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Scientific Computing
WITH CASE STUDIES
Scientific Computing
WITH CASE STUDIES

Dianne P. O’Leary
University of Maryland
College Park, Maryland

Society for Industrial and Applied Mathematics

Philadelphia
Copyright © 2009 by the Society for Industrial and Applied Mathematics and the Mathematical
Programming Society

10 9 8 7 6 5 4 3 2 1

All rights reserved. Printed in the United States of America. No part of this book may be repro-
duced, stored, or transmitted in any manner without the written permission of the publisher. For
information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street,
6th Floor, Philadelphia, PA, 19104-2688 USA.

Trademarked names may be used in this book without the inclusion of a trademark symbol.
These names are used in an editorial context only; no infringement of trademark is intended.

MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product information,
please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA, 508-647-
7000, Fax: 508-647-7101, info@mathworks.com, www.mathworks.com.

Mathematica is a registered trademark of Wolfram Research, Inc.

Maple is a registered trademark of Waterloo Maple, Inc.

The images in Figure 1.1 were taken from http://nightglow.gsfc.nasa.gov/eric_journal_files/

sydney_bridge.jpg and http://www.cpsc.gov/cpscpub/prerel/prhtml07/07267a.jpg

Figure 26.1 (http://www.myrmecos.net/insects/Tribolium1.html) is owned by Alex Wild.

Figures 11.1 and 11.2 were taken by Timothy O’Leary.

Library of Congress Cataloging-in-Publication Data

O'Leary, Dianne P.
Scientific computing with case studies / Dianne P. O'Leary.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-898716-66-5
1. Mathematical models--Data processing--Case studies. I. Title.
QA401.O44 2008
510.285--dc22
2008031493

is a registered trademark.
 
To Gene H. Golub, my first research mentor.
To my parents, Raymond and Anne Prost.
To my husband, Timothy.
To my children, Theresa, Thomas, and Brendan.
With love.


November 20, 2008 10:52 sccsbook Sheet number 1 Page number vii cyan magenta yellow black

Contents

Preface xiii

I Preliminaries: Mathematical Modeling, Errors, Hardware, and Software 1

1 Errors and Arithmetic 5

1.1 Sources of Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Computational Science and Scientific Computing . . . . . . . . . . . . 7
1.3 Computer Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 How Errors Propagate . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Mini Case Study: Avoiding Catastrophic Cancellation . . . . . . . . . . 15
1.6 How Errors Are Measured . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 Conditioning and Stability . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Sensitivity Analysis: When a Little Means a Lot 23

2.1 Sensitivity Is Measured by Derivatives. . . . . . . . . . . . . . . . . . . 23
2.2 Condition Numbers Give Bounds on Sensitivity. . . . . . . . . . . . . . 24
2.3 Monte Carlo Experiments Can Estimate Sensitivity. . . . . . . . . . . . 27
2.4 Confidence Intervals Give Insight into Sensitivity . . . . . . . . . . . . 28

3 Computer Memory and Arithmetic: 31

A Look Under the Hood
3.1 A Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Memory Management . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Determining Hardware Parameters . . . . . . . . . . . . . . . . . . . . 34
3.4 Speed of Computer Arithmetic . . . . . . . . . . . . . . . . . . . . . . 36

4 Design of Computer Programs: 39

Writing Your Legacy
4.1 Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Software Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Validation and Debugging . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

vii
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viii Contents

II Dense Matrix Computations 45

5 Matrix Factorizations 49
5.1 Basic Tools for Matrix Manipulation: The B LAS . . . . . . . . . . . . . 50
5.2 The LU and Cholesky Decompositions . . . . . . . . . . . . . . . . . . 52
5.3 The QR Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3.1 QR Decomposition by Givens Rotations . . . . . . . . . . . . 58
5.3.2 QR by Gram–Schmidt Orthogonalization . . . . . . . . . . . . 60
5.3.3 Computing and Using the QR Decomposition . . . . . . . . . . 62
5.3.4 Mini Case Study: Least Squares Data Fitting . . . . . . . . . . 65
5.4 The Rank-Revealing QR Decomposition (RR-QR) . . . . . . . . . . . . 67
5.5 Eigendecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.5.1 Computing the Eigendecomposition . . . . . . . . . . . . . . . 68
5.5.2 Mini Case Study: Stability Analysis of a Linear Control System 71
5.5.3 Other Uses for Eigendecompositions . . . . . . . . . . . . . . 72
5.6 The Singular Value Decomposition (SVD) . . . . . . . . . . . . . . . . 73
5.6.1 Computing and Using the SVD . . . . . . . . . . . . . . . . . 73
5.6.2 Mini Case Study: Solving Ill-Conditioned and Rank-Deficient
Least Squares Problems . . . . . . . . . . . . . . . . . . . . . 74
5.7 Some Matrix Tasks to Avoid . . . . . . . . . . . . . . . . . . . . . . . 76
5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6 Case Study: Image Deblurring: I Can See Clearly Now 81

