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

This textbook is written primarily for undergraduate mathematicians and also appeals to students working at an advanced level in other disciplines. The text begins with a clear motivation for the study of numerical analysis based on real-world problems. The authors then develop the necessary machinery including iteration, interpolation, boundary-value problems and finite elements. Throughout, the authors keep an eye on the analytical basis for the work and add historical notes on the development of the subject. There are numerous exercises for students "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."--BOOK JACKET. 1 Discrete Time-Space Models 1 -- 1.1 Newton Cooling Models 1 -- 1.2 Heat Diffusion in a Wire 9 -- 1.3 Diffusion in a Wire with Little Insulation 17 -- 1.4 Flow and Decay of a Pollutant in a Stream 25 -- 1.5 Heat and Mass Transfer in Two Directions 32 -- 1.6 Convergence Analysis 42 -- 2 Steady State Discrete Models 51 -- 2.1 Steady State and Triangular Solves 51 -- 2.2 Heat Diffusion and Gauss Elimination 59 -- 2.3 Cooling Fin and Tridiagonal Matrices 68 -- 2.4 Schur Complement 77 -- 2.5 Convergence to Steady State 86 -- 2.6 Convergence to Continuous Model 91 -- 3 Poisson Equation Models 99 -- 3.1 Steady State and Iterative Methods 99 -- 3.2 Heat Transfer in 2D Fin and SOR 107 -- 3.3 Fluid Flow in a 2D Porous Medium 116 -- 3.4 Ideal Fluid Flow 122 -- 3.5 Deformed Membrane and Steepest Descent 130 -- 3.6 Conjugate Gradient Method 138 -- 4 Nonlinear and 3D Models 145 -- 4.1 Nonlinear Problems in One Variable 145 -- 4.2 Nonlinear Heat Transfer in a Wire 152 -- 4.3 Nonlinear Heat Transfer in 2D 159 -- 4.4 Steady State 3D Heat Diffusion 166 -- 4.5 Time Dependent 3D Diffusion 171 -- 4.6 High Performance Computations in 3D 179 -- 5 Epidemics, Images and Money 189 -- 5.1 Epidemics and Dispersion 189 -- 5.2 Epidemic Dispersion in 2D 197 -- 5.3 Image Restoration 204 -- 5.4 Restoration in 2D 213 -- 5.5 Option Contract Models 219 -- 5.6 Black-Scholes Model for Two Assets 228 -- 6 High Performance Computing 237 -- 6.1 Vector Computers and Matrix Products 237 -- 6.2 Vector Computations for Heat Diffusion 244 -- 6.3 Multiprocessors and Mass Transfer 249 -- 6.4 MPI and the IBM/SP 258 -- 6.5 MPI and Matrix Products 263 -- 6.6 MPI and 2D Models 268 -- 7 Message Passing Interface 275 -- 7.1 Basic MPI Subroutines 275 -- 7.2 Reduce and Broadcast 282 -- 7.3 Gather and Scatter 288 -- 7.4 Grouped Data Types 294 -- 7.5 Communicators 301 -- 7.6 Fox Algorithm for AB 307 -- 8 Classical Methods for Ax = d 313 -- 8.1 Gauss Elimination 313 -- 8.2 Symmetric Positive Definite Matrices 318 -- 8.3 Domain Decomposition and MPI 324 -- 8.4 SOR and P-regular Splittings 328 -- 8.5 SOR and MPI 333 -- 8.6 Parallel ADI Schemes 339 -- 9 Krylov Methods for Ax = d 345 -- 9.1 Conjugate Gradient Method 345 -- 9.2 Preconditioners 350 -- 9.3 PCG and MPI 356 -- 9.4 Least Squares 360 -- 9.5 GMRES 365 -- 9.6 GMRES(m) and MPI 372