# An Introduction To Technical Problem Solving

Document Sample

```					An Introduction To
Technical Problem Solving
with MATLAB v.7
Jon Sticklen, PhD
M. Taner Eskil, PhD
Introduction
Chapter 1
Introduction

What is Technical Problem Solving?
The Path to Becoming a Good Technical
Problem Solver
What You Must Do to Master the Material

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1-1 What is Technical Problem
Solving?
Good common sense applied to technical
problems
Quantitative in nature
Basis for making many decisions
Rooted in numerical calculations
Decompose the problem
Mainstay of what an engineer does
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1-2 The Path to Becoming a
Good Technical Problem Solver
Master the conceptual subject matter of a
given technical area
Demonstrate what you have learned
Master the tools of the trade
MATLAB
Mathematica

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1-3 What You Must Do to Master
the Material
Be prepared to spend substantial time
learning this material
Read assigned sections of the text with
Work the problems hands-on

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A Framework for
Technical Problem Solving
Chapter 2
A Framework for Technical
Problem Solving

Steps in a Framework for Technical
Problem Solving
An Example Using the Framework

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2-1 Steps in a Framework for
Technical Problem Solving
Step 1: Refine and Structure
Arrive at a precise problem statement
Give the problem initial structure:
Problem inputs
Computational output
Step 2: Make a sketch or diagram
Visualize the physical situation

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2-1 Steps in a Framework for
Technical Problem Solving
Step 3: Assemble and Organize
What do you need to know to solve the
problem?
Get the needed information:
Internet search engines such as Yahoo or Google
Library
Step 4: Simplify
Find if there are suitable approximations
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2-1 Steps in a Framework for
Technical Problem Solving
Step 5: Decompose
Simpler problems
Reduce the complexity of the solution
Step 6: Dimensional Analysis
Are the mathematical relationships you intend
to apply in your solution flawed?
Substitute the units for each variable and
algebraically simplify
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2-1 Steps in a Framework for
Technical Problem Solving
Step 7: Compute and Discuss
Perform the computations needed to obtain a
solution
Use a computational tool such as MATLAB
Examine and understand the results

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2-2 An Example Using the
Framework
Problem:
What is the optimum firing angle we should set
for a catapult whose purpose is to hurl hay
bales to a herd of starving caribou, given the
initial velocity of hay bales as they exit the
catapult?

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2-2.1 Refine and Structure
Clarify:
The ultimate source of information is the person setting
the problem
Refine:
What is the firing angle of a catapult in order to hurl
projectiles a maximum horizontal distance given the
initial velocity of the object?
What are the Input(s) and Output(s)?
Initial speed, angle, maximized distance

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2-2.2 Sketch or Diagram
What is the firing angle measured from the
horizontal we should set for a catapult to hurl
projectiles a maximum horizontal distance given
the initial velocity of the object?

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2-2.3 Background Knowledge
For our example:
General knowledge of physics
General knowledge of one way in which
optimizing problems may be solved
Seek the value of the firing angle f that
maximizes the total horizontal distance D
traveled given an initial speed of the hay
bale as it comes out of the catapult
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2-2.4 Assumptions and
Approximations
Many times in technical problem solving, the path
the problem and then solving the simplified
version of the problem.
Negligible air resistance
No difference between a hay bale and a snowball
Be aware of the assumptions you are making and
communicate them to the person who originally
set the problem for you.
Importance of good documentation
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2-2.5 Decomposing /Recursive
Structuring
Break the problem into pieces, each with its
own well defined input and output
Develop technical solutions for each piece
Work backwards to find what you need

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2-2.6 Dimensional Analysis

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2-2.7 Putting It All Together
f (degrees)               0          10         20        30           40
Unit Distance (feet)       0.0000    0.0107     0.0201    0.0271       0.0308
f (degrees)               90         80         70        60           50
Unit Distance (feet)      0.0000     0.0107     0.0201    0.0271       0.0308

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2-2 Synopsis

Identify input and output variables correctly
Create a sketch of the physical situation
Generalize the problem
Apply general background knowledge
General approach to optimization problems

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MATLAB Basics: Scalars
Chapter 3
Scalars

The First Time You Bring Up MATLAB
MATLAB as a Calculator for Scalars
Fetching and Setting Scalar Variables
MATLAB Built-in Functions, Operators,
and Expressions
Problem Sets for Scalars

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3-1 The First Time You Bring Up
MATLAB
Basic windows in MATLAB are:
Command - executes single-line commands
Workspace - keeps track of all defined variables
Command History - keeps a running record of all single
line programs you have executed
Current Folder - lists all files that are directly available for
MATLAB use
Array Editor - allows direct editing of MATLAB arrays
Preferences - for setting preferences for the display of
results, fonts used, and many other aspects of how
MATLAB looks to you

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3-2 MATLAB as a Calculator for
Scalars
A scalar is simply a number…

In science the term scalar is used as opposed to a vector,
i.e. a magnitude having no direction.
