Computational mathematics: models, methods and analysis with MATLAB and MPI

Robert E. White9781584883647, 1584883642

Computational science supplements the traditional laboratory and theoretical methods of scientific investigation by providing mathematical models whose solutions can be approximated by computer simulations. By adjusting a model and running more simulations, we gain insight into the application under investigation. 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.

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Table of contents :
COMPUTATIONAL MATHEMATICS: Models, Methods, and Analysis with MATLAB and MPI……Page 2
Contents……Page 5
List of Figures……Page 8
List of Tables……Page 10
Preface……Page 11
Table of Contents……Page 0
Introduction……Page 13
1.1.1 Introduction……Page 15
1.1.2 Applied Area……Page 16
1.1.4 Method……Page 17
1.1.5 Implementation……Page 18
1.1.6 Assessment……Page 20
1.1.7 Exercises……Page 22
1.2.2 Applied Area……Page 23
1.2.3 Model……Page 24
1.2.5 Implementation……Page 26
1.2.6 Assessment……Page 28
1.3.2 Applied Area……Page 31
1.3.3 Model……Page 32
1.3.4 Method……Page 33
1.3.5 Implementation……Page 34
1.3.6 Assessment……Page 35
1.3.7 Exercises……Page 38
1.4.2 Applied Area……Page 39
1.4.3 Model……Page 40
1.4.5 Implementation……Page 42
1.4.6 Assessment……Page 45
1.5.1 Introduction……Page 46
1.5.3 Model……Page 47
1.5.5 Implementation……Page 49
1.5.7 Exercises……Page 55
1.6.3 Model……Page 56
1.6.5 Implementation……Page 57
1.6.6 Assessment……Page 59
1.6.7 Exercises……Page 63
2.1.1 Introduction……Page 64
2.1.2 Applied Area……Page 66
2.1.4 Method……Page 68
2.1.5 Implementation……Page 69
2.1.6 Assessment……Page 71
2.2.1 Introduction……Page 72
2.2.3 Model……Page 75
2.2.4 Method……Page 77
2.2.5 Implementation……Page 78
2.2.6 Assessment……Page 80
2.3.1 Introduction……Page 81
2.3.2 Applied Area……Page 82
2.3.4 Method……Page 84
2.3.5 Implementation……Page 85
2.3.6 Assessment……Page 88
2.3.7 Exercises……Page 89
2.4.3 Model……Page 90
2.4.4 Method……Page 92
2.4.5 Implementation……Page 94
2.4.7 Exercises……Page 98
2.5.2 Vector and Matrix Norms……Page 99
2.5.3 Application to the Cooling Wire……Page 102
2.5.4 Application to Pollutant in a Stream……Page 103
2.6.1 Introduction……Page 104
2.6.3 Model……Page 105
2.6.5 Implementation……Page 106
2.6.6 Assessment……Page 107
2.6.7 Exercises……Page 110
3.1.1 Introduction……Page 111
3.1.3 Model……Page 112
3.1.4 Method……Page 113
3.1.5 Implementation……Page 115
3.1.6 Assessment……Page 116
3.2.2 Applied Area……Page 119
3.2.4 Method……Page 121
3.2.5 Implementation……Page 123
3.2.6 Assessment……Page 124
3.2.7 Exercises……Page 127
3.3.3 Model……Page 128
3.3.4 Method……Page 129
3.3.5 Implementation……Page 130
3.3.7 Exercises……Page 133
3.4.2 Applied Area……Page 134
3.4.3 Model……Page 135
3.4.4 Method……Page 136
3.4.5 Implementation……Page 137
3.4.6 Assessment……Page 140
3.4.7 Exercises……Page 141
3.5.3 Model……Page 142
3.5.4 Method……Page 145
3.5.5 Implementation……Page 147
3.5.6 Assessment……Page 148
3.6.2 Method……Page 150
3.6.3 Implementation……Page 153
3.6.4 Assessment……Page 155
3.6.5 Exercises……Page 156
4.1.1 Introduction……Page 157
4.1.3 Method: Picard……Page 158
4.1.5 Implementation……Page 160
4.1.6 Assessment……Page 162
4.2.2 Applied Area……Page 164
4.2.4 Method……Page 165
