Introduction to computational mathematics:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey
World Scientific
2015
|
| Edition: | 2nd edition |
| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (page 319-324) and index |
| Physical Description: | xii, 329 pages illustrations |
| ISBN: | 9789814635776 9814635774 9789814635783 9814635782 |
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Record in the Search Index
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| adam_text | INTRODUCTION TO
Computational
Mathematics
This unique book provides a comprehensive
introduction to computational mathematics,
which forms an essential part of contemporary
numerical algorithms, scientific computing
and optimization. It uses a theorem-free
approach with just the right balance between
mathematics and numerical algorithms.
This edition covers all major topics in
computational mathematics with a wide
range of carefully selected numerical
algorithms, ranging from the root-finding
algorithm, numerical integration, numerical
methods of partial differential equations, finite
element methods, optimization algorithms,
stochastic models, nonlinear curve-fitting to
data modelling, bio-inspired algorithms and
swarm intelligence. This book is especially
suitable for both undergraduates and
graduates in computational mathematics,
numerical algorithms, scientific computing,
mathematical programming, artificial
intelligence and engineering optimization.
Thus, it can be used as a textbook and/or
reference book.
Review of the First Edition
This book is excellent and
can be recommended
for undergraduates and
graduates to acquire rapidly
the basic knowledge in
computational mathematics.
Zentralblatt MATH
I
World Scientific
www.worldscientific.com
9404 he
ISBN 978-981-4635-77-6
789814
635776՛
Contents
Preface v
I Mathematical Foundations 1
1. Mathematical Foundations 3
1.1 The Essence of an Algorithm............................. 3
1.2 Big-O Notations ........................................ 5
1.3 Differentiation and Integration ........................ 6
1.4 Vector and Vector Calculus............................. 10
1.5 Matrices and Matrix Decomposition...................... 15
1.6 Determinant and Inverse................................ 20
1.7 Matrix Exponential..................................... 24
1.8 Hermitian and Quadratic Forms.......................... 26
1.9 Eigenvalues and Eigenvectors........................... 28
1.10 Definiteness of Matrices............................... 31
2. Algorithmic Complexity, Norms and Convexity 33
2.1 Computational Complexity .............................. 33
2.2 NP-Complete Problems .................................. 34
2.3 Vector and Matrix Norms................................ 35
2.4 Distribution of Eigenvalues ........................... 37
2.5 Spectral Radius of Matrices............................ 44
2.6 Hessian Matrix......................................... 47
2.7 Convexity.............................................. 48
vii
viii Introduction to Computational Mathematics
3. Ordinary Differential Equations 51
3.1 Ordinary Differential Equations............................ 51
3.2 First-Order ODEs .......................................... 52
3.3 Higher-Order ODEs ......................................... 53
3.4 Linear System.............................................. 56
3.5 Sturm-Liouville Equation .................................. 58
4. Partial Differential Equations 59
4.1 Partial Differential Equations............................. 59
4.1.1 First-Order Partial Differential Equation.......... 60
4.1.2 Classification of Second-Order Equations........... 61
4.2 Mathematical Models........................................ 61
4.2.1 Parabolic Equation................................. 61
4.2.2 Poisson’s Equation................................. 61
4.2.3 Wave Equation...................................... 62
4.3 Solution Techniques ....................................... 64
4.3.1 Separation of Variables............................ 65
4.3.2 Laplace Transform.................................. 67
4.3.3 Similarity Solution................................ 68
II Numerical Algorithms 71
5. Roots of Nonlinear Equations 73
5.1 Bisection Method........................................... 73
5.2 Simple Iterations.......................................... 75
5.3 Newton’s Method............................................ 76
5.4 Iteration Methods.......................................... 78
5.5 Numerical Oscillations and Chaos........................... 81
6. Numerical Integration 85
6.1 Trapezium Rule............................................. 86
6.2 Simpson’s Rule............................................. 87
6.3 Gaussian Integration....................................... 89
7. Computational Linear Algebra 95
7.1 System of Linear Equations................................. 95
7.2 Gauss Elimination.......................................... 97
Contents ix
7.3 LU Factorization........................................ 101
7.4 Iteration Methods....................................... 103
