كتاب Introduction to Modeling and Simulation with MATLAB and Python
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منتدى هندسة الإنتاج والتصميم الميكانيكى
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وتسعدونا بالأراء والمساهمات
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أو وإذا كانت هذة زيارتك الأولى للمنتدى فنتشرف بإنضمامك لأسرتنا
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 كتاب Introduction to Modeling and Simulation with MATLAB and Python

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Introduction to Modeling and Simulation with MATLAB and Python
Steven I. Gordon, Brian Guilfoos

كتاب Introduction to Modeling and Simulation with MATLAB and Python  I_t_m_13
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Contents
Preface, xiii
Authors, xvii
Chapter 1 ◾ Introduction to Computational Modeling 1
1.1 THE IMPORTANCE OF COMPUTATIONAL SCIENCE 1
1.2 HOW MODELING HAS CONTRIBUTED
TO ADVANCES IN SCIENCE AND ENGINEERING 3
1.2.1 Some Contemporary Examples 8
1.3 THE MODELING PROCESS 9
1.3.1 Steps in the Modeling Process 11
1.3.2 Mathematical Modeling Terminology and
Approaches to Simulation 14
1.3.3 Modeling and Simulation Terminology 14
1.3.4 Example Applications of Modeling and Simulation 15
EXERCISES 17
REFERENCES 18
Chapter 2 ◾ Introduction to Programming Environments 21
2.1 THE MATLAB® PROGRAMMING ENVIRONMENT 21
2.1.1 The MATLAB® Interface 21
2.1.2 Basic Syntax 23
2.1.2.1 Variables and Operators 23
2.1.2.2 Keywords 25
2.1.2.3 Lists and Arrays 26
2.1.3 Common Functions 28viii ◾ Contents
2.1.4 Program Execution 28
2.1.5 Creating Repeatable Code 29
2.1.6 Debugging 30
2.2 THE PYTHON ENVIRONMENT 30
2.2.1 Recommendations and Installation 30
2.2.2 The Spyder Interface 31
2.2.3 Basic Syntax 32
2.2.3.1 Variables and Operators 32
2.2.3.2 Keywords 34
2.2.3.3 Lists and Arrays 35
2.2.4 Loading Libraries 38
2.2.5 Common Functions 39
2.2.6 Program Execution 40
2.2.7 Creating Repeatable Code 40
2.2.8 Debugging 41
EXERCISES 42
Chapter 3 ◾ Deterministic Linear Models 45
3.1 SELECTING A MATHEMATICAL REPRESENTATION
FOR A MODEL 45
3.2 LINEAR MODELS AND LINEAR EQUATIONS 46
3.3 LINEAR INTERPOLATION 49
3.4 SYSTEMS OF LINEAR EQUATIONS 51
3.5 LIMITATIONS OF LINEAR MODELS 51
EXERCISES 52
REFERENCES 53
Chapter 4 ◾ Array Mathematics in MATLAB® and Python 55
4.1 INTRODUCTION TO ARRAYS AND MATRICES 55
4.2 BRIEF OVERVIEW OF MATRIX MATHEMATICS 56
4.3 MATRIX OPERATIONS IN MATLAB® 58
4.4 MATRIX OPERATIONS IN PYTHON 59
EXERCISES 60Contents    ◾    ix
Chapter 5 ◾ Plotting 61
5.1 PLOTTING IN MATLAB® 61
5.2 PLOTTING IN PYTHON 68
EXERCISES 76
Chapter 6 ◾ Problem Solving 79
6.1 OVERVIEW 79
6.2 BOTTLE FILLING EXAMPLE 80
6.3 TOOLS FOR PROGRAM DEVELOPMENT 81
6.3.1 Pseudocode 82
6.3.2 Top–Down Design 82
6.3.3 Flowcharts 83
6.4 BOTTLE FILLING EXAMPLE CONTINUED 84
EXERCISES 85
Chapter 7 ◾ Conditional Statements 87
7.1 RELATIONAL OPERATORS 87
7.2 LOGICAL OPERATORS 88