In MATLAB, scalar is used as opposed to arrays, i.e. a
single number.
Since we have not covered arrays (tables of numbers)
yet, we will be dealing with scalars in MATLAB.

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Using the Command History
Window

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3-3 Fetching and Setting Scalar
Variables
Think of computer
variables as named
containers.
We can perform 2
types of operations
on variables:

we can set the value held in the container: x = 22
we can look at the value held in the container: x
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The Assignment Operator (=)
The equal sign is the assignment operator in
MATLAB.
>> x = 22
places number 22 in container x
>> x = x + 1
mathematics and the assignment operator in
MATLAB!
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3-3.2 An Example - Setting
Variables for the Hay Bale Problem
>> clear
>> accelGravity = 32; % units: ft/sec/sec
>> speedInitial = 50; % units: ft/sec
>> phi = 10; % units: degrees

clear deletes variables from workspace, you
should use it before starting new work
The percent symbol is used for putting reminders
(comments) for ourselves, ignored by MATLAB
Command lines ending with semicolon do not
display the results.
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3-4 MATLAB Built-in Functions,
Operators, and Expressions
MATLAB comes with a large number of built-
in functions (e.g.. sin, cos, tan, log10, log, exp)
A special subclass of often-used MATLAB
functions is called operators
Assignment operator (=)
Arithmetic operators (+, -, *, /, ^)
Relational operators (<, <=, = =, ~=, >=, >)
Logical operators (&, |, ~)

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Example – Arithmetic Operators

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Example – Relational and
Logical Operators

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3-4.2 Rules for Forming
Expressions
MATLAB expressions consist of:
Numerical values or variables
Logical values or variables
Legal applications of MATLAB functions or
operators
A combination of MATLAB expressions
What is the error in the following MATLAB
expression?
>> sin(pi/2, pi/8)
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Order of Precedence
If two operators are at
the same level of
precedence, the
evaluation is carried out
from left to right
An Example – Compute

for x = 4, y = 2
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Applying Scalar Computations to
a Problem

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Synopsis for Chapter 3
A MATLAB variable can be thought of as a named container. The value of a
MATLAB variable then is the contents of the box.
Setting the variable value is done using the assignment operator; fetching the
value of a variable is done by typing the name of the variable.
In MATLAB the type of a variable is defined by the way the variable is used.
Scalars are simple numbers.
Logical values can be TRUE (any non-zero number) or FALSE (0).
MATLAB operators are a special subclass of MATLAB built-in functions.
Arithmetic operators take numerical variables as input and output a numerical
result.
Relational operators take numerical variables as input and output a logical
result.
Logical operators take logical variables as input and output a logical result.
MATLAB expressions are blueprints for performing computations.
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Saving MATLAB Work
Chapter 4
Saving MATLAB Work

The MATLAB “Current Directory”
Saving MATLAB Commands in Script
Files
Saving MATLAB Commands in User-
Defined Function Files
Testing and Debugging MATLAB Script
and Function Files
Problem Sets for “Saving Your Work”
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4-1 The MATLAB Current
Directory
MATLAB starts
up with the
default folder
connected
It is the default folder MATLAB will save files
It is the first folder MATLAB will attempt to load files
It may be changed interactively using the Current
Directory Window or built-in commands

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4-2 Saving MATLAB Commands
in Script Files
Script files are lines of code just like you
would type in to the Command Window
It is good programming practice to include
The location of the file
Variables used in the script but defined outside
Results produced by the script
Units of values calculated in the script
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An Example – The Water Tower
Problem
compute water tower cost
and volume
cylinder caped by
hemisphere
diameter and height of
cylinder known
known cost/m2 for
hemisphere and for
cylinder

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Synopsis for MATLAB Scripts
A script file consists of groups of MATLAB
commands bundled together into a module
Scripts files have a DOT-M extension
When executing inside a script file, all
variables in the workspace are available
Variables created in a script are available at
the Command Window and in other scripts
Making a sketch of a problem is important
in making the problem context concrete
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4-3 Saving MATLAB Commands
in User-Defined Function Files
Scripts - these are lines of code exactly like
you could type in to the command window
Functions - are “computational boxes.” You
give them a set of input values, and they
calculate a set of output values. Purpose of
functions is the same as the purpose of scripts
+
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Central Points…
The only way of getting a variable‟s value into a
function is for that variable to be input to the
function.
