Análisis de libros de Matlab

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© 2004 by Chapman & Hall/CRC

No claim to original U.S. Government works International Standard Book Number 1-58488-364-2 Library of Congress Card Number 2003055207 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

Library of Congress Cataloging-in-Publication Data

White, R. E. (Robert E.) Computational mathematics : models, methods, and analysis with MATLAB and MPI / Robert E. White. p. cm. Includes bibliographical references and index. ISBN 1-58488-364-2 (alk. paper) 1. Numerical analysis. 2. MATLAB. 3. Computer interfaces. 4. Parallel programming (Computer science) I. Title.

QA297.W495 2003 519.4—dc21 2003055207

© 2004 by Chapman & Hall/CRC

Visit the CRC Press Web site at www.crcpress.com

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

R. E. White Department of Mathematics North Carolina State University

Updated on August 3, 2003 To Be Published by CRC Press in 2003

© 2004 by Chapman & Hall/CRC

white@math.ncsu.edu

Contents

List of Figures ix

List of Tables xi

Preface xiii

Introduction xv

1 Discrete Time-Space Models 1 1.1 Newton Cooling Models . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Heat Diusion in a Wire . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Diusion 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 Diusion 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

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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 Diusion . . . . . . . . . . . . . . . . . . . 166 4.5 Time Dependent 3D Diusion . . . . . . . . . . . . . . . . . . . . 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 Diusion . . . . . . . . . . . . . . 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

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