7.4.1 Jacobi Iteration Method......................... 103
7.4.2 Gauss-Seidel Iteration.......................... 107
7.4.3 Relaxation Method .............................. 108
7.5 Newton֊Raphson Method................................... 109
7.6 QR Decomposition........................................ 110
7.7 Conjugate Gradient Method............................... 115
8. Interpolation 117
8.1 Spline Interpolation.................................... 117
8.1.1 Linear Spline Functions......................... 117
8.1.2 Cubic Spline Functions.......................... 118
8.2 Lagrange Interpolating Polynomials...................... 123
8.3 Bézier Curve ........................................... 125
III Numerical Methods of PDEs 127
9. Finite Difference Methods for ODEs 129
9.1 Integration of ODEs..................................... 129
9.2 Euler Scheme............................................ 130
9.3 Leap-Frog Method........................................ 131
9.4 Runge-Kutta Method...................................... 132
9.5 Shooting Methods ....................................... 134
10. Finite Difference Methods for PDEs 139
10.1 Hyperbolic Equations.................................... 139
10.2 Parabolic Equation...................................... 142
10.3 Elliptical Equation..................................... 143
10.4 Spectral Methods........................................ 146
10.5 Pattern Formation....................................... 148
10.6 Cellular Automata....................................... 150
11. Finite Volume Method 153
11.1 Concept of the Finite Volume............................ 153
11.2 Elliptic Equations...................................... 154
11.3 Parabolic Equations..................................... 155
11.4 Hyperbolic Equations.................................. 156
x Introduction to Computational Mathematics
12. Finite Element Method 157
12.1 Finite Element Formulation........................... 157
12.1.1 Weak Formulation.............................. 157
12.1.2 Galerkin Method............................... 158
12.1.3 Shape Functions............................... 159
12.2 Derivatives and Integration.......................... 163
12.2.1 Derivatives................................... 163
12.2.2 Gauss Quadrature.............................. 164
12.3 Poisson’s Equation................................... 165
12.4 Transient Problems................................... 169
IV Mathematical Programming 171
13. Mathematical Optimization 173
13.1 Optimization......................................... 173
13.2 Optimality Criteria.................................. 175
13.3 Unconstrained Optimization........................... 177
13.3.1 Univariate Functions ......................... 177
13.3.2 Multivariate Functions ....................... 178
13.4 Gradient-Based Methods............................... 180
13.4.1 Newton’s Method............................... 181
13.4.2 Steepest Descent Method....................... 182
14. Mathematical Programming 187
14.1 Linear Programming................................... 187
14.2 Simplex Method ...................................... 189
14.2.1 Basic Procedure............................... 189
14.2.2 Augmented Form ............................... 191
14.2.3 A Case Study ................................. 192
14.3 Nonlinear Programming................................ 196
14.4 Penalty Method....................................... 196
14.5 Lagrange Multipliers................................. 197
14.6 Karush-Kuhn-Tucker Conditions........................ 199
14.7 Sequential Quadratic Programming..................... 200
14.7.1 Quadratic Programming ........................ 200
14.7.2 Sequential Quadratic Programming.............. 200
14.8 No Free Lunch Theorems .............................. 202
Contents xi
V Stochastic Methods and Data Modelling 205
15. Stochastic Models 207
15.1 Random Variables ......................................... 207
15.2 Binomial and Poisson Distributions.........................209
15.3 Gaussian Distribution..................................... 211
15.4 Other Distributions........................................213
15.5 The Central Limit Theorem................................. 215
15.6 Weibull Distribution...................................... 216
16. Data Modelling 221
16.1 Sample Mean and Variance.................................. 221
16.2 Method of Least Squares................................... 223
16.2.1 Maximum Likelihood.................................223
16.2.2 Linear Regression..................................223
16.3 Correlation Coefficient................................... 226
16.4 Linearization............................................. 227
16.5 Generalized Linear Regression............................. 229
16.6 Nonlinear Regression...................................... 233
16.7 Hypothesis Testing.........................................237
16.7.1 Confidence Interval................................237
16.7.2 Student’s ¿-Distribution.......................... 238