The only way of getting a value out of a function is
for that variable to be output from the function.
A variable used inside a function that is not an input
or an output variable is not visible outside the
function.
An example:
>> x = pi;
>> y = sin(x)
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Syntax of a User Defined Function

Functions are saved in DOT M files - just
like the script files
Same rules for the connected directory
applies
First line of the file defines
Name of the function
Input variable(s)
Output variable(s)
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Defining a function – the first line
function <outputVars> = <function name>(<inputVars>)

0, 1, or more vars. If 0,
0, 1, or more vars. If                     then the enclosing parens
more than 1, put in                        are not needed.
square brackets.
required

You choose the name - see text or
ML help for valid name - mostly
exactly as appears                 character and have no spaces.

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Formal Parameters
...
calling program:
z = myFun3(a,b)

...
the function called:
function out33 = myFun3(x,y)

When the function myFun3 is “called”…
1. The formal input variables (x,y) take the values given in the
calling line (a,b)
2. The function “runs”
3. The output variables in the function are given back to the calling
program’s variable.
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There are no uniformly agreed upon rules for
It is always good programming practice to include
comment lines indicating:
the purpose of the function
the inputs to the function
the outputs from the function
any assumptions

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Examples
function <outputVars> = <function name>(<inputVars>)

For each of the following function definitions,
how many input and output variables are
there?
function x = myFun1
function z = myFun2(y)
function out33 = myFun3(x,y)
function [a,b] = myFun4(q,r)
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Examples

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Examples

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Synopsis for User-Defined
Functions
The workspace of a function is insulated from the outside.
between the actual parameters in the call and the formal
parameters in the function definition.
The number of actual input parameters must be the same as
the number of formal input parameters. This is also valid
for the output parameters.
The first line of a MATLAB function begins with the
keyword function, and the rest of the first line looks like an
assignment statement.
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4-4 Testing and Debugging
MATLAB Script and Function Files
Types of errors in programs:
Syntax errors
Results from incorrect application of MATLAB rules
MATLAB aborts the computation and points of the
error
Runtime errors
Results from incorrect logic and MATLAB not doing
what you intend
For this type error MATLAB debugging facilities are
useful
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Examples

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Examples

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Synopsis for Debugging
Identify the set of inputs you will use for the test
Determine what you expect for each of the test
input sets to produce
Compare what you expect to what MATLAB
produces to identify runtime errors in the function
Identify and correct the line(s) of code that are
causing the runtime error

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Vector Operations
Chapter 5
Vector Operations

Vector Creation
Accessing Vector Elements
Row Vectors and Column Vectors, and the
Transpose Operator
Vector Built-in Functions, Operators, and
Expressions

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5-1 Vector Creation
Vectors are defined in square brackets;
temperaturesMonThu = [32 31 29 33];
temperaturesFriSat = [35 33];
You can concatenate a vector with a scalar;
temperaturesFriSun = [35 33 27];
or concatenate 2 vectors;
weeklyTemperatures = [temperaturesMonThu, temperaturesFriSun];
To find the size of a vector, we use length;
numTemperatures = length(weeklyTemperatures);
We could find the average temperature by typing;
avgTemperature = mean(dailyTemperatures)
or by using sum and length;
totalTemperature = sum(dailyTemperatures)/length(dailyTemperatures);
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Some Useful Vector Functions
brackets (e.g. [27 36 41]): Creates vectors.
colon operator (e.g. [0:5:30]): Creates linearly spaced vectors.
linspace (e.g. linspace(0,100,21)): Creates linearly spaced vectors.
length (e.g. length([0:5:30])): Finds the length of a vector.
zeros (e.g. zeros(1,5)): Creates vectors filled with zeroes.
ones (e.g. ones(1,5): Creates vectors filled with ones.
sum (e.g. sum([5 3 6 2])): Sums up the contents of a vector.
sort (e.g. sort([5 3 6 2])): Sorts the contents of a vector.
mean (e.g. mean([5 3 6 2])): Finds the average of contents.

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5.2 – Accessing Vector Elements –
Examples
1. Create a row vector x consisting of the numbers in the ordered
set: {1 4 7 10} using the colon operator.
x = [1:3:10]

2. Set a variable y to be the length of x.
y = length(x)

3. Set variable y to be the 1st element of x.
y = x(1)

4. Set variable y to be the 1st, 2nd, and 3rd elements of x.
y = x([1,2,3]) OR y = x(1:3)

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Accessing Vector Elements –
Example cont’d.
