Numerical Analysis for Applied Science
Autor Myron B Allen, Eli L Isaacsonen Limba Engleză Hardback – 19 mar 2019
Numerical analysis is a core subject in data science and an essential tool for applied mathematicians, engineers, and physical and biological scientists. This updated and expanded edition of Numerical Analysis for Applied Science follows the tradition of its precursor by providing a modern, flexible approach to the theory and practical applications of the field. As before, the authors emphasize the motivation, construction, and practical considerations before presenting rigorous theoretical analysis. This approach allows instructors to adapt the textbook to a spectrum of uses, ranging from one-semester, methods-oriented courses to multi-semester theoretical courses.
The book includes an expanded first chapter reviewing useful tools from analysis and linear algebra. Subsequent chapters include clearly structured expositions covering the motivation, practical considerations, and theory for each class of methods. The book includes over 250 problems exploring practical and theoretical questions and 32 pseudocodes to help students implement the methods. Other notable features include:
- A preface providing advice for instructors on using the text for a single semester course or multiple-semester sequence of courses
- Discussion of topics covered infrequently by other texts at this level, such as multidimensional interpolation, quasi-Newton methods in several variables, multigrid methods, preconditioned conjugate-gradient methods, finite-difference methods for partial differential equations, and an introduction to finite-element theory
- New topics and expanded treatment of existing topics to address developments in the field since publication of the first edition
- More than twice as many computational and theoretical exercises as the first edition.
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Specificații
ISBN-13: 9781119245469
ISBN-10: 111924546X
Pagini: 592
Dimensiuni: 155 x 229 x 33 mm
Greutate: 1.07 kg
Ediția:2nd edition
Editura: Wiley
Locul publicării:Hoboken, United States
ISBN-10: 111924546X
Pagini: 592
Dimensiuni: 155 x 229 x 33 mm
Greutate: 1.07 kg
Ediția:2nd edition
Editura: Wiley
Locul publicării:Hoboken, United States
Notă biografică
Myron B. Allen III, PhD, is Professor of Mathematics in the Department of Mathematics and Statistics at the University of Wyoming, Laramie, USA. His research focuses on the numerical analysis of fluid flows in porous media.
The Late Eli L. Isaacson, PhD, was Professor Emeritus of Mathematics in the Department of Mathematics at the University of Wyoming, Laramie, USA. His work includes analytic and numerical methods for solving systems of hyperbolic conservation laws, including front-tracking methods.
The Late Eli L. Isaacson, PhD, was Professor Emeritus of Mathematics in the Department of Mathematics at the University of Wyoming, Laramie, USA. His work includes analytic and numerical methods for solving systems of hyperbolic conservation laws, including front-tracking methods.
Cuprins
Preface v
1 Some Useful Tools 1
1.1 Introduction 1
1.2 Bounded Sets 4
1.3 Normed Vector Spaces 8
1.4 Eigenvalues and Matrix Norms 19
1.5 Results from Calculus 26
1.6 Problems 33
2 Approximation of Functions 37
2.1 Introduction 37
2.2 Polynomial Interpolation 38
2.3 Piecewise Polynomial Interpolation 48
2.4 Hermite Interpolation 55
2.5 Interpolation in Two Dimensions 63
2.6 Splines 78
2.7 Least-squares Methods 95
2.8 Trigonometric Interpolation 104
2.9 Problems 118
3 Direct Methods for Linear Systems 125
3.1 Introduction 125
3.2 The Condition Number of a Linear System 127
3.3 Gauss Elimination 131
3.4 Variants of Gauss Elimination 148
3.5 Band Matrices 155
3.6 Iterative Improvement 167