(coauthored by James G. Nagy)

7 Case Study: Updating and Downdating Matrix Factorizations: 87

A Change in Plans

8 Case Study: The Direction-of-Arrival Problem 97

III Optimization and Data Fitting 105

9 Numerical Methods for Unconstrained Optimization 109

9.1 Fundamentals for Unconstrained Optimization . . . . . . . . . . . . . . 109
9.1.1 How Do We Recognize a Solution? . . . . . . . . . . . . . . . 110
9.1.2 Geometric Conditions for Optimality . . . . . . . . . . . . . . 112
9.1.3 The Basic Minimization Algorithm . . . . . . . . . . . . . . . 113
9.2 The Model Method: Newton . . . . . . . . . . . . . . . . . . . . . . . 114
9.2.1 How Well Does Newton’s Method Work? . . . . . . . . . . . . 116
9.2.2 Making Newton’s Method Safe: Modified Newton Methods . . 117
9.3 Descent Directions and Backtracking Linesearches . . . . . . . . . . . 119
9.4 Trust Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
9.5 Alternatives to Newton’s Method . . . . . . . . . . . . . . . . . . . . . 122
9.5.1 Methods that Require Only First Derivatives . . . . . . . . . . 123
9.5.2 Low-Storage First-Derivative Methods . . . . . . . . . . . . . 126
9.5.3 Methods that Require No Derivatives . . . . . . . . . . . . . . 129
9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
November 20, 2008 10:52 sccsbook Sheet number 3 Page number ix cyan magenta yellow black

Contents ix

10 Numerical Methods for Constrained Optimization 135

10.1 Fundamentals for Constrained Optimization . . . . . . . . . . . . . . . 135
10.1.1 Optimality Conditions for Linear Constraints . . . . . . . . . . 136
10.1.2 Optimality Conditions for the General Case . . . . . . . . . . . 138
10.2 Solving Problems with Bound Constraints . . . . . . . . . . . . . . . . 139
10.3 Solving Problems with Linear Equality Constraints: Feasible Directions 140
10.4 Barrier and Penalty Methods for General Constraints . . . . . . . . . . 141
10.5 Interior-Point Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 144
10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

11 Case Study: Classified Information: 149

The Data Clustering Problem
(coauthored by Nargess Memarsadeghi)

12 Case Study: Achieving a Common Viewpoint: 157

Yaw, Pitch, and Roll
(coauthored by David A. Schug)

13 Case Study: Fitting Exponentials: An Interest in Rates 163

14 Case Study: Blind Deconvolution: Errors, Errors Everywhere 169

15 Case Study: Blind Deconvolution: A Matter of Norm 175

IV Monte Carlo Computations 183

16 Monte Carlo Principles 187

16.1 Random Numbers and Their Generation . . . . . . . . . . . . . . . . . 188
16.2 Properties of Probability Distributions . . . . . . . . . . . . . . . . . . 190
16.3 The World Is Normal . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
16.4 Pseudorandom Numbers and Their Generation . . . . . . . . . . . . . . 192
16.5 Mini Case Study: Testing Random Numbers . . . . . . . . . . . . . . . 193

17 Case Study: Monte Carlo Minimization and Counting: 195

One, Two, Too Many
(coauthored by Isabel Beichl and Francis Sullivan)