5.   Set variable y to be the 3rd through the last element of x - and do so
such that your solution works no matter how long x is.
y = x(3:end)

6.   Set variable y to be the next-to-last and last element of x - and do so
such that your solution works no matter how long x is.
y = x([end-1,end])

7.   Change the 2nd element of x to be 3.
x(2) = 3

8.   Change the 2nd element of x to be 102 and the 4th element of x be
205.
x([2,4]) = [102, 205]

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Synopsis for Fetching and Setting
Elements in Vectors
Accessing element(s) in a vector is done by indexing into
the vector.
To delete element(s) in a vector, empty square brackets are
used.
To find the length of a vector V, use the length built-in
function length(V).
When setting elements of a vector, the number of elements
being set must be equal to the number of elements in the
vector on the right hand side of the assignment operation.
The exception is that a scalar on the right-hand side can be
used to set multiple vector elements.
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5-3 Row Vectors and Column
Vectors, and the Transpose Operator
Row and column vectors are represented as single
rows and columns of values, respectively.
When creating a column vector with square
brackets, you may use the semicolon operator:
temp = [35; 33; 27];
or you may use the transpose operator;
temp = [35 33 27]‟;
When creating an equally spaced column vector,
you need to use the transpose operator;
springConstants = [10:10:100]‟;
springConstants = linspace(10,100,10)‟;
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5-4 Vector Built-in Functions,
Operators, and Expressions

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Sample Problem – Vector Built-
in Functions

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Sample Problem – Vector
Arithmetic Operators

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Sample Problem – Vector
Arithmetic Operators

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Sample Problem – Vector
Arithmetic Operators

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Vector Relational Operators

Think of them as comparing numbers…
<, >, = =, >=, <=
A relational operator can be used to
compare the values of two variables
a>b
But… remember MATLAB is for matrices
what are you testing?
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What are you testing?
Number (scalar) vs. Number
Number vs. Vector (or Matrix)
A scalar is compared to each element of the vector…
5<[1:10]
Vector vs. Vector
Each corresponding element of the two vectors is
compared…
[1:10]<=[10:-1:1]

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Sample Problem – Vector
Relational Operators

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Vector Logical Operators
They operate on the results of relational operators
How many elements in vector x are in range
(6,10)?
How many elements in x are
… greater than 6
AND
… less than 10?
We use logical operators…
AND (&), OR (|), NOT (~)
any, all

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Sample Problem – Vector
Logical Operators

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Synopsis for Vector Operators
There are functions that work in a cell-by-cell fashion (like sin) and functions
that aggregate (like sum).
Cell-by-cell vector operators apply the indicated operation to the
corresponding elements of the two vectors.
For cell-by-cell operations, the two arguments must be the same type of vector
(row or column) and be of the same length, or one of the arguments must be a
scalar.
Cell-by-cell vector operators include the classes (assignment, colon and
transpose operators), the vector arithmetic cell-by-cell operators (Table 5-2),
the vector relational operators (Table 5-4), and vector cell-by-cell logical
operators (Table 5-5).
Logical computations are extended via built-in logical functions (Table 5-6).
The built-in logical function find is useful because it enables a type of content
The operator precedence table was updated to include new possibilities (Table
5-7).

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2-D Plotting and Help in
MATLAB
Chapter 6
2-D Plotting and Help in
MATLAB
Using EZPLOT to Plot Functions
Using Vectors to Plot Numerical Data
Overlay plots and subplots
Other 2-D plot types in MATLAB
Problem Sets for 2-D Plotting

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6-1 Using EZPLOT to Plot
Functions

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Getting Help

You can‟t possibly learn everything there is
… and you don‟t need to.
It is crucial to develop the ability to

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Getting Help cont’d

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Getting Help cont’d
Click the tab in the navigation pane labeled
Search.
Then type into the Search field the name ezplot.

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Using EZPLOT to Plot Functions
There are three forms of ezplot:
f(x)                e.g., f(t) = 3e-2tcos(5t)
ezplot('3*exp(-2*t)*cos(5*t)')
f(t), g(t)          e.g., f(t) = 3t2 + 2; g(t) = sin(5t)
ezplot('3*t^2 + 2', 'sin(5*t)')
f(x,y) = 0          e.g., f(x,y) = 3xy + y2 + 55 = 0
ezplot('3*x*y + y^2 + 55',[-30,30,-20,20])

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Sample Problem - EZPLOT

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Graphing with MATLAB
Use ezplot to make a quick and dirty chart of
functions.
Optional arguments allow changing the default
functional domain [-2π, 2π].
Use xlabel, ylabel, and title built-in functions to
refine labeling the plots made by ezplot.