7.3 CONDITIONAL STATEMENTS 89
7.3.1 MATLAB® 89
7.3.2 Python 92
EXERCISES 95
Chapter 8 ◾ Iteration and Loops 97
8.1 FOR LOOPS 97
8.1.1 MATLAB® Loops 97
8.1.2 Python Loops 98
8.2 WHILE LOOPS 99
8.2.1 MATLAB® While Loops 99
8.2.2 Python While Loops 99
8.3 CONTROL STATEMENTS 100
8.3.1 Continue 100
8.3.2 Break 100
EXERCISES 100x ◾ Contents
Chapter 9 ◾ Nonlinear and Dynamic Models 101
9.1 MODELING COMPLEX SYSTEMS 101
9.2 SYSTEMS DYNAMICS 101
9.2.1 Components of a System 102
9.2.2 Unconstrained Growth and Decay 104
9.2.2.1 Unconstrained Growth Exercises 106
9.2.3 Constrained Growth 108
9.2.3.1 Constrained Growth Exercise 110
9.3 MODELING PHYSICAL AND SOCIAL PHENOMENA 111
9.3.1 Simple Model of Tossed Ball 112
9.3.2 Extending the Model 113
9.3.2.1 Ball Toss Exercise 114
REFERENCES 115
Chapter 10 ◾ Estimating Models from Empirical Data 117
10.1 USING DATA TO BUILD FORECASTING MODELS 117
10.1.1 Limitations of Empirical Models 118
10.2 FITTING A MATHEMATICAL FUNCTION TO DATA 120
10.2.1 Fitting a Linear Model 122
10.2.2 Linear Models with Multiple Predictors 125
10.2.3 Nonlinear Model Estimation 126
10.2.3.1 Limitations with Linear
Transformation 130
10.2.3.2 Nonlinear Fitting and Regression 130
10.2.3.3 Segmentation 131
EXERCISES 131
FURTHER READINGS 132
REFERENCES 132
Chapter 11 ◾ Stochastic Models 133
11.1 INTRODUCTION 133
11.2 CREATING A STOCHASTIC MODEL 134Contents    ◾    xi
11.3 RANDOM NUMBER GENERATORS IN
MATLAB® AND PYTHON 136
11.4 A SIMPLE CODE EXAMPLE 137
11.5 EXAMPLES OF LARGER SCALE STOCHASTIC
MODELS 139
EXERCISES 142
FURTHER READINGS 143
REFERENCES 143
Chapter 12 ◾ Functions 145
12.1 MATLAB® FUNCTIONS 145
12.2 PYTHON FUNCTIONS 147
12.2.1 Functions Syntax in Python 147
12.2.2 Python Modules 148
EXERCISES 149
Chapter 13 ◾ Verification, Validation, and Errors 151
13.1 INTRODUCTION 151
13.2 ERRORS 152
13.2.1 Absolute and Relative Error 152
13.2.2 Precision 153
13.2.3 Truncation and Rounding Error 153
13.2.4 Violating Numeric Associative and
Distributive Properties 155
13.2.5 Algorithms and Errors 155
13.2.5.1 Euler’s Method 156
13.2.5.2 Runge–Kutta Method 158
13.2.6 ODE Modules in MATLAB®
and Python 159
13.3 VERIFICATION AND VALIDATION 159
13.3.1 History and Definitions 160
13.3.2 Verification Guidelines 162xii ◾ Contents
13.3.3 Validation Guidelines 163
13.3.3.1 Quantitative and Statistical
Validation Measures 164
13.3.3.2 Graphical Methods 166
EXERCISES 166
REFERENCES 167
Chapter 14 ◾ Capstone Projects 169
14.1 INTRODUCTION 169
14.2 PROJECT GOALS 170
14.3 PROJECT DESCRIPTIONS 171
14.3.1 Drug Dosage Model 171
14.3.2 Malaria Model 172
14.3.3 Population Dynamics Model 174
14.3.4 Skydiver Project 176
14.3.5 Sewage Project 178
14.3.6 Empirical Model of Heart Disease Risk Factors 180
14.3.7 Stochastic Model of Traffic 180
14.3.8 Other Project Options 181
REFERENCE 181
INDEX, 183
Index
Note: Page numbers followed by f and t refer to figures and tables, respectively.
