Computational Mathematics: Models, Methods, and Analysis with MATLAB and MPI explores and illustrates this process. Each section of the first six chapters is motivated by a specific application. The author applies a model, selects a numerical method, implements computer simulations, and assesses the ensuing results. These chapters include an abundance of MATLAB code. By studying the code instead of using it as a "black box, " you take the first step toward more sophisticated numerical modeling. The last four chapters focus on multiprocessing algorithms implemented using message passing interface (MPI). These chapters include Fortran 9x codes that illustrate the basic MPI subroutines and revisit the applications of the previous chapters from a parallel implementation perspective. All of the codes are available for download from www4.ncsu.edu./~white.This book is not just about math, not just about computing, and not just about applications, but about all three--in other words, computational science. Whether used as an undergraduate textbook, for self-study, or for reference, it builds the foundation you need to make numerical modeling and simulation integral parts of your investigational toolbox.
Author(s): Robert E. White
Edition: 1
Publisher: Chapman and Hall\/CRC
Year: 2003
Language: English
Pages: 402
Cover......Page 1
Computational Mathematics: Models, Methods, and Analysis with MATLAB and MPI......Page 2
Copyright......Page 3
Contents......Page 6
List of Figures......Page 10
List of Tables......Page 12
Preface......Page 14
Introduction......Page 16
1.1 Newton Cooling Models......Page 18
1.2 Heat Diffusion in a Wire......Page 26
1.3 Diffusion in a Wire with Little Insulation......Page 34
1.4 Flow and Decay of a Pollutant in a Stream......Page 42
1.5 Heat and Mass Transfer in Two Directions......Page 49
1.6 Convergence Analysis......Page 59
2.1 Steady State and Triangular Solves......Page 68
2.2 Heat Diffusion and Gauss Elimination......Page 76
2.3 Cooling Fin and Tridiagonal Matrices......Page 85
2.4 Schur Complement......Page 94
2.5 Convergence to Steady State......Page 103
2.6 Convergence to Continuous Model......Page 108
3.1 Steady State and Iterative Methods......Page 116
3.2 Heat Transfer in 2D Fin and SOR......Page 124
3.3 Fluid Flow in a 2D Porous Medium......Page 133
3.4 Ideal Fluid Flow......Page 139
3.5 Deformed Membrane and Steepest Descent......Page 147
3.6 Conjugate Gradient Method......Page 155
4.1 Nonlinear Problems in One Variable......Page 162
4.2 Nonlinear Heat Transfer in aWire......Page 169
4.3 Nonlinear Heat Transfer in 2D......Page 176
4.4 Steady State 3D Heat Diffusion......Page 183
4.5 Time Dependent 3D Diffusion......Page 188
4.6 High Performance Computations in 3D......Page 196
5.1 Epidemics and Dispersion......Page 206
5.2 Epidemic Dispersion in 2D......Page 214
5.3 Image Restoration......Page 221
5.4 Restoration in 2D......Page 230
5.5 Option Contract Models......Page 236
5.6 Black-Scholes Model for Two Assets......Page 245
6.1 Vector Computers and Matrix Products......Page 254
6.2 Vector Computations for Heat Diffusion......Page 261
6.3 Multiprocessors and Mass Transfer......Page 266
6.4 MPI and the IBM/SP......Page 275
6.5 MPI and Matrix Products......Page 280
6.6 MPI and 2DModels......Page 285
7.1 Basic MPI Subroutines......Page 292
7.2 Reduce and Broadcast......Page 299
7.3 Gather and Scatter......Page 305
7.4 Grouped Data Types......Page 311
7.5 Communicators......Page 318
7.6 Fox Algorithm for AB......Page 324
8.1 Gauss Elimination......Page 330
8.2 Symmetric Positive Definite Matrices......Page 335
8.3 Domain Decomposition and MPI......Page 341
8.4 SOR and P-regular Splittings......Page 345
8.5 SOR and MPI......Page 350
8.6 Parallel ADI Schemes......Page 356
9.1 Conjugate Gradient Method......Page 362
9.2 Preconditioners......Page 367
9.3 PCG and MPI......Page 373
9.4 Least Squares......Page 377
9.5 GMRES......Page 382
9.6 GMRES(m) and MPI......Page 389
Bibliography......Page 396
Index......Page 398