When needed, use grid to activate a grid on a plot
created.
If you would like to keep the existing graph and
generate a new one, use figure.
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6-2 Using Vectors to Plot
Numerical Data
Mostly from observed data - your goal is to understand the
relationship between the variables of a system.
Speed (mi/hr)                20 30 40           50      60     70
Stopping Distance (ft) 46 75 128 201 292 385

Determine the independent and dependent variables and plot:
speed = 20:10:70;
stopDis = [46,75,128,201,292,385];
plot(speed, stopDis, '-ro') % note the „-ro‟ switch
Don‟t forget to properly label your graphs:
title('Stopping Distance versus Vehicle Speed', 'FontSize', 14)
xlabel('vehicle speed (mi/hr)', 'FontSize', 12)
ylabel('stopping distance (ft)', 'FontSize', 12)
grid on

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Sample Problem – Plotting
Numerical Data

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Plotting Functions Numerically
ezplot is a great tool for plotting functions, but it has several
it doesn‟t provide as much control as plot, e.g. dotted lines.
you must fill in values for any constants, e.g.

When you need more control, plot numerically with plot:
d = 4;
h = linspace(1,10); % Step 1 - create vector for independent variable
V = pi*d^2/4*h; % Step 2 – compute vector for dependent variable
plot(h,V,'-r') % Step 3 - plot and label
xlabel('height (m)', 'FontSize', 12)
ylabel('Volume (m^3)', 'FontSize', 12)
title('Volume of a cylinder versus its height','FontSize', 14)
grid on

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Sample Problem – Plotting
Functions Numerically
A function G(x,y,z) of three independent variables is
defined as:

Write a function that takes no inputs or outputs but
creates a plot of G(x,y,z), subject to:
0.1 < x < 4
y = 5, z = 3

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Synopsis for ezplot and plot
The first argument to plot should be the vector of values
for the independent variable (going on the x-axis); the
second argument should be the vector of values for the
dependent variable (going on the y-axis).
An optional third argument plot is the line spec which
specifies the type of line used (solid, dotted, etc.), the
color of the line used, and the type of data marker (if any).
For plotting numerical data from experimentation or
observation, use data markers.
For plotting numerical data that are computed from a
mathematical relationship, data markers must not be used.

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6-3 Overlay Plots and Subplots

Allows putting more than one                    For multiple dependent variables
relationship directly into the same             whose data are not of the same type,
plotting window.                                e.g. acceleration, speed and distance
Two key functions: hold and legend              Key function to learn: subplot

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Sample Problems – Overlay Plots
and Subplots

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Sample Problems – Overlay Plots
and Subplots

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Synopsis for Overlay Plots and
Subplots
Overlay plots are used to show a family of parameterized
results
hold on is the key MATLAB command needed to turn on
overlays
Subplots are used to display plots of different independent
variables usually from one experimental data set or from
one set of equations for a single physical system.
subplot is the key MATLAB command needed to identify
the target for a created plot.

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Arrays
Chapter 7
So far we have learned…

Using MATLAB for scalar computations (Ch. 3)
Saving your work in MATLAB user-defined
functions (Ch. 4)
Debugging MATLAB functions (Ch. 4)
Using MATLAB for vector operations (Ch. 5)
Using MATLAB to make 2-D plots (Ch. 6)
Using the MATLAB Help facility to let you
extend what you know (Ch. 6)
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95
The Pattern…

Scalars – numbers
MatLab Vectors – Ordered, linear groupings
of scalars
Simple extension – MatLab Arrays
Instead of having a one-dimensional grouping
of scalars as in vectors, MATLAB arrays are
two-dimensional groups of scalars.

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7-1 Array Creation

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Creating Arrays
A semicolon as punctuation in the square bracket
operator tells MATLAB to start a new row
>> A = [1, 2, 3; 10, 20, 30]
linspace and the colon operator can be used to
create vectors that are subsequently composed into
an array:
>> A = [1:3:15; linspace(0,1,5)]
or…
>> A = [(1:3:15)', linspace(0,1,5)']

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Things to know…
A typical mistake – trying to concatenate incompatible
vectors:
>> B1 = [1, 2, 3];
>> B2 = [10, 11];
>> stackedUpDown = [B1; B2]
??? Error using ==> vertcat
All rows in the bracketed expression must have
the same number of columns.

Creating an array whose elements are all value 0 (or 1) :
>> twoByFourZeros = zeros(2,4)

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Synopsis for Creating Arrays
Semicolon punctuation inside the square bracket operator
indicates to MATLAB that a new row is to be created.
