James F. Epperson

An Introduction to Numerical Methods and Analysis (eBook, ePUB)

• Format: ePub

Produktbeschreibung

The new edition of the popular introductory textbook on numerical approximation methods and mathematical analysis, with a unique emphasis on real-world application An Introduction to Numerical Methods and Analysis helps students gain a solid understanding of a wide range of numerical approximation methods for solving problems of mathematical analysis. Designed for entry-level courses on the subject, this popular textbook maximizes teaching flexibility by first covering basic topics before gradually moving to more advanced material in each chapter and section. Throughout the text, students are provided clear and accessible guidance on a wide range of numerical methods and analysis techniques, including root-finding, numerical integration, interpolation, solution of systems of equations, and many others. This fully revised third edition contains new sections on higher-order difference methods, the bisection and inertia method for computing eigenvalues of a symmetric matrix, a completely re-written section on different methods for Poisson equations, and spectral methods for higher-dimensional problems. New problem sets--ranging in difficulty from simple computations to challenging derivations and proofs--are complemented by computer programming exercises, illustrative examples, and sample code. This acclaimed textbook: * Explains how to both construct and evaluate approximations for accuracy and performance * Covers both elementary concepts and tools and higher-level methods and solutions * Features new and updated material reflecting new trends and applications in the field * Contains an introduction to key concepts, a calculus review, an updated primer on computer arithmetic, a brief history of scientific computing, a survey of computer languages and software, and a revised literature review * Includes an appendix of proofs of selected theorems and a companion website with additional exercises, application models, and supplemental resources An Introduction to Numerical Methods and Analysis, Third Edition is the perfect textbook for upper-level undergraduate students in mathematics, science, and engineering courses, as well as for courses in the social sciences, medicine, and business with numerical methods and analysis components.

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 672
• Erscheinungstermin: 21. Juli 2021
• Englisch
• ISBN-13: 9781119604747
• Artikelnr.: 62322311

Autorenporträt

JAMES F. EPPERSON, PHD, has been Associate Editor of Mathematical Reviews, published by the American Mathematical Society, since 2001, and will be retiring from this position in July 2021. He was previously Associate Professor in the Department of Mathematical Sciences at the University of Alabama in Huntsville. Dr. Epperson is the author of the previous editions of An Introduction to Numerical Methods and Analysis and their accompanying solutions manuals.

Inhaltsangabe

Preface xiii

1 Introductory Concepts and Calculus Review 1

1.1 Basic Tools of Calculus 2

1.1.1 Taylor's Theorem 2

1.1.2 Mean Value and Extreme Value Theorems 9

1.2 Error, Approximate Equality, and Asymptotic Order Notation 14

1.2.1 Error 14

1.2.2 Notation: Approximate Equality 15

1.2.3 Notation: Asymptotic Order 16

1.3 A Primer on Computer Arithmetic 20

1.4 A Word on Computer Languages and Software 29

1.5 A Brief History of Scientific Computing 32

1.6 Literature Review 36

References 36

2 A Survey of Simple Methods and Tools 39

2.1 Horner's Rule and Nested Multiplication 39

2.2 Difference Approximations to the Derivative 44

2.3 Application: Euler's Method for Initial Value Problems 52

2.4 Linear Interpolation 58

2.5 Application--The Trapezoid Rule 64

2.6 Solution of Tridiagonal Linear Systems 75

2.7 Application: Simple Two-Point Boundary Value Problems 81

3 Root-Finding 87

3.1 The Bisection Method 88

3.2 Newton's Method: Derivation and Examples 95

3.3 How to Stop Newton's Method 101

3.4 Application: Division Using Newton's Method 104

3.5 The Newton Error Formula 108

3.6 Newton's Method: Theory and Convergence 113

3.7 Application: Computation of the Square Root 117

3.8 The Secant Method: Derivation and Examples 120

3.9 Fixed-Point Iteration 124

3.10 Roots of Polynomials, Part 1 133

3.11 Special Topics in Root-finding Methods 141

3.11.1 Extrapolation and Acceleration 141

3.11.2 Variants of Newton's Method 145

3.11.3 The Secant Method: Theory and Convergence 149

3.11.4 Multiple Roots 153

3.11.5 In Search of Fast Global Convergence: Hybrid Algorithms 157

3.12 Very High-order Methods and the Efficiency Index 162

3.13 Literature and Software Discussion 166

References 166

4 Interpolation and Approximation 169

4.1 Lagrange Interpolation 169

4.2 Newton Interpolation and Divided Differences 175

4.3 Interpolation Error 185

4.4 Application: Muller's Method and Inverse Quadratic Interpolation 190

4.5 Application: More Approximations to the Derivative 194

4.6 Hermite Interpolation 196

4.7 Piecewise Polynomial Interpolation 200

4.8 An Introduction to Splines 207

4.8.1 Definition of the Problem 207

4.8.2 Cubic B-Splines 208

4.9 Tension Splines 223

4.10 Least Squares Concepts in Approximation 229

4.10.1 An Introduction to Data Fitting 229

4.10.2 Least Squares Approximation and Orthogonal Polynomials 233

4.11 Advanced Topics in Interpolation and Approximation 246

4.11.1 Stability of Polynomial Interpolation 247

4.11.2 The Runge Example 249

4.11.3 The Chebyshev Nodes 253

4.11.4 Spectral Interpolation 257

4.12 Literature and Software Discussion 265

References 266

5 Numerical Integration 269

5.1 A Review of the Definite Integral 270

5.2 Improving the Trapezoid Rule 272

5.3 Simpson's Rule and Degree of Precision 277

5.4 The Midpoint Rule 288

5.5 Application: Stirling's Formula 292

5.6 Gaussian Quadrature 294

5.7 Extrapolation Methods 306

5.8 Special Topics in Numerical Integration 313

5.8.1 Romberg Integration 313

5.8.2 Quadrature with Non-smooth Integrands 318

5.8.3 Adaptive Integration 323