices 85

3.3 Local Equispaced Differentiation 93

3.4 Accuracy of finite differences 96

3.5 Chebyshev differentiation 102

Problems and exercises 108

4 Numerical integration 111

4.1 Introduction 111

4.2 Interpolatory quadratures 112

4.2.1 Newton-Cotes quadratures 117

4.2.2 Composite quadrature rules 120

4.3 Accuracy of quadrature formulas 124

4.4 Clenshaw-Curtis quadrature 132

4.5 Integration of periodic functions 140

4.6 Improper integrals 144

4.6.1 Improper integrals of the first kind 145

4.6.2 Improper integrals of the second kind 148

Problems and exercises 155

Part II 159

5 Numerical Linear Algebra 161

5.1 Introduction 161

5.2 Direct Linear Solvers 162

5.2.1 Diagonal and triangular systems 163

5.2.2 The Gaussian elimination method 167

5.3 LU factorization of a matrix 173

5.3.1 Solving systems with LU 180

5.3.2 Accuracy of LU 182

5.4 LU with partial pivoting 185

5.5 The Least Squares Problem 199

5.5.1 QR factorization 202

5.5.2 Linear data fitting 216

5.6 Matrix norms and conditioning 222

5.7 Gram-Schmidt orthonormalization 229

5.7.1 Instability of CGS: reorthogonalization 234

5.8 Matrix-free Krylov solvers 242

Problems and exercises 256

6 Systems of Nonlinear Equations 261

6.1 Newton’s method for nonlinear systems 263

6.2 Nonlinear systems with parameters 274

6.3 Numerical Continuation (homotopy) 278

Problems and exercises 288

7 Numerical Fourier Analysis 293

7.1 The Discrete Fourier Transform 294

7.1.1 Time-Frequency windows 303

7.1.2 Aliasing 307

7.2 Fourier Differentiation 312

Problems and exercises 322

8 Ordinary Differential Equations 325

8.1 Boundary Value Problems 326

8.1.1 Bounded domains 327

8.1.2 Periodic domains 344

8.1.3 Unbounded domains 346

8.2 The Initial Value Problem 351

8.2.1 Runge-Kutta One-Step Formulas 353

8.2.2 Linear Multistep Formulas 360

8.2.3 Convergence of time-steppers 374

8.2.4 A-Stability 377

8.2.5 A-Stability in nonlinear systems: stiffness 395

Problems and exercises 417

Solutions to problems and exercises 422

Chapter 1 422

Chapter 2 424

Chapter 3 427

Chapter 4 430

Chapter 5 435

Chapter 6 440

Chapter 7 445

Chapter 8 447

Bibliography 456

Index 460