When concatenating arrays, their dimensions must be
consistent.
ones and zeros are built-in functions that create arrays
whose elements are all value 1 or all value 0, respectively.
ones and zeros take two arguments: the number of rows
and the number of columns in the array that will be
created.

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7-2 Accessing Array Elements

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Fetching Elements
Create an array A by the following:
>> A = [1,2,3,4; 10,11,12,13; 20, 21,22,23]
Pull out the value of the element at the second row, third column:
>> x = A(2,3)
Fetch the second and third elements in the second row of A:
>> V = A(2, [2,3])
Extract the entire second column:
>> X2 = A(:, 2)
Fetch the entire third and fourth columns:
>> partOfB = A(:, [3,4])
Fetch the first and second elements in the second and third columns:
>> anotherPartOfB = A([1,2], [2,3])

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Fetching Elements cont’d

How would you address to number 0?

Row first, column next;
>> A(2,4)

>> A(3,2)
Intro to Technical Problem Solving with MATLAB v.7
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Fetching Elements cont’d

How can we extract the collection of numbers in the dotted box?
That is, the numbers in the 1st through 3rd rows, 2nd through 4th
columns…
Specify the row and column numbers by counting them…
A(1:3, 2:4)

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Setting Elements
Create an array A by the following:
>> A = [1,2,3,4; 10,11,12,13; 20, 21,22,23]
Set the element at row two and column 4 to 100:
>> A(2,4) = 100;
Create a new array B that is identical to modified A, except that the
second and third columns are interchanged.
>> B = A;
>> B(:,[2,3]) = A(:, [3,2])
The shape of the array to be set must be the same as the shape of the
array that holds the new values.

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105
Built-in Functions end and size
Create an array A by the following:
>> A = [1,2,3,4; 10,11,12,13; 20, 21,22,23]
Replace the last and next to last row/column elements with [100, 101;
200, 201]
>> A(end-1:end,end-1:end) = [100, 101; 200, 201]
For vectors, we had length to return the number of elements.
For arrays, size built-in function is used:
>> [numRows, numCols] = size(A)

Intro to Technical Problem Solving with MATLAB v.7
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Synopsis for Setting Elements
Array access operations (fetch and set) are directly
analogous to vector access operations.
For array setting, the part of an array to be set and the
elements which will be inserted must be the same shape.
The colon may be used as an index element to indicate all.
end is used in array access as it is used in vector access.
To determine the number of rows and columns in an array,
use size.

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7-3 Transpose Applied to Arrays

Intro to Technical Problem Solving with MATLAB v.7
108
The Transpose Operator
The transpose operator is used to flip an array.
More formally, if A is an NxM vector, then A' will be an
MxN array whose elements are defined by A'(i,j) = A(j,i).
>> D = [1,2,3,4; 10,11,12,13; 20, 21,22,23]
>> transposeD = D‟
The effect of applying the transpose operator to an array is
to flip rows and columns.
What was a row is now a column, and what was a column
is now a row.

Intro to Technical Problem Solving with MATLAB v.7
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7-4 Array Built-in Functions,
Operators, and Expressions

Intro to Technical Problem Solving with MATLAB v.7
110
Built-in Functions and Operators
The same types in Vectors exist – with new
possibilities
>> D = [1,10; 100,110]
>> sumOverColumns = sum(D,1)
>> sumOverRows = sum(D,2)

Intro to Technical Problem Solving with MATLAB v.7
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Cell-by-Cell Operators
Arrays A and B are defined as:
>> A = [2:4; 20:10:40]
>> B = [1:3; 1:3]
Find cell-by-cell product of A and B:
>> A .* B
Find A raised to the power B, cell-by-cell:
>> A .^ B
Find A/B, cell-by-cell:
>> A ./ B
A./B stands for          , whereas B./A stands for                ,
cell-by-cell
Intro to Technical Problem Solving with MATLAB v.7
112
Example Problem 1
Cell-by-Cell Operators
The ABC electronics factory makes four different items: a
48-inch HDTV, a 32-inch regular TV, a computer called
the M2 model, and a DVD player called the R2 model.

Compute:
(a) the total cost for materials used on
all four product lines for each
quarter and

(b) the total yearly cost for
materials used in each of
four product lines.

Intro to Technical Problem Solving with MATLAB v.7
113
Example Problem cont’d
Find by hand quarter 1 to material costs from
the HDTV product line.
532 * \$892 = \$474,544.
Think through the problem statement.
This problem is not conceptually difficult but is
tedious.