18 Case Study: Multidimensional Integration: 203

Partition and Conquer

19 Case Study: Models of Infection: Person to Person 213

V Ordinary Differential Equations 221

20 Solution of Ordinary Differential Equations 225

20.1 Initial Value Problems for Ordinary Differential Equations . . . . . . . 226
20.1.1 Standard Form . . . . . . . . . . . . . . . . . . . . . . . . . . 226
20.1.2 Solution Families and Stability . . . . . . . . . . . . . . . . . 228
November 20, 2008 10:52 sccsbook Sheet number 4 Page number x cyan magenta yellow black

x Contents

20.2 Methods for Solving IVPs for ODEs . . . . . . . . . . . . . . . . . . . . 232

20.2.1 Euler’s Method, Stability, and Error . . . . . . . . . . . . . . . 232
20.2.2 Predictor-Corrector Methods . . . . . . . . . . . . . . . . . . . 237
20.2.3 The Adams Family . . . . . . . . . . . . . . . . . . . . . . . . 239
20.2.4 Some Ingredients in Building a Practical ODE Solver . . . . . . 240
20.2.5 Solving Stiff Problems . . . . . . . . . . . . . . . . . . . . . . 243
20.2.6 An Alternative to Adams Formulas: Runge–Kutta . . . . . . . 243
20.3 Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
20.4 Differential-Algebraic Equations . . . . . . . . . . . . . . . . . . . . . 247
20.4.1 Some Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
20.4.2 Numerical Methods for DAEs . . . . . . . . . . . . . . . . . . 249
20.5 Boundary Value Problems for ODEs . . . . . . . . . . . . . . . . . . . 250
20.5.1 Shooting Methods . . . . . . . . . . . . . . . . . . . . . . . . 253
20.5.2 Finite Difference Methods . . . . . . . . . . . . . . . . . . . . 254
20.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

21 Case Study: More Models of Infection: It’s Epidemic 259

22 Case Study: Robot Control: Swinging Like a Pendulum 265

(coauthored by Yalin E. Sagduyu)

23 Case Study: Finite Differences and Finite Elements 273

Getting to Know You

VI Nonlinear Equations and Continuation Methods 281

24 Nonlinear Systems 285

24.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
24.2 Nonlinear Least Squares Problems . . . . . . . . . . . . . . . . . . . . 287
24.3 Newton-like Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
24.3.1 Newton’s Method for Nonlinear Equations . . . . . . . . . . . 288
24.3.2 Alternatives to Newton’s Method . . . . . . . . . . . . . . . . 289
24.4 Continuation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
24.4.1 The Theory behind Continuation Methods . . . . . . . . . . . 293
24.4.2 Following the Solution Path . . . . . . . . . . . . . . . . . . . 294

25 Case Study: Variable-Geometry Trusses 297

26 Case Study: Beetles, Cannibalism, and Chaos 301

VII Sparse Matrix Computations,

with Application to Partial Differential Equations 307

27 Solving Sparse Linear Systems 311

Taking the Direct Approach
27.1 Storing and Factoring Sparse Matrices . . . . . . . . . . . . . . . . . . 311
27.2 What Matrix Patterns Preserve Sparsity? . . . . . . . . . . . . . . . . . 313
27.3 Representing Sparsity Structure . . . . . . . . . . . . . . . . . . . . . . 314
November 20, 2008 10:52 sccsbook Sheet number 5 Page number xi cyan magenta yellow black

Contents xi

27.4 Some Reordering Strategies for Sparse Symmetric Matrices . . . . . . . 314

27.5 Reordering Strategies for Nonsymmetric Matrices . . . . . . . . . . . . 321

28 Iterative Methods for Linear Systems 323

28.1 Stationary Iterative Methods ( SIMs) . . . . . . . . . . . . . . . . . . . 324
28.2 From SIMs to Krylov Subspace Methods . . . . . . . . . . . . . . . . . 326
28.3 Preconditioning CG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
28.4 Krylov Methods for Symmetric Indefinite Matrices and for Normal
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
28.5 Krylov Methods for Nonsymmetric Matrices . . . . . . . . . . . . . . . 331
28.6 Computing Eigendecompositions and SVDs with Krylov Methods . . . 333

29 Case Study: Elastoplastic Torsion: Twist and Stress 335

30 Case Study: Fast Solvers and Sylvester Equations 341

Both Sides Now

31 Case Study: Eigenvalues: Valuable Principles 347

32 Multigrid Methods: Managing Massive Meshes 353

Bibliography 361

Index 373
November 20, 2008 10:52 sccsbook Sheet number 6 Page number xii cyan magenta yellow black
November 20, 2008 10:52 sccsbook Sheet number 7 Page number xiii cyan magenta yellow black