2D plotting, 61
command, 61, 68
in MATLAB, 62f
in Python, 69f
tools and functions, 68
2005 Toyota Avalon, design, 9
A
abs() function, 28t, 39t
Absolute error, 152–153, 164
Acceleration (a), 113–114, 176
Algorithms and errors, 155–159
Euler’s method, 156–158
vs. analytic solution, 157f
RK4 method, 158–159
American Society of Mechanical
Engineers (ASME), 161
Anaconda, programming language,
31, 68
Array, 26
lists and, 35–38
MATLAB® and Python, 55–60
arrays and matrices, 55–56
matrix mathematics, 56–58
one-dimensional, 27, 36
Python, 37
two-dimensional, 27, 36
array() function, 37
ASME (American Society of Mechanical
Engineers), 161
Axis function, 68, 75
B
Ball toss exercise, 114–115
Biochemical oxygen demand (BOD), 178
BlenX, programming language, 141
Blood plasma, 171–172
Blue Waters, 7
BOD (biochemical oxygen demand), 178
Break command, 100
Breakpoints, 30, 41
Brownian motion, 141
Built-in functions, 127
MATLAB, 28t
Python, 39t
Business systems model, 102
C
Calculation verification, 161
Capstone projects, 169–181
descriptions, 171–181
drug dosage model, 171–172, 171f
heart disease risk factors, empirical
model, 180
malaria model, 172–174, 173f
options, 181
population dynamics model,
174–176, 174f, 175f
sewage project, 178–179, 178f
skydiver project, 176–177
traffic, stochastic model, 180–181
goals, 170–171
overview, 169–170184 ◾ Index
Carbon dioxide, 107
Carrying capacity (C), 109
Centers for Disease Control and
Prevention, 172, 180
Cmap, 12, 12f
Code verification, 161
Coefficient of determination, 124–125, 165
Coin toss simulation, 138
colon() function, 27
Command Window
MATLAB, 22
programs execution, 29
run command, 30
Spyder, 31
variable, 24
Computational modeling, 1–17
computational science
importance, 1–3
modeling process, 9–17
mathematical modeling
terminology, 14
and simulation terminology,
14–15
steps in, 11–14, 11f
in science and engineering, 3–9
Computational science, 1–3
variables in, 24, 34
Computer power and scientific modeling,
4t, 5t–6t
Concept map/concept-mapping, 12
drug dosage model, 171, 171f
Moose–Wolf population
dynamics, 175f
tools, 12
Conceptual model, 12, 160
Conditional statements, 87–94
logical operators, 88
MATLAB®, 89–92
Python, 92–94
relational operators, 87–88
Constrained growth, 108–111
exercises, 110–111, 111t
Continue command, 100
Continuous model, 15
Control statements, 100
Cosmic rays, 107
Cray-1 supercomputer, 2
D
Def keyword, 147
Del command, 33
Demographer forecasting, 106
Deoxygenation rate, 178
Department of Defense (DoD), 161
Deterministic linear models, 45–52
linear interpolation, 49–51, 50t
linear models/linear equations, 46–49
limitations, 51–52
systems, 51
mathematical representation, 45–46
observe/experiment, 45
screening model, 45
Deterministic model, 14–16, 163
disp() function, 28t
Dissolved oxygen (DO), 178–179
divmod() function, 39t
DO (dissolved oxygen), 178–179
DoD (Department of Defense), 161
Drag coefficient (Cd), 114
Drug dosage model, 171–172
concept map, 171f
Drug screening, 8
Dynamic model, 15–16
nonlinear and, 101–115
E
Empirical data, estimating model, 117–131
build forecasting models, 117–120
limitations, 118–120
fitting mathematical function to data,
120–131
fitting linear model, 122–125
linear models with multiple
predictors, 125–126
nonlinear model estimation,
126–131
ENIAC, computer, 3, 7
Errors, scientific research and modeling,
152–159
absolute and relative, 152–153
algorithms and, 155–159
Euler’s method, 156–158
RK4 method, 158–159Index    ◾    185
numeric associative and distributive
properties, violation, 155
ODE Modules in MATLAB® and
Python, 159
precision, 153
truncation and rounding, 153–155
Euler’s method, 156–158
vs. analytic solution, 157f
Exogenous parameters, 102
Exponential function, 104, 106
Extending model, 113–115
ball toss exercise, 114–115
eye() function, 27, 37
F
F distribution, 124
Figure function, 74
Fitted regression line and residuals, 123f
float() function, 39t
Flowchart(s), 83–84
bottle filling, 85f
if-elif-else, 93, 93f
if-elseif-else-end, 90, 90f
symbols, 84f
tipping, 91f, 94f