16.7.3 Student’s ¿-Test.................................. 240
17. Data Mining, Neural Networks and Support Vector Machine 243
17.1 Clustering Methods.........................................243
17.1.1 Hierarchy Clustering.............................. 243
17.1.2 ¿-Means Clustering Method ........................ 244
17.2 Artificial Neural Networks................................ 247
17.2.1 Artificial Neuron..................................247
17.2.2 Artificial Neural Networks ........................248
17.2.3 Back Propagation Algorithm........................ 250
17.3 Support Vector Machine.................................... 251
17.3.1 Classifications................................... 251
17.3.2 Statistical Learning Theory....................... 252
17.3.3 Linear Support Vector Machine .................... 253
17.3.4 Kernel Functions and Nonlinear SVM.................256
Introduction to Computational Mathematics
xii
18. Random Number Generators and Monte Carlo Method 259
18.1 Linear Congruential Algorithms......................... 259
18.2 Uniform Distribution................................... 260
18.3 Generation of Other Distributions...................... 262
18.4 Metropolis Algorithms.................................. 266
18.5 Monte Carlo Methods.................................... 267
18.6 Monte Carlo Integration................................ 270
18.7 Importance of Sampling................................. 273
18.8 Quasi-Monte Carlo Methods.............................. 275
18.9 Quasi-Random Numbers................................... 276
VI Computational Intelligence 279
19. Evolutionary Computation 281
19.1 Introduction to Evolutionary Computation............... 281
19.2 Simulated Annealing.................................... 282
19.3 Genetic Algorithms..................................... 286
19.3.1 Basic Procedure................................. 287
19.3.2 Choice of Parameters............................ 289
19.4 Differential Evolution................................. 291
20. Swarm Intelligence 295
20.1 Introduction to Swarm Intelligence..................... 295
20.2 Ant and Bee Algorithms................................. 296
20.3 Particle Swarm Optimization............................ 297
20.4 Accelerated PSO........................................ 299
20.5 Binary PSO............................................. 301
21. Swarm Intelligence: New Algorithms 303
21.1 Firefly Algorithm...................................... 303
21.2 Cuckoo Search ......................................... 306
21.3 Bat Algorithm ......................................... 310
21.4 Flower Algorithm....................................... 313
21.5 Other Algorithms....................................... 317
Bibliography 319
Index 325
|
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| discipline | Mathematik |
| edition | 2nd edition |
| format | Book |
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| spelling | Yang, Xin-She 1965- Verfasser (DE-588)1043733906 aut Introduction to computational mathematics Xin-She Yang (Middlesex University London, UK) 2nd edition New Jersey World Scientific 2015 xii, 329 pages illustrations txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references (page 319-324) and index Numerical analysis Algorithms Mathematical analysis Foundations Programming (Mathematics) Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Computermathematik (DE-588)4788218-9 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Computermathematik (DE-588)4788218-9 s Numerische Mathematik (DE-588)4042805-9 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028045420&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028045420&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Yang, Xin-She 1965- Introduction to computational mathematics Numerical analysis Algorithms Mathematical analysis Foundations Programming (Mathematics) Numerische Mathematik (DE-588)4042805-9 gnd Computermathematik (DE-588)4788218-9 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4788218-9 (DE-588)4123623-3 |
| title | Introduction to computational mathematics |
| title_auth | Introduction to computational mathematics |
| title_exact_search | Introduction to computational mathematics |
| title_full | Introduction to computational mathematics Xin-She Yang (Middlesex University London, UK) |
| title_fullStr | Introduction to computational mathematics Xin-She Yang (Middlesex University London, UK) |
| title_full_unstemmed | Introduction to computational mathematics Xin-She Yang (Middlesex University London, UK) |
| title_short | Introduction to computational mathematics |
| title_sort | introduction to computational mathematics |
| topic | Numerical analysis Algorithms Mathematical analysis Foundations Programming (Mathematics) Numerische Mathematik (DE-588)4042805-9 gnd Computermathematik (DE-588)4788218-9 gnd |
| topic_facet | Numerical analysis Algorithms Mathematical analysis Foundations Programming (Mathematics) Numerische Mathematik Computermathematik Lehrbuch |
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