Preface

A master carpenter does not need to know how her hammer was designed or what
Newton’s laws say about the force that the hammer applies. But she does need to know
how to use the hammer, when to use a ball-peen hammer instead, and what to do when
things go wrong, for example, when a nail bends as it is driven.
We take the same viewpoint in this book. Although there are fascinating stories to
tell in the details of how basic numerical algorithms are designed and how they operate, we
view them as tools in our virtual toolbox, discussing the innards just enough to be able to
master their uses. Instead we focus on how to choose the most appropriate algorithm, how
to make use of it, how to evaluate the results, and what to do when things go wrong.
This viewpoint frees us to explore many diverse applications of our tools, and through
such case studies we practice the analysis and experimentation that are the mainstays of
computational science.
The reader should have background knowledge equivalent to a first course in scien-
tific computing or numerical analysis. Excellent textbooks for learning this information
include those by Michael Heath [71], Cleve Moler [108], and Charles Van Loan [148].
Examples and illustrations use the MATLAB R
programming language. Standard
MATLAB functions provide us with our basic numerical algorithms, and the graphics in-
terface is quite useful. For some problems, we make use of some of the MATLAB tool-
boxes, in particular, the Optimization Toolbox. If you do not have access to MATLAB, the
basic numerical algorithms can also be obtained from N ETLIB and other sources noted in
the text. Sample programs for each case study are available at the website
www.cs.umd.edu/users/oleary/SCCS/
No single book can give a computational scientist all of the background needed for a
career. In fact, computational science is primarily a collaborative enterprise, since it is rare
that a single individual has all of the computational and scientific background necessary
to complete a project. My hope is that this particular slice of knowledge will prove use-
ful in your work and will lead you to further study, exciting applications, and productive
collaborations.
I’m grateful to my many mentors, collaborators, and students, who through their
probing questions forced me to seek deeper understanding and clearer explanations. May
you too be blessed with good colleagues.

xiii
November 20, 2008 10:52 sccsbook Sheet number 8 Page number xiv cyan magenta yellow black

xiv Preface

Notes to Students
This book is written as a textbook for a second course in scientific computing, so it assumes
that you have had a semester (or equivalent) of background using a standard textbook such
as that by Heath [71], Moler [108], Van Loan [148], or equivalent. The Basics box at the
beginning of each unit tells you what part of this material you might want to review in
preparation for the unit. The Mastery box is a checklist of points to master in working
through the unit.
The basic premise behind this book is that people learn by doing. Therefore, the
book is best read with a pencil, paper, and MATLAB window close at hand. Challenges
are sprinkled throughout the text, and they are meant be worked as they are encountered,
or at least before the end of the chapter. Answers are provided for most challenges at
www.cs.umd.edu/users/oleary/SCCS/
There you can see examples of how someone else worked through the challenges. Mastery
will be best if the answers are used to verify and refine your own approach to the problem.
Merely reading the answer, though tempting, is (unfortunately) no substitute for trying to
work the challenge on your own.
Pointers give important information and references to additional literature and soft-
ware. I hope the content of this book leads you to want to learn more about scientific
computing.

Notes to Instructors
The material in this book has been used for a semester and a half in a graduate level course
in the applied mathematics program at the University of Maryland.
• I lecture from the introductory material in each unit, with material from the Case
Studies used to occasionally provide extra information and motivation. Students can
become quite passionate about some of the Case Studies, especially the more visual
ones such as the image deblurring problem (Chapter 6), the data clustering problem
(Chapter 11), and the epidemiology models (Chapter 19 and 21).
• For quizzes and exams, I derive problems from the Mastery points at the beginning
of each unit.
• If possible, I like to allow “laboratory time” in class for students to work on some of
the Challenges. The opportunity to see how other people solve problems is helpful
even to the best students. This is especially true if, as at the University of Maryland,
the students in this course come from backgrounds in mathematics, computer sci-
ence, and engineering. This provides a remarkably diverse set of viewpoints on the
material and enriches the dialog.
• Many of the Case Studies were originally homeworks.
• For a term project, I often ask students to develop a Case Study, using the tools
presented in the course to solve a problem in their application area. Such projects can
then be adapted for use in later terms. My students Nargess Memarsadeghi, David
A. Schug, and Yalin E. Sagduyu developed particularly interesting case studies, and
adapted versions of them are included here.
November 20, 2008 10:52 sccsbook Sheet number 9 Page number xv cyan magenta yellow black

Preface xv

• There are not many unsolved exercises in this book. In the age of the Internet, there
are very few textbook problems for which solutions cannot be found somewhere,
and providing solutions here at least puts all students on equal footing. Some un-
solved exercises and Case Studies are available on the book’s website, and I would
be grateful for your contribution of additional ones to post there.

