Intro Tech Matlab

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An Introduction To

Technical Problem Solving

with MATLAB v.7

Jon Sticklen, PhD

M. Taner Eskil, PhD



IntroductionChapter 1



Intro to Technical Problem Solving with MATLAB v.7 3

Introduction

What is Technical Problem Solving?

The Path to Becoming a Good Technical

Problem Solver

What You Must Do to Master the Material




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




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

MathCad




1-3 What You Must Do to Master

the MaterialBe prepared to spend substantial time

learning this material

Read assigned sections of the text with

MATLAB at your side

Work the problems hands-on



A Framework for

Technical Problem SolvingChapter 2




A Framework for Technical

Problem Solving

Steps in a Framework for Technical

Problem Solving

An Example Using the Framework




2-1 Steps in a Framework for

Technical Problem SolvingStep 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 diagramVisualize the physical situation





Technical Problem SolvingStep 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

LibraryStep 4: Simplify

Find if there are suitable approximations





Technical Problem SolvingStep 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





Technical Problem SolvingStep 7: Compute and Discuss

Perform the computations needed to obtain a

solutionUse a computational tool such as MATLAB

Examine and understand the results

Be ready to explain them!




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 haybales to a herd of starving caribou, given the

initial velocity of hay bales as they exit the

catapult?




2-2.1 Refine and Structure

Clarify:Think about the problem

The ultimate source of information is the person setting

the problemRefine:

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




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?

h

D

L1

L2

g

s




2-2.3 Background Knowledge

For our example:

General knowledge of physics

General knowledge of one way in whichoptimizing 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




2-2.4 Assumptions and

Approximations

Many times in technical problem solving, the pathto a solution leads to making assumptions aboutthe problem and then solving the simplifiedversion of the problem.

Negligible air resistance

No difference between a hay bale and a snowball

Be aware of the assumptions you are making andcommunicate them to the person who originallyset the problem for you.

Importance of good documentation




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




2-2.6 Dimensional Analysis

g

s

Do )cos()sin(22 f f

2

2

s

ft

s

ft

ft

f t f t




2-2.7 Putting It All Together

2

max 031.0 os D

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




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



MATLAB Basics: ScalarsChapter 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




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 singleline programs you have executed

Current Folder - lists all files that are directly available forMATLAB 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 howMATLAB looks to you




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.

,4,

7

22,5.3




Using the Command History

Window




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




The Assignment Operator (=)

The equal sign is the assignment operator inMATLAB.

>> x = 22

places number 22 in container x

How about:

>> x = x + 1

Note the difference between the equal sign inmathematics and the assignment operator inMATLAB!



Intro to Technical Problem Solving with MATLAB v.729

3-3.2 An Example - Setting

Variables for the Hay Bale Problem

clear deletes variables from workspace, youshould use it before starting new work

The percent symbol is used for putting reminders

(comments) for ourselves, ignored by MATLABCommand lines ending with semicolon do not

display the results.

>> clear>> accelGravity = 32; % units: ft/sec/sec>> speedInitial = 50; % units: ft/sec>> phi = 10; % units: degrees




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 MATLABfunctions is called operators

Assignment operator (=)

Arithmetic operators (+, -, *, /, ^)

Relational operators (<, <=, = =, ~=, >=, >)

Logical operators (&, |, ~)




Example – Arithmetic Operators




Example – Relational and

Logical Operators




3-4.2 Rules for Forming

Expressions

MATLAB expressions consist of:

Numerical values or variables

Logical values or variablesLegal applications of MATLAB functions or

operators

A combination of MATLAB expressions

What is the error in the following MATLABexpression?

>> sin(pi/2, pi/8)




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




Applying Scalar Computations to

a Problem




Synopsis for Chapter 3A MATLAB variable can be thought of as a named container. The value of aMATLAB 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.