Force of drag (Fd), 114
For loop, 97–99
MATLAB®, 97–98
Python, 98–99
Forrester, Jay, 101, 108–109
Free and Open-Source Software (FOSS), 30
Frictional force, 114
Function(s), 145–149
2D plotting tools and, 68
abs(), 28t, 39t
array(), 37
axis, 68, 75
built-in, 127
MATLAB, 28t
Python, 39t
colon(), 27
curve_fit, 130
disp(), 28t
divmod(), 39t
exponential, 104, 106
eye(), 27, 37
figure, 74
float(), 39t
globals(), 39t
legend, 68, 76
linspace(), 27, 37
MATLAB®, 145–147
ones(), 27, 37
open(), 28t, 39t
plot/plotting, 63, 64f, 70, 73f, 74
print(), 39t
Python, 147–149
code reusability, 148
modules, 148–149
syntax, 147–148
variable-length argument lists, 148
title, 68, 75
G
Galaxy formation, 8
Gametocytes, 172–173
Generalized linear model, 130
globals() function, 39t
Guidelines, 162–166
validation, 163–166
graphical methods, 166
quantitative and statistical
validation measures, 164–166
verification, 162–163
H
Healthy villagers, 173
Heart disease risk factors, empirical
model, 180
Helloworld.m file, 22, 29
Hold command, 66, 67f
Household heating system, 102
Human-managed systems, 108
I
IBI (index of biotic integrity), 121, 122f
Identity matrix, 57–58
IDEs. See Integrated development
environments (IDEs)
If-elif-else flowchart, 93, 93f186 ◾ Index
If-elseif-else-end flowchart, 89–90, 90f
If statements, 89
Immune villagers, 173
Index of biotic integrity (IBI), 121, 122f
Industrial Dynamics (book), 101
Integrated development environments
(IDEs), 30
free, 30
Spyder, 31
Intersection conflict time delays and
probabilities, 135t
int() function, 39t
iPhone 5s, 2
IPython, 31, 38, 40, 68, 69f
Isle Royale, 174
K
Keywords, 25, 34
MATLAB, 25–26, 25t
Python, 34–35, 35t
L
Legend function, 68, 76
len() function, 39t
Libby, Willard, 107
Light intensity, 61, 68, 126, 127f
The Limits to Growth (book), 108
Linear equation, 47, 48f, 49
Linear interpolation, 49–51, 50t
Linear model, 47–48
coefficients, 125
fitting, 122–125, 124t
generalized, 130
with multiple predictors, 125–126
spring, 48
standard, 129
Linear regression, 122, 124, 127
Linear transformation
limitations, 130
nonlinear data, 126t
Line specifiers, 63, 70
in MATLAB, 63, 64f
in Python, 70, 71f
linspace() function, 27, 37
Llight, variable, 127
Local variables, 24, 33
Logical operators, 88
Lotka–Volterra equation, 175
Low sampling resolution
in MATLAB, 65f
in Python, 73f
M
Malaria model, 172–174, 173f
Mantissa, 153
Mathematical model(ing), 10, 14, 172
MATLAB®/MATLAB, 23–24
2D plot, 62f
built-in functions, 28t
code, 121
coin toss simulation, 138
Command Window, 22
conditional statements, 89–92
curve_fit functions, 130
curve fitting app, 129f
functions, 145–147
hello world script, 29f
keywords, 25–26
reserved, 25t
linear/nonlinear model, procedures, 128t
line specifiers, 63, 64f
loops, 97–98
low sampling resolution, 65f
mathematic operators, 24t, 25t
matrix operations in, 58–60
plotting in, 61–68, 66f
programming environment, 21–30
basic syntax, 23–28, 24t, 25t
breakpoints, 30
built-in functions, 28t
Command Window, 22
debugging, 30
defined, 21, 23
interface, 21–22, 22f
program execution, 28–29
repeatable code creation, 29–30
reserved keywords, 25t
scalar operation in, 24–25
and Python
array mathematics, 55–60
ODE modules, 159, 159t
random number generators,
136–137, 137tIndex    ◾    187
R2016a, 21
while loops, 99
Matplotlib, 68, 74
Matrix, 27, 36
algebra, 55
identity, 57–58
mathematics, 56–58
in MATLAB®, operations, 58–60
addition/subtraction, 58
multiplication, 57, 59–60
in Python, operations, 59–60
addition, 59
import numpy as np, 59
subtraction, 59
max() function, 28t, 39t
Mind Map Maker, 12, 13f
min() function, 28t, 39t
Mississippi River Basin Model, 10, 10f
Model(ing), 9–17
auto manufacturers, 9
business systems, 102
classification, 14–15
complex systems, 101
computational. See Computational
modeling
computer power and scientific, 4t,
5t–6t
conceptual, 12, 160
continuous, 15
deterministic, 14–16, 163
linear. See Deterministic linear
models
drug dosage, 171–172, 171f
dynamic, 15–16
nonlinear, 101–115
empirical data, estimating.
