An introduction to computational science

✍ Scribed by Holder A., Eichholz J

• Publisher: Springer
• Year: 2019
• Tongue: English
• Leaves: 475
• Category: Library

✦ Table of Contents

Foreword......Page 7
Preface......Page 9
Acknowledgments......Page 11
Contents......Page 12
Part I Computational Methods......Page 16
1 Solving Single Variable Equations......Page 17
1.1 Bisection......Page 18
1.2 Linear Interpolation......Page 21
1.3 The Method of Secants......Page 22
1.4 Newton's Method......Page 27
1.4.1 Improving Efficiency with Polynomials......Page 29
1.4.2 Convergence of Newton's Method......Page 31
1.4.3 MATLAB Functions......Page 34
1.5 Exercises......Page 35
2 Solving Systems of Equations......Page 46
2.1.1 Upper- and Lower-Triangular Linear Systems......Page 47
2.1.2 General mn Linear Systems......Page 49
2.2 Special Structure: Positive Definite Systems......Page 57
2.2.1 Cholesky Factorization......Page 60
2.2.2 The Method of Conjugate Directions......Page 62
2.3 Newton's Method for Systems of Equations......Page 65
2.4 MATLAB Functions......Page 68
2.5 Exercises......Page 71
3.1 Linear Models and the Method of Least Squares......Page 79
3.1.1 Lagrange Polynomials: An Exact Fit......Page 84
3.2 Linear Regression: A Statistical Perspective......Page 86
3.2.1 Random Variables......Page 87
3.2.2 Stochastic Analysis and Regression......Page 95
3.3 Cubic Splines......Page 110
3.4 Principal Component Analysis......Page 115
3.4.1 Principal Component Analysis and the Singular Value Decomposition......Page 123
3.5 Exercises......Page 124
4.1 Unconstrained Optimization......Page 134
4.1.1 The Search Direction......Page 138
4.1.2 The Line Search......Page 142
4.1.3 Example Algorithms......Page 143
4.2 Constrained Optimization......Page 147
4.2.1 Linear and Quadratic Programming......Page 162
4.3 Global Optimization and Heuristics......Page 175
4.3.1 Simulated Annealing......Page 176
4.3.2 Genetic Algorithms......Page 182
4.4 Exercises......Page 188
5 Ordinary Differential Equations......Page 201
5.1 Euler Methods......Page 202
5.2 Runge-Kutta Methods......Page 208
5.3 Quantifying Error......Page 212
5.4 Stiff Ordinary Differential Equations......Page 217
5.5 Adaptive Methods......Page 221
5.6 Exercises......Page 229
6 Stochastic Methods and Simulation......Page 246
6.1 Simulation......Page 247
6.2 Numerical Integration......Page 248
6.2.1 Simpson's Rule......Page 250
6.2.2 Monte Carlo Integration......Page 254
6.2.3 Bootstrapping......Page 258
6.2.4 Deterministic or Stochastic Approximation......Page 260
6.3 Random Models......Page 263
6.3.1 Simulation and Stochastic Differential Equations......Page 264
6.3.2 Simulation and Stochastic Optimization Models......Page 268
6.4 Exercises......Page 271
7 Computing Considerations......Page 278
7.1 Language Choice......Page 279
7.2 C/C++ Extensions......Page 280
7.3 Parallel Computing......Page 285
7.3.1 Taking Advantage of Built-In Commands......Page 286
7.3.2 Parallel Computing in MATLAB and Python......Page 287
7.3.3 Parallel Computing in Python......Page 293
7.3.4 Pipelining......Page 295
7.3.5 Ahmdal's Law......Page 300
7.3.6 GPU Computing......Page 301
7.4 Exercises......Page 305
Part II Computational Modeling......Page 313
8 Modeling with Matrices......Page 317
8.1 Signal Processing and the Discrete Fourier Transform......Page 318
8.1.1 Linear Time Invariant Filters......Page 319
8.1.2 The Discrete Fourier Transform......Page 323
8.1.3 The Fast Fourier Transform......Page 328
8.1.4 Filtering Signals......Page 332
8.1.5 Exercises......Page 333
8.2.1 A Radiobiological Model to Calculate Dose......Page 334
8.2.2 Treatment Design......Page 340
8.2.3 Exercises......Page 346
8.3 Aeronautic Lift......Page 347
8.3.1 Air Flow......Page 348
8.3.2 Flow Around a Wing......Page 351
8.3.3 Numerical Examples......Page 356
8.3.4 Exercises......Page 361
9.1 Couette Flows......Page 362
9.1.1 Exercises......Page 369
9.2 Pharmacokinetics: Insulin Injections......Page 370
9.2.1 Exercises......Page 377
9.3 Chemical Reactions......Page 378
9.3.1 Exercises......Page 382
10 Modeling with Delay Differential Equations......Page 384
10.1 Is a Delay Model Necessary or Appropriate?......Page 385
10.2 Epidemiology Models......Page 386
10.3 The El-Niño–La-Niña Oscillation......Page 390
10.4 Exercises......Page 392
11.1 The Heat Equation......Page 395
11.2 Explicit Solutions by Finite Differences......Page 398
11.3 The Wave Equation......Page 403
11.4 Exercises......Page 407
12.1 Stock Pricing and Portfolio Selection......Page 409
12.1.1 Stock Pricing......Page 410
12.1.2 Portfolio Selection......Page 414
12.1.3 Exercises......Page 421
12.2 Magnetic Phase Transitions......Page 424
12.2.1 The Gibbs Distribution of Statistical Mechanics......Page 425
12.2.2 Simulation and the Ising Model......Page 427
12.2.3 Exercises......Page 434
13 Regression Modeling......Page 437
13.1 Stepwise Regression......Page 441
13.2 Qualitative Inputs and Indicator Variables......Page 443
13.3 Exercises......Page 447
A.1 Matrix Algebra Motivated with Polynomial Approximation......Page 452
A.2 Properties of Matrix–Matrix Multiplication......Page 458
A.3 Solving Systems, Eigenvalues, and Differential Equations......Page 460
A.3.1 The Nature of Solutions to Linear Systems......Page 461
A.3.2 Eigenvalues and Eigenvectors......Page 465
A.4 Some Additional Calculus......Page 468
Index......Page 470