3.7 Problems 169
4 Solution of Nonlinear Equations 175
4.1 Introduction 175
4.2 Bisection 179
4.3 Successive Substitution in One Variable 183
4.4 Newton's Method in One Variable 192
4.5 The Secant Method 203
4.6 Successive Substitution: Several Variables 211
4.7 Newton's Method: Several Variables 219
4.8 Problems 233
5 Iterative Methods for Linear Systems 239
5.1 Introduction 239
5.2 Conceptual Foundations 243
5.3 Matrix-Splitting Techniques 248
5.4 Successive Overrelaxation 266
5.5 Multigrid Methods 280
5.6 The Conjugate-Gradient Method 293
5.7 Problems 311
6 Eigenvalue Problems 317
6.1 More About Eigenvalues 318
6.2 Power Methods 323
6.3 The QR Decomposition 328
6.4 The QR Algorithm for Eigenvalues 338
6.5 Singular Value Decomposition 352
6.6 Problems 358
7 Numerical Integration 363
7.1 Introduction 363
7.2 Newton-Cotes Formulas 364
7.3 Romberg and Adaptive Quadrature 373
7.4 Gauss Quadrature 385
7.5 Problems 399
8 Ordinary Differential Equations 403
8.1 Introduction 403
8.2 One-Step Methods 406
8.3 Multistep Methods: Consistency and Stability 420
8.4 Multistep Methods: Convergence 438
8.5 Problems 448
9 Difference Methods for PDEs 453
9.1 Introduction 453
9.2 The Poisson Equation 462
9.3 The Advection Equation 475
9.4 Other Time-Dependent Equations 489
9.5 Problems 505
10 Introduction to Finite Elements 511
10.1 Introduction and Background 511
10.2 A Steady-State Problem 517
10.3 A Transient Problem 537
10.4 Problems 547
A Divided Differences 549
B Local Minima 553
C Chebyshev Polynomials 555
References 559
Index 563
1 Some Useful Tools 1
1.1 Introduction 1
1.2 Bounded Sets 4
1.3 Normed Vector Spaces 8
1.4 Eigenvalues and Matrix Norms 19
1.5 Results from Calculus 26
1.6 Problems 33
2 Approximation of Functions 37
2.1 Introduction 37
2.2 Polynomial Interpolation 38
2.3 Piecewise Polynomial Interpolation 48
2.4 Hermite Interpolation 55
2.5 Interpolation in Two Dimensions 63
2.6 Splines 78
2.7 Least-squares Methods 95
2.8 Trigonometric Interpolation 104
2.9 Problems 118
3 Direct Methods for Linear Systems 125
3.1 Introduction 125
3.2 The Condition Number of a Linear System 127
3.3 Gauss Elimination 131
3.4 Variants of Gauss Elimination 148
3.5 Band Matrices 155
3.6 Iterative Improvement 167
3.7 Problems 169
4 Solution of Nonlinear Equations 175
4.1 Introduction 175
4.2 Bisection 179
4.3 Successive Substitution in One Variable 183
4.4 Newton's Method in One Variable 192
4.5 The Secant Method 203
4.6 Successive Substitution: Several Variables 211
4.7 Newton's Method: Several Variables 219
4.8 Problems 233
5 Iterative Methods for Linear Systems 239
5.1 Introduction 239
5.2 Conceptual Foundations 243
5.3 Matrix-Splitting Techniques 248
5.4 Successive Overrelaxation 266
5.5 Multigrid Methods 280
5.6 The Conjugate-Gradient Method 293
5.7 Problems 311
6 Eigenvalue Problems 317
6.1 More About Eigenvalues 318
6.2 Power Methods 323
6.3 The QR Decomposition 328
6.4 The QR Algorithm for Eigenvalues 338
6.5 Singular Value Decomposition 352
6.6 Problems 358
7 Numerical Integration 363
7.1 Introduction 363
7.2 Newton-Cotes Formulas 364
7.3 Romberg and Adaptive Quadrature 373
7.4 Gauss Quadrature 385
7.5 Problems 399
8 Ordinary Differential Equations 403
8.1 Introduction 403
8.2 One-Step Methods 406
8.3 Multistep Methods: Consistency and Stability 420
8.4 Multistep Methods: Convergence 438
8.5 Problems 448
9 Difference Methods for PDEs 453
9.1 Introduction 453
9.2 The Poisson Equation 462
9.3 The Advection Equation 475
9.4 Other Time-Dependent Equations 489
9.5 Problems 505
10 Introduction to Finite Elements 511
10.1 Introduction and Background 511
10.2 A Steady-State Problem 517
10.3 A Transient Problem 537
10.4 Problems 547
A Divided Differences 549
B Local Minima 553
C Chebyshev Polynomials 555
References 559
Index 563