4.2.5 Implementation……Page 168
4.2.7 Exercises……Page 170
4.3.3 Model……Page 171
4.3.4 Method……Page 172
4.3.5 Implementation……Page 173
4.3.7 Exercises……Page 177
4.4.3 Model……Page 178
4.4.5 Implementation……Page 180
4.4.7 Exercises……Page 182
4.5.2 Applied Area……Page 183
4.5.4 Method……Page 184
4.5.5 Implementation……Page 185
4.5.7 Exercises……Page 189
4.6.2 Methods via Red-Black Reordering……Page 191
4.6.3 Methods via Domain Decomposition Reordering……Page 194
4.6.4 Implementation of Gaussian Elimination via Domain Decomposition……Page 196
4.6.5 Exercises……Page 198
5.1.1 Introduction……Page 200
5.1.2 Application……Page 201
5.1.3 Model……Page 202
5.1.4 Method……Page 203
5.1.5 Implementation……Page 204
5.1.6 Assessment……Page 206
5.1.7 Exercises……Page 207
5.2.2 Application……Page 208
5.2.4 Method……Page 209
5.2.5 Implementation……Page 211
5.2.6 Assessment……Page 214
5.3.2 Application……Page 215
5.3.3 Model……Page 216
5.3.4 Method……Page 218
5.3.5 Implementation……Page 221
5.3.7 Exercises……Page 223
5.4.2 Application……Page 224
5.4.3 Model……Page 225
5.4.5 Implementation……Page 226
5.4.7 Exercises……Page 229
5.5.1 Introduction……Page 230
5.5.2 Application……Page 231
5.5.3 Model……Page 232
5.5.4 Method……Page 234
5.5.5 Implementation……Page 235
5.5.6 Assessment……Page 238
5.6.3 Model……Page 239
5.6.5 Implementation……Page 241
5.6.7 Exercises……Page 246
6.1.1 Introduction……Page 248
6.1.3 Model……Page 251
6.1.4 Implementation……Page 252
6.1.5 Assessment……Page 254
6.2.2 Applied Area……Page 255
6.2.4 Method……Page 256
6.2.5 Implementation……Page 257
6.2.6 Assessment……Page 259
6.3.1 Introduction……Page 260
6.3.3 Model……Page 262
6.3.4 Method……Page 264
6.3.5 Implementation……Page 265
6.3.7 Exercises……Page 268
6.4.2 IBM/SP Computer……Page 269
6.4.3 Basic MPI……Page 270
6.5.2 Matrix-vector Products……Page 274
6.5.3 Matrix-matrix Products……Page 276
6.6.2 Heat Diffusion in Two Directions……Page 279
6.6.4 Exercises……Page 284
7.1.1 Introduction……Page 286
7.1.2 Syntax for mpi_send() and mpi_recv()……Page 287
7.1.3 First MPI Code……Page 288
7.1.4 Application to Dot Product……Page 291
7.2.1 Syntax for mpi_reduce() and mpi_bcast()……Page 293
7.2.2 Illustrations of mpi_reduce()……Page 294
7.2.3 Illustrations of mpi_bcast()……Page 297
7.3.1 Introduction……Page 299
7.3.3 Illustrations of mpi_scatter()……Page 300
7.3.4 Illustrations of mpi_gather()……Page 302
7.4.2 Count Type……Page 305
7.4.3 Derived Type……Page 307
7.4.4 Packed Type……Page 310
7.5.1 Introduction……Page 312
7.5.2 A Grid Communicator……Page 313
7.5.3 Illustration gridcommpi.f……Page 314
7.6.3 Parallel Fox Algorithm……Page 318
7.6.4 Illustration foxmpi.f……Page 319
7.6.5 Exercises……Page 322
8.1 Gauss Elimination……Page 323
8.2 Symmetric Positive Definite Matrices……Page 328
8.2.1 Exercises……Page 333
8.3 Domain Decomposition and MPI……Page 334
8.4 SOR and P-regular Splittings……Page 338
8.5 SOR and MPI……Page 343
8.5.1 Exercises……Page 348
8.6 Parallel ADI Schemes……Page 349
8.6.1 Exercises……Page 353
9.1 Conjugate Gradient Method……Page 354
9.1.1 Exercises……Page 358
9.2 Preconditioners……Page 359
9.2.1 Exercises……Page 364
9.3 PCG and MPI……Page 365
9.4 Least Squares……Page 369
9.5 GMRES……Page 374
9.6 GMRES(m) and MPI……Page 381
9.6.1 Exercises……Page 385
Bibliography……Page 387