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 Scriptand Function Files

Problem Sets for “Saving Your Work”




4-1 The MATLAB Current

Directory

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

MATLAB starts

up with the

default folderconnected




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 includecomments for:

The location of the file

Variables used in the script but defined outsideResults produced by the script

Units of values calculated in the script




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 forcylinder




Synopsis for MATLAB Scripts

A script file consists of groups of MATLABcommands bundled together into a module

Scripts files have a DOT-M extension

When executing inside a script file, allvariables in the workspace are available

Variables created in a script are available at

the Command Window and in other scriptsMaking a sketch of a problem is important

in making the problem context concrete




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.” Yougive 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

+

modularize your code




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)




Syntax of a User Defined Function

Functions are saved in DOT M files - just

like the script files

Same rules for the connected directoryapplies

First line of the file defines

Name of the functionInput variable(s)

Output variable(s)




Defining a function – the first line

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

Keyword - has to beexactly as appears

0, 1, or more vars. If

more than 1, put in

square brackets.

You choose the name - see text or

ML help for valid name - mostly

just start with a regular

character and have no spaces.

0, 1, or more vars. If 0,

then the enclosing parens

are not needed.

required




Linkage Between Actual and

Formal Parameters

...

z = myFun3(a,b)

...

calling program:

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 thecalling line (a,b)

2. The function “runs”

3. The output variables in the function are given back to the callingprogram’s variable.




Comments in Functions

There are no uniformly agreed upon rules for

inserting comments into functions

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 functionany assumptions




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)




50

Examples




51

Examples




52

Synopsis for User-Defined

Functions

The workspace of a function is insulated from the outside.

When a MATLAB function is called, a linkage is made

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 thekeyword function, and the rest of the first line looks like an

assignment statement.




53

4-4 Testing and Debugging

MATLAB Script and Function Files

Types of errors in programs:

Syntax errors

Results from incorrect application of MATLAB rulesMATLAB aborts the computation and points of the

error

Runtime errors

Results from incorrect logic and MATLAB not doingwhat you intend

For this type error MATLAB debugging facilities are

useful




54

Examples




55

Examples




56

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 produceCompare what you expect to what MATLAB

produces to identify runtime errors in the function

Identify and correct the line(s) of code that arecausing the runtime error



Vector Operations

Chapter 5




58

Vector Operations

Vector Creation

Accessing Vector Elements

Row Vectors and Column Vectors, and theTranspose Operator

Vector Built-in Functions, Operators, and

Expressions




59

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);




60

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.




61

5.2 – Accessing Vector Elements –

Examples

1. Create a row vector x consisting of the numbers in the orderedset: {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)

A l




62

Accessing Vector Elements –

Example cont’d.

5. Set variable y to be the 3rd through the last element of x - and do sosuch 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 sosuch 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 be205.

x([2,4]) = [102, 205]

S i f F hi d S i




63

Synopsis for Fetching and Setting

Elements in VectorsAccess to whole vector is similar to scalar access.

Accessing element(s) in a vector is done by indexing intothe 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-infunction length(V).

When setting elements of a vector, the number of elements

being set must be equal to the number of elements in thevector on the right hand side of the assignment operation.The exception is that a scalar on the right-hand side can beused to set multiple vector elements.

5 3 R V d C l




64

5-3 Row Vectors and Column

Vectors, and the Transpose Operator Row and column vectors are represented as singlerows 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)‟;




65

5-4 Vector Built-in Functions,


S l P bl V t B ilt




66

Sample Problem – Vector Built-

in Functions

S l P bl V t




67

Sample Problem – Vector

Arithmetic Operators

S l P bl V t




68



S l P bl V t




69






70

Vector Relational Operators

Think of them as comparing numbers…

<, >, = =, >=, <=

A relational operator can be used tocompare the values of two variables

a>b

But… remember MATLAB is for matrices

what are you testing?




71

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]

Sample Problem Vector




72


Relational Operators




73

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

Sample Problem Vector




74


Logical Operators




75

Synopsis for Vector Operators

There are functions that work in a cell-by-cell fashion (like sin) and functionsthat aggregate (like sum).

Cell-by-cell vector operators apply the indicated operation to thecorresponding 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 andtranspose operators), the vector arithmetic cell-by-cell operators (Table 5-2),the vector relational operators (Table 5-4), and vector cell-by-cell logicaloperators (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

addressing.The operator precedence table was updated to include new possibilities (Table5-7).



2-D Plotting and Help in

MATLABChapter 6

2 D Plotting and Help in




77

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




78

6-1 Using EZPLOT to Plot

Functions




79

Getting Help

You can‟t possibly learn everything there is

to know about MATLAB,

… and you don‟t need to. It is crucial to develop the ability to

augment your knowledge in MATLAB

toward accomplishing a given task.