See Empirical data, estimating
model
generalized linear, 130
heart disease risk factors, empirical,
180
linear, 47–48
coefficients, 125
fitting, 122–125
generalized, 130
with multiple predictors, 125–126
spring, 48
standard, 129
malaria, 172–174, 173f
mathematical, 10, 14
Mississippi River Basin, 10, 10f
molecular, 9
multiple regression, 180
nonlinear and dynamic, 101–115
modeling complex systems, 101
physical and social phenomena,
111–115
systems dynamics, 101–111
one-compartment, 172
physical, 9–10
physical and social phenomena,
111–115
extending model, 113–115
tossed ball, model, 112–113
population dynamics, 174–176, 174f,
175f
predator–prey, 110, 174–175, 174f
initial parameters, 111t
probabilistic, 14
regression, 130
screening, 46
spatial, 102
steady-state, 15
steps in, 11–14, 11f
computer model creation, 13
conceptual model, 12
partial concept map, 12, 12f
partial mind map, 12f, 13
problem analyze and objective,
11–12
simplifying assumptions, 13
stochastic, 17, 133–141
creation, 134–136
definition, 133
larger scale, example of, 139–141
random number generators,
136–134
simple code example, 137–139
traffic, 180–181
Streeter–Phelps, 178
systems dynamics, 16–17
Modeling and simulation (M&S), 8, 161
application, 15–17
benefits, 9
terminology, 14–15
concepts and, 160f188 ◾ Index
mod() function, 28t
Modules, 38
Math and SciPy, 127
Python, 148–149
and MATLAB®, ODE, 159
Molecular modeling, 9
Monte Carlo modeling, 139–140
Moose–Wolf population dynamics, 175, 175f
M&S. See Modeling and simulation (M&S)
Mules, 9
Multiple regression, 126
model, 180
Municipal sewage treatment plants, 178
N
Newton’s second law of motion, 113–114, 176
Nonlinear model
and dynamic, 101–115
modeling complex systems, 101
physical and social phenomena,
111–115
systems dynamics, 101–111
estimation, 126–131, 126t
limitations with linear
transformation, 130
nonlinear fitting/regression, 130–131
segmentation, 131
Numerical errors, 155
Numeric associative and distributive
properties, 155
NumPy, library, 36–37, 137
O
Object-oriented programming, 145
ODE. See Ordinary differential equation
(ODE)
One-compartment model, 172
One-dimensional array, 27, 36
ones() function, 27, 37
open() function, 28t, 39t
Operators
“:”, 27
logical, 88
MATLAB, 24–25, 24t
Python, 34
mathematic, 34t
relational, 87–88
variable and, 23–25, 32–34
Optional starting code, 170–171
Ordinary differential equation (ODE), 158
in MATLAB® and Python, 159
modules, 158
solvers in, 159t
Oxygen sag curve, 178, 179f
P
Partial concept/mind map, 12–13, 12f
Pass by reference, 147
Pass by value, 146
Physical and social phenomena, modeling,
111–115
extending model, 113–115
ball toss exercise, 114–115
tossed ball, model, 112–113
Plot function, 63, 70, 73f, 74
Plotting, 61–76
in MATLAB, 61–68, 62f
2D plot command, 61, 62f
function ploting, 64f
help plot command, 63
hold command, 66, 67f
line specifiers, 63, 64f
low sampling resolution, 65f
multiple curves in single plot, 66f
in Python, 68–76
2D plot, 69f
IPython graphics backend setting,
69f
line specifiers, 70, 71f
matplotlib, 68
multiple curves, 74f
multiple plot commands, 75f
simple plot, 71f
Population dynamics model, 174–176,
174f, 175f
Precision, 153
Predator–prey model, 110, 174
classic, 175
initial parameters, 111t
with Yellowstone National Park, 174f
print() function, 39t
Probabilistic model, 14–15
Problem solving, 79–85Index    ◾    189
bottle filling example, 80–81,
84–85, 85f
overview, 79–80
program development, tools, 81–84
flowchart, 83–84, 84f
pseudocode, 82
top–down design, 82–83
Pseudocode, 82
Python, 92–94, 137
array, 37. See also Array, MATLAB®
and Python
code, 122, 138
conditional statements, 92–94
environment, 30–42
code libraries, 38
debugging, 41–42
defined, 30
keywords, 34–35
libraries, 38–39
lists and arrays, 35–39
mathematic operators, 34t
program execution, 40
recommendations and installation,
30–31
repeatable code creation, 40, 41f