There is a great deal of flexibility in choice and ordering of units, except that the
optimization unit should be covered before nonlinear equations, and dense matrix compu-
tations should be discussed before optimization. The first six units form the syllabus for
a one semester course at Maryland, while the final one is combined with a textbook in
numerical solution of partial differential equations for the second semester.

Acknowledgments
I am grateful for the help of many, including the following:

• Computing in Science and Engineering, published by the American Institute of Physics

and the IEEE Computer Society, for permission to include chapters derived from the
case studies published there: Chapters
1 (Vol. 8, No. 5, 2006, pp. 86–90),
3 (Vol. 8, No. 3, 2006, pp. 86–89),
4 (Vol. 7, No. 6, 2006, pp. 78–80),
6 (Vol. 5, No. 3, 2003, pp. 82–85),
7 (Vol. 8, No. 2, 2006, pp. 66–70),
8 (Vol. 5, No. 6, 2003, pp. 60–63),
11 (Vol. 5, No. 5, 2003, pp. 54–57),
12 (Vol. 6, No. 5, 2004; pp. 60–62),
13 (Vol. 6, No. 3, 2004, pp. 66–69),
14 (Vol. 7, No. 1, 2005, pp. 56–59),
15 (Vol. 7, No. 2, 2005, pp. 60–62),
17 (Vol. 9, No. 1, 2007, pp. 72–76),
18 (Vol. 6, No. 6, 2004; pp. 58–62),
19 (Vol. 6, No. 1, 2004, pp. 68–70),
21 (Vol. 6, No. 2, 2004, pp. 50–53),
22 (Vol. 5, No. 4, 2003, pp. 68–71),
23 (Vol. 7, No. 3, 2005, pp. 20–23),
26 (Vol. 9, No. 2, 2007, pp. 96–99),
27 (Vol. 7, No. 5, 2005, pp. 62–67),
28 (Vol. 8, No. 4, 2006, pp. 74–78),
29 (Vol. 6, No. 4, 2004, pp. 74–76),
30 (Vol. 7, No. 6, 2005, pp. 74–77),
31 (Vol. 7, No. 4, 2005, pp. 68–70),
32 (Vol. 8, No. 5, 2006, pp. 86–90).

• Jennifer Stout, Lead Editor of Computing in Science and Engineering, who patiently
edited the case studies.

• Mei Huang, for her work on Chapter 18.

November 20, 2008 10:52 sccsbook Sheet number 10 Page number xvi cyan magenta yellow black

xvi Preface

• Jin Hyuk Jung, who as a teaching assistant wrote supplementary lecture notes from
which some of the figures were taken, particularly those in Chapters 5, 9, and 24.
• Nargess Memarsadeghi, David Schug, and Yalin Sagduyu, whose term projects were
so interesting that they led to case studies included here.
• Staff in the Technical Support Department at The MathWorks, for discussions about
the sources of overhead in MATLAB interpreted and compiled instructions.
• James G. Nagy, a master teacher, who inspired the case studies and coauthored the
first one.
• The National Science Foundation and the National Institute of Standards and Tech-
nology, for supporting my research into many of the problems discussed in the case
studies.
• Timothy O’Leary for the photo of Charlie in Chapter 11.
• Students in the University of Maryland courses Scientific Computing I and II: (espe-
cially Samuel Lamphier) for their patience and debugging as the notes were devel-
oped.
• G. W. Stewart, for his example of clearly written textbooks and for the privilege of
being his colleague at Maryland.
• Howard Elman, David Gilsinn, Vadim Kavalerov, Tamara Kolda, Samuel Lamphier,
K.J.R. Liu, Brendan O’Leary, Bert Rust, Simon P. Schurr, Elisa Sotelino, G. W.
Stewart, and Layne T. Watson for helpful comments.
The images in Figure 1.1 were taken from http://nightglow.gsfc.nasa.
gov/eric_journal_files/sydney_bridge.jpg and http://www.cpsc.gov/
cpscpub/prerel/prhtml07/07267a.jpg, and that in Figure 26.1 (http://www.
myrmecos.net/insects/Tribolium1.html) is owned by Alex Wild.
November 20, 2008 10:52 sccsbook Sheet number 11 Page number 1 cyan magenta yellow black

Unit I
Preliminaries:
Mathematical Modeling, Errors,
Hardware and Software
November 20, 2008 10:52 sccsbook Sheet number 12 Page number 2 cyan magenta yellow black
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