Computational Mathematics: Models, Methods, and Analysis with MATLAB and MPI

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.

Table of Contents:
INTRODUCTION DISCRETE TIME-SPACE MODELS Newton Cooling Models Heat Diffusion in a Wire Diffusion in a Wire with Little Insulation Flow and Decay of a Pollutant in a Stream Heat and Mass Transfer in Two Directions Convergence Analysis STEADY STATE DISCRETE MODELS Steady State and Triangular Solves Heat Diffusion and Gauss Elimination Cooling Fin and Tridiagonal Matrices Schur Complement Convergence to Steady State Convergence to Continuous Model POISSON EQUATION MODELS Steady State and Iterative Methods Heat Transfer in 2D Fin and SOR Fluid Flow in a 2D Porous Medium Ideal Fluid Flow Deformed Membrane and Steepest Descent Conjugate Gradient Method NONLINEAR AND 3D MODELS Nonlinear Problems in One Variable Nonlinear Heat Transfer in a Wire Nonlinear Heat Transfer in 2D Steady State 3D Heat Diffusion Time Dependent 3D Diffusion High Performance Computations in 3D EPIDEMICS, IMAGES AND MONEY Epidemics and Dispersion Epidemic Dispersion in 2D Image Restoration Restoration in 2D Option Contract Models Black-Scholes Model for Two Assets HIGH PERFORMANCE COMPUTING Vector Computers and Matrix Products Vector Computations for Heat Diffusion Multiprocessors and Mass Transfer MPI and IBM/SP MPI and Matrix Products MPI and 2D Models MESSAGE PASSING INTERFACE Basic MPI Subroutines Reduce and Broadcast Gather and Scatter Grouped Data Types Communicators Fox Algorithm for AB CLASSICAL METHODS FOR Ax = D Gauss Elimination Symmetric Positive Definite Matrices Domain Decomposition and MPI SOR and P-Regular Splittings SOR and MPI Parallel ADI Schemes KRYLOV METHODS FOR Ax = D Conjugate Gradient Method Preconditioners PCG and MPI Least Squares GMRES GMRES(m) and MPI BIBLIOGRAPHY INDEX