80

Getting Help cont’d




81

Getting Help cont’d

Click the tab in the navigation pane labeled

Search.

Then type into the Search field the name ezplot.




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])




Sample Problem - EZPLOT




Graphing with MATLAB

Use ezplot to make a quick and dirty chart of functions.

Optional arguments allow changing the defaultfunctional domain [-2π, 2π].

Use xlabel , ylabel , and title built-in functions torefine labeling the plots made by ezplot.

When needed, use grid to activate a grid on a plotcreated.

If you would like to keep the existing graph andgenerate a new one, use figure.

6-2 Using Vectors to Plot




6-2 Using Vectors to Plot

Numerical Data

Mostly from observed data - your goal is to understand therelationship between the variables of a system.

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

Speed (mi/hr) 20 30 40 50 60 70

Stopping Distance (ft) 46 75 128 201 292 385

Sample Problem – Plotting




Sample Problem Plotting

Numerical Data




Plotting Functions Numerically

ezplot is a great tool for plotting functions, but it has severaldisadvantages:

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

hd

V cylinder 4

2

Sample Problem – Plotting




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






6-3 Overlay Plots and Subplots

Allows putting more than onerelationship directly into the sameplotting window.

Two key functions: hold and legend

For multiple dependent variableswhose data are not of the same type,e.g. acceleration, speed and distance

Key function to learn: subplot

Sample Problems – Overlay Plots




Sample Problems Overlay Plots

and Subplots

Sample Problems – Overlay Plots




Sample Problems Overlay Plots

and Subplots

Synopsis for Overlay Plots and




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

overlaysSubplots 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 identifythe target for a created plot.



Arrays

Chapter 7

S f h l d




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)

Th P




The Pattern…

Scalars – numbers

MatLab Vectors – Ordered, linear groupings

of scalarsSimple extension – MatLab Arrays

Instead of having a one-dimensional grouping

of scalars as in vectors, MATLAB arrays aretwo-dimensional groups of scalars.




7-1 Array Creation

C i A




Creating Arrays

A semicolon as punctuation in the square bracketoperator tells MATLAB to start a new row>> A = [1, 2, 3; 10, 20, 30]

linspace and the colon operator can be used tocreate vectors that are subsequently composed intoan array:>> A = [1:3:15; linspace(0,1,5)]

or… >> A = [(1:3:15)', linspace(0,1,5)']

Thi k




Things to know…

A typical mistake – trying to concatenate incompatiblevectors:>> 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)

S i f C i A




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 rowsand the number of columns in the array that will be

created.




7-2 Accessing Array Elements

F t hi El t




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])

F t hi El t t’d




Fetching Elements cont’d

3547

1821

0198

6745

A

How would you address to number 0?

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

How about 2?>> A(3,2)

F t hi El t t’d




Fetching Elements cont’d

3547

1821

0198

6745

A

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)

S tti El t




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 thearray that holds the new values.

B ilt i F ti d d i




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)

S i f S tti El t




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.




7-3 Transpose Applied to Arrays

Th T O t




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.




7-4 Array Built-in Functions,


Built in Functions and Operators




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)

Cell by Cell Operators




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

B

A

A

BA./B stands for , whereas B./A stands for ,cell-by-cell

Example Problem 1




p

Cell-by-Cell OperatorsThe 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.

Example Problem cont’d




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…

Matrix Operators




Matrix Operators

Matrix multiplication operation is defined as:

1. The number of columns in A must be equal to thenumber 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.

Matrix Multiplication




(2,1) (2,2)

(1,1) (1,2)

Matrix Multiplication

1 2

3 4

9 7

8 6

2 X 2 2 X 2

=X

2 X 2

=

1*9 + 2*8

3*9 + 4*8

1*7 + 2*6

3*7 + 4*6

25 1959 45

Example Problem 2




Matrix Operators

67

89

35

61

42

,

B A

where

B A find

Revisiting Example Problem 1




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.

Example Problem cont’d




Example Problem cont d

=?