MATLAB provides a better way…

Intro to Technical Problem Solving with MATLAB v.7
114
Matrix Operators
Matrix multiplication operation is defined as:

1. The number of columns in A must be equal to the
number of rows in B. Otherwise, this is not a legal
operation.
2. Assuming Rule 1 is met the number of rows in C will be
equal to the number of rows in A.
3. Likewise, the number of columns in C will be equal to
the number of columns in B.
Intro to Technical Problem Solving with MATLAB v.7
115
Matrix Multiplication

1   2          9         7                  (1,1)
1*9 + 2*8              (1,2)
1*7 + 2*6
3   4
X    8         6
=          (2,1)
3*9 + 4*8              (2,2)
3*7 + 4*6
2X2               2X2                                              2X2
25                19
=              59                45

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Example Problem 2
Matrix Operators

Intro to Technical Problem Solving with MATLAB v.7
117
Revisiting Example Problem 1
The ABC electronics factory makes four different items: a
48-inch HDTV, a 32-inch regular TV, a computer called
the M2 model, and a DVD player called the R2 model.

Compute:
(a) the total cost for materials used on
all four product lines for each
quarter and

(b) the total yearly cost for
materials used in each of
four product lines.

Intro to Technical Problem Solving with MATLAB v.7
118
Example Problem cont’d

?                         =

Intro to Technical Problem Solving with MATLAB v.7
119
Example Problem 3
Matrix Operators

Intro to Technical Problem Solving with MATLAB v.7
120
Matrix Left Division
A linear system of equations can be modeled as:

In other words…

Intro to Technical Problem Solving with MATLAB v.7
121
Matrix Left Division cont’d

Can be solved for x as follows:

Or in MatLab by left division:

Intro to Technical Problem Solving with MATLAB v.7
122
Example Problem 4
Matrix Left Division
Jeanie, Juan, and Alexander each have some fruit. Each has a
number of apples, oranges, and pears.
All apples have the same weight, all oranges have the same
weight, and all pears have the same weight.
Jeanie has 3 apples, 2 oranges, and 1 pear. The total weight of
fruit that Jeanie has is 52 ounces.
Juan has 2 apples, 3 oranges, and 1 pear. The total weight of
fruit that Juan has is 50 ounces.
Alexander has 1 apple, 2 oranges, and 3 pears. The total
weight of fruit that Alexander has is 56 ounces.

What is the weight of each apple, orange, and pear?
Intro to Technical Problem Solving with MATLAB v.7
123
Example Problem 5
Relational and Logical Operators

1.   Using the find function, find and display:
a.   the row and column numbers of elements in A that are less than
zero
b.   elements that are less than zero
c.   elements that are greater than -4 but less than 4
2. Using the all or any functions, determine:
a.   if all elements in A are greater than -8
b.   if any elements in A are less than -5
Intro to Technical Problem Solving with MATLAB v.7
124
Synopsis
Arrays are indexed by giving the row and column locations.
All cell-by-cell operations are generalizations of the corresponding
vector operation.
Matrix multiplication can be very advantageous when the problem you
are solving involves a sum of scalar multiplication operations.
Matrix left division is often used to solve systems of linear
simultaneous equations.
Values in an array that meet some relational test may be extracted
using find as an indexing term.
Two output values are returned when the find function is applied to an
array: a vector of row index values and a corresponding vector of
column index values.
Intro to Technical Problem Solving with MATLAB v.7
125
Conditional and Iterative
Programming
Chapter 7
8-1 Program Flow

1. Straight Line Code

One line of code after
another… just in
sequence. Also called
“sequential code”.

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127
Program Flow cont’d

2. Conditional Code

NO              YES                Based on a test,
test
perform one alternative
set of code and not
another…

Intro to Technical Problem Solving with MATLAB v.7
128
Program Flow cont’d

3. Iterative code
Execute the same
block of code
again and again …

repeat…

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129
Synopsis for Program Flow
Types
There are three major types of program control: straight
line control, conditional control, and iterative control.
Programming constructs for conditional control and
iterative control should be considered “modules,” meaning
there is one point of entrance into the construct and one
point of exit.
Straight line code executes in the order it is written in a
program.
Conditional code executes one alternative of a number of
possibilities, selecting the alternative to run based on a
relational/logical test of program variables.
Iterative code executes the same block of code a number of
times.
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8-2 Iterative Program Flow:
FOR
General form:

A FOR loop must end with a line containing
end.
Intro to Technical Problem Solving with MATLAB v.7
131
Questions for the Iterative Case

How many times does
it repeat?
What controls how
many times it repeats?