reserved keywords, 35t
Spyder interface, 31–32, 31f
variables and operators, 32–34
functions, 147–149
built-in, 39t
code reusability, 148
modules, 148–149
syntax, 147–148
title, 75
variable-length argument lists, 148
hello world script, 41f
keywords, 34–35
reserved, 35t
line specifiers, 70, 71f
for loop, 98–99
low sampling resolution in, 73f
MATLAB®/MATLAB
ODE modules, 159, 159t
random number generators,
136–137, 137t
matrix operations in, 59–60
addition, 59
import numpy as np, 59
modules, 148–149
operators, 34, 34t
plotting in, 68–76
2D plot, 69f
IPython graphics backend
setting, 69f
line specifiers, 70, 71f
matplotlib, 68
multiple curves, 74f
multiple plot commands, 75f
simple plot, 71f
procedures, 128t
variables, 32–33
while loops, 99
Q
Quantitative and statistical validation
measures, 164–166
R R
2 (R squared), 124
Radiocarbon age, 107
range() function, 39t, 98
Regression model, 130, 166
Relational operators, 87–88
Relative error, 152–153, 164–165
Return keyword, 147
RK4 (Runge–Kutta 4) method, 158–159
RMSE (root mean square error), 165
Rng command, 137
Root mean square error (RMSE), 165
Rounding error, 153–154
R squared (R2), 124
Runge–Kutta 4 (RK4) method, 158–159
S
Scalar variable, 24
Scipy.integrate.odeint, 159, 159t
Scope, 24, 33
Screening model, 46
Segmentation, 131
Sewage project, 178–179, 178f
SIAM (Society for Industrial and Applied
Mathematics), 2
Sick villagers, 173190 ◾ Index
Simulation
coin toss, MATLAB and Python code,
138
computer, 155, 160
defined, 1, 14
Hooke’s Law (HTML5), 49
modeling and, 14–15
applications, 15–17
concepts and terminology, 160f
terminology and approaches,
mathematical, 14
society, 160
Single precision numbers, 153, 155
size() function, 28t
Skydiver project, 176–177
Slices, array, 28, 38
Society for Industrial and Applied
Mathematics (SIAM), 2
Spatial model, 102
Sporozoites, 172–173
Spring constant, 49
Spyder, 30–31
advantage over testing code, 41
Editor, 40
interface, 31–32
default, 31f
stats() functions, 145–146, 148
Steady-state model, 15
Stochastic models, 17, 133–141, 165
creation, 134–136
definition, 133
larger scale, example of, 139–141
MATLAB® and Python, random
number generators, 136–137,
137t
overview, 133–134
simple code example, 137–139
traffic, 180–181
Streeter–Phelps model and formulation,
178–179
Supercomputers, 1–2, 8
Cray-1, 2
Switch-case structures, 91–92
Systems dynamics, 101–111
components, 102–104
constrained growth, 108–111
exercises, 110–111, 111t
models, 16–17
unconstrained growth and decay,
104–108, 105f
exercises, 106–108
T
Thermostat control, 102–103
Tipping flowchart, 91f, 94f
Title function, 68, 75
Top–down design, 82–83
Toxic effect, 172
Traffic control devices, 13, 180
Truck loading data, 47t
Truncation
code, 154
error, 153, 155
rounding, 153–155
Truncation.m/truncation.py program, 154
T test, 124
Tuple, 148
Two-dimensional array, 27, 36
U
Unconstrained growth and decay,
104–108, 105f
exercises, 106–108
Uniform random number scheme, 135
V
Varargin and varargout, variables, 146
Variable(s), 147–148
Command Window, 24
in computational science, 24, 34
Explorer button, 32
homogeneous, 24
llight, 127
local, 24, 33
MATLAB, 23–24
operators and, 23–25, 32–34
Python, 32–33
scalar, 24
varargin and varargout, 146
Vector, 27, 36
Vector-borne disease, 172Index    ◾    191
Velocity, 113
Verification and validation (V&V), 159–166
definition, 152, 160
guidelines, 162–166
graphical methods, 166
quantitative and statistical
validation measures, 164–166
overview, 160–162
Verification, validation, and accreditation
(VV&A), 161
von Neumann, John, 3
V&V. See Verification and validation (V&V)
VV&A (verification, validation, and
accreditation), 161
W
While loops, MATLAB® and
Python, 99
whos() function, 28t
Y
Yellowstone National Park, 174, 174f
Z
Zero-indexed array, 37–38
zeros() function, 27, 37

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