Example Problem 3




Matrix Operators



Matrix Left Division cont’d




Matrix Left Division cont d

Can be solved for x as follows:

Or in MatLab by left division:

y x A

y A x 1

y A x

\

Example Problem 4




Matrix Left Division

Jeanie, Juan, and Alexander each have some fruit. Each has anumber of apples, oranges, and pears.

All apples have the same weight, all oranges have the sameweight, 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?

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

Synopsis




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.



Conditional and Iterative

ProgrammingChapter 7

8-1 Program Flow




8 1 Program Flow

1. Straight Line Code

One line of code after

another… just in

sequence. Also called

“sequential code”.

Program Flow cont’d




Program Flow cont d

2. Conditional Code

Based on a test,

perform one alternative

set of code and notanother…

testYESNO

Program Flow cont’d




Program Flow cont d

3. Iterative code

Execute the same

block of codeagain and again …

repeat…

Synopsis for Program Flow




Types

There are three major types of program control: straightline control, conditional control, and iterative control.

Programming constructs for conditional control anditerative control should be considered “modules,” meaning

there is one point of entrance into the construct and onepoint of exit.

Straight line code executes in the order it is written in aprogram.

Conditional code executes one alternative of a number of

possibilities, selecting the alternative to run based on arelational/logical test of program variables.

Iterative code executes the same block of code a number of times.

8-2 Iterative Program Flow:




FOR

General form:

A FOR loop must end with a line containing

end.

Questions for the Iterative Case




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”

repeat…repeat…

Iteration over Elements of a




Code to be repeated

Values taken on by changing var

Name of var that changes

f

Row Vector mySum = 0;

for itemThisTime = [1 0 -5 78]

mySum = mySum + itemThisTime;

display(itemThisTime);

display(mySum );

disp('====')

end

disp('***********************')

mySum

Iteration over Columns of an




Code to be repeated

Values taken on by changing var

Name of var that changes

f

ArraysumColumnProducts = 0;

for oneCol = [1 0 -5 78; 61 9 44 10]

sumColumnProducts = sumColumnProducts +

oneCol(1)*oneCol(2);display(oneCol);

display(sumColumnProducts);

disp('====')

end

disp('***********************')

sumColumnProducts

Nested FOR Loops




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

Synopsis for FOR




Synopsis for FOR

FOR loops are used in cases where you need more controlover computations than allowed in cell-by-cell operations.

FOR loops start with a code line that begins with keyword

for and end with keyword end.FOR loops iterate over a code block body using successivevalues of supplied vector or array.

If a FOR loop is supplied with an array, then successivevalues 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 “stepthrough” the loop.

8-4 Conditional Program Flow




8 4 Conditional Program Flow

Form 1: IFif <conditions>

<statements>

end;

J&T Computers is planning to give a

$3,000 holiday bonus to everyemployee provided ALL

employees have a performance

evaluation higher than 3/5.

For the dataset, find if bonus will begiven.

Employee Numbers,

Performance Ratings, Salaries

Conditional Program Flow

’d




cont’d

Form 2: IF/ELSE

if <conditions>

<statements> else

<statements>

end;

Instead of giving bonus to allemployees, consider the following

scenario:

every employee with a

performance rating of 4 or 5 gets a 5percent holiday bonus while all

other employees will get a 2

percent bonus.Employee Numbers,



’d




cont’d

Form 3: IF/ELSEIF

if <conditions>

<statements> elseif

<statements>

end;

New scenario:Employees with performance

ratings of 5 get a 4% bonus, those

with performance ratings of 4 get

a2% bonus, and those with

performance ratings of 3 get a 1%

bonus.Employee Numbers,



’d if <conditions>




cont’d

Form 4:

IF/ELSEIF/ELSE

if <conditions>

<statements>

elseif<statements>

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 performanceratings of 3 get a 2% bonus, and

everyone else gets a 1% bonus Employee Numbers,


S l P bl




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?

Synopsis



y pIF-THEN-ELSE can be used to express conditional

program control. It is best understood in four distinctforms.1. IF: In this form, one relational/logical test exists. During

execution, if the test results in true, then the commands in thefollowing block are run. If the test results in false, then thecommands 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 (atmost) code block following a test will be run, which will be theone following the first test that results in true.

4. IF-ELSEIF-ELSE: This form is a combination of the second andthird forms.

The key to effective use is to correctly match the problem

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