How are you going to
set parameter values in
a handy way for each
“pass”

Intro to Technical Problem Solving with MATLAB v.7
132
Iteration over Elements of a
Row Vector
Name of var that changes
mySum = 0;                          Values taken on
for itemThisTime = [1 0 -5 78]      by changing var
mySum = mySum + itemThisTime;
display(itemThisTime);
display(mySum );
disp('====')      Code to be repeated
end

disp('***********************')
mySum         Intro to Technical Problem Solving with MATLAB v.7
133
Iteration over Columns of an
Array Name of var that changes
sumColumnProducts = 0;                   Values taken on
for oneCol = [1 0 -5 78; 61 9 44 10]     by changing var
sumColumnProducts = sumColumnProducts +
oneCol(1)*oneCol(2);
display(oneCol);
display(sumColumnProducts);
disp('====')        Code to be repeated
end

disp('***********************')
sumColumnProducts
Intro to Technical Problem Solving with MATLAB v.7
134
Nested FOR Loops

function cellSum = prob8_A_11(A,B)
% Calculates cell-by-cell sum of two arrays
% Input: 2 arrays of dimensions nxm
% Output: Cell-by-cell product array nxm
[nRows,nCols] = size(A);
cellSum = zeros(nRows, nCols);
for i=1:nRows
for j=1:nCols
cellSum(i,j)=A(i,j)+B(i,j);
end
end

Intro to Technical Problem Solving with MATLAB v.7
135
Synopsis for FOR
FOR loops are used in cases where you need more control
over computations than allowed in cell-by-cell operations.
for and end with keyword end.
FOR loops iterate over a code block body using successive
values of supplied vector or array.
If a FOR loop is supplied with an array, then successive
values of the columns of the array are set to the value of
the loop variable.
To understand a FOR loop, a good strategy is to “step
through” the loop.
Intro to Technical Problem Solving with MATLAB v.7
136
8-4 Conditional Program Flow
if <conditions>
Form 1: IF                    <statements>
end;

J&T Computers is planning to give a
\$3,000 holiday bonus to every
employee provided ALL
employees have a performance
evaluation higher than 3/5.
For the dataset, find if bonus will be
given.
Employee Numbers,
Performance Ratings, Salaries
Intro to Technical Problem Solving with MATLAB v.7
137
Conditional Program Flow
cont’d       if <conditions>
<statements>
Form 2: IF/ELSE             else
<statements>
end;

Instead of giving bonus to all
employees, consider the following
scenario:
every employee with a
performance rating of 4 or 5 gets a 5
percent holiday bonus while all
other employees will get a 2                          Employee Numbers,
percent bonus.                                   Performance Ratings, Salaries
Intro to Technical Problem Solving with MATLAB v.7
138
Conditional Program Flow
cont’d         if <conditions>
<statements>
Form 3: IF/ELSEIF                 elseif
<statements>
end;

New scenario:
Employees with performance
ratings of 5 get a 4% bonus, those
with performance ratings of 4 get
a
2% bonus, and those with
performance ratings of 3 get a 1%                     Employee Numbers,
bonus.                                           Performance Ratings, Salaries
Intro to Technical Problem Solving with MATLAB v.7
139
Conditional Program Flow
cont’d         if <conditions>
<statements>
elseif
<statements>
Form 4:                          else
IF/ELSEIF/ELSE                         <statements>
end;

One last scenario:
employees with performance ratings
of 5 get a 4% bonus, those with
performance ratings of 4 get a 3%
bonus, those with performance
ratings of 3 get a 2% bonus, and
everyone else gets a 1% bonus                         Employee Numbers,
Performance Ratings, Salaries
Intro to Technical Problem Solving with MATLAB v.7
140
Sample Problem

I will be depositing \$5,000 in the beginning
of every year in my bank account. The
bank offers an interest rate of 4%.

When will I be a millionaire?
How much savings will I have after 10 years?

Intro to Technical Problem Solving with MATLAB v.7
141
Synopsis
IF-THEN-ELSE can be used to express conditional
program control. It is best understood in four distinct
forms.
1. IF: In this form, one relational/logical test exists. During
execution, if the test results in true, then the commands in the
following block are run. If the test results in false, then the
commands are not run.
2. IF-ELSE: This form performs a relational/logical test and, if true,
then runs a set of commands. If false, an alternative set of
commands is run.
3. IF-ELSEIF: There can be multiple ELSEIF clauses. Only one (at
most) code block following a test will be run, which will be the
one following the first test that results in true.
4. IF-ELSEIF-ELSE: This form is a combination of the second and
third forms.
The key to effective use is to correctly match the problem
situation you have with one of the appropriate four forms.
Intro to Technical Problem Solving with MATLAB v.7
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