Lecture 14: M/O/F/ for
Engineering Applications Part 2
BJ Furman
28NOV2011
The Plan for Today

Matlab/Octave/FreeMat (M/O/F) for
engineering applications – part 2






Recap M/O/F key concepts
Element-by-element operations (dot operator)
Function examples
2D graphs
File IO
Resources for more information
Learning Objectives

Distinguish between matrix and array
arithmetic, and use them appropriately

Explain the differences between script
files and functions

Explain the basic elements of creating a
2D graph

Explain how to read from and write to
files
Last Lecture



Overview of Matlab/Octave
Useful commands
The array as a fundamental element



creating arrays
indexing elements
vectors





extracting sub-arrays using indexing and the colon operator
special matrices


colon operator
linspace
logspace
ones(), zeros(), diag(), eye()
Introduction to plotting in M/O/F via script file

comparison to Ch and Excel
Arrays, Vectors, and Matrices

Array :


Vector:


A collection of data elements
arranged in rows and columns
A one-dimensional array
(single row or single column
of data elements)
>> A=[1:9]
>> B=[1:9]’
row or column?
column vector
Matrix :

A two-dimensional array
(more than one row and/or
column of data elements)
>> C=[1:3; 4:6; 7:9]
C=
1
4
7
2
5
8
3
6
9
Colon Operator vs. linspace()

What will the following do?



D=0 : 3 : 10 base : increment : limit
F=linspace(0, 10, 11) start : end : n
Observations about the two methods:


both methods produce vectors with equally spaced
elements
colon operator method allows you to specify the first
element and the interval spacing between elements,
but not the number of elements


If just start : end, then spacing is 1
linspace method allows you to specify the first and
last elements of the vector, but not the spacing
between elements
Review of length() and size()

What will the following do?

length(ones(1,3))


length returns the number of elements of the vector
size(zeros(2,3))

size returns the size of the dimensions of its
argument
Review of Array Manipulation

Given G=[2 4 6; 8 10 12], what is:

G(2,3)
 G(: , 2)
 G(4)
 G(1,1)=0
 G(1, :)=0
Observations:





Array indexing begins with 1 (contrast with C)
: means “all of” the elements in that dimension
Extract elements by indexing
Extract sub-arrays using vectors as the indexing
arguments
More Array Manipulation

Suppose H=1:9 (what will this produce?)

How could we form into a 3x3 matrix?

Reshape function



J=I(3:-1:1,:)
Reverse the order of the columns of I (how?)


I = I’
Reverse the order of the rows of I


I=reshape(H,3,3)
How to transpose the rows to be the columns?
K=I(:, 3:-1:1)
Reverse the order of all the elements of I (how?)

Pseudocode:


Index I in reverse order
L = reshape(I(9:-1:1),3,3)
Reshape
Matrix and Array Arithmetic

Arithmetic operators:
 +
*

/
\
^
’
add. sub. mult. right div. left div. expon.
algebr. transpose
addition and subtraction are done element-by-element (same for
matrix and array arithmetic)

Unless one is a scalar, the operands must be of the same size


scalar  (matrix or array) --> ?
(matrix or array)  (matrix or array) --> ?
 (matrix or arrays) must be of the same size
For the other operators, need to distinguish between matrix and array
operation
n columns
B
 Matrix arithmetic operations

mn 

per rules of linear algebra
A
n rows n  p
 rows and columns must conform



For example, A x B: must have column and row agreement
Array arithmetic operations

element-by-element

Denote with dot operator: .* ./ .\ .^ .’ (array transpose)
 
Matrix and Array Arithmetic Examples

Scalar and matrix operands

If L=ones(1,5) and M=ones(1,4)



N = 2*L --> ?
N – 1 --> ?
Non-scalar operations

If O = [ 1:5 ]

O + M --> ?
??? Error using ==> +
Matrix dimensions must agree.

L * O --> ?
??? Error using ==> * (1x5 * 1x5 does not work!)
Inner matrix dimensions must agree.

L* O’ --> ?
15 Same as sum(L .* O)
(1x5 * 5x1 works! Inner matrix dimensions agree. Results in a 1x1)
Array Operations

Element-by-element array operation

Ex: Given a set of distances and times, calculate
average speeds and maximum of averages


How would you do this in C?
Pseudocode:



Calculate avg. speeds: speed[i] = distance[i] / time[i], for i=1 to 4
Determine maximum speed
M/O/F (vectorize!):





distances=[120, 213, 87, 35] (in miles)
times=[ 2, 3.8, 0.9, 0.6] (in hours)
speeds=distances ./ times %( note: ‘dot /’  divide element-by-element)
max_speed=max(speeds)
To get the maximum speed and its index:
 [max_speed, i] = max(speeds)
Circuit Analysis Equations
i2
i1
R2

+V
i3
R3
Matrix operations
R1

i1  i2  i3  0
Matrix division

0i1  R2  R3 i2  0i3  V
Recall the circuit analysis




0i1  0i2  i3 R1  V
R1=10k
R2=R3=5k
V=10V
Matrix solution
1
0

0
1
Ri  V R 1Ri  R 1V  i  R V
1
0
R2  R3 
R
 1  i1   0 
R1  i2   V 
   
0  i3  V 
i V
use ' left' division to solve for i
i  R \ V Think of it like inverting R and multiplying on the left side of V
If we had iR = V instead, we’d use ‘right’ division to solve for i: ( i = R / V )
Think of it like inverting R and multiplying on the right side of V: i = VR-1
Circuit Analysis Solution

Circuit analysis solution:


Build R, build V, solve for i
Build R

all at once





eq1 = [ 1 -1 1]
eq2 = [0 0 10e3]
eq3 = [0 10e3 0]
R = [eq1; eq2; eq3]
Build V


R=[1 -1 1; 0 0 10e3; 0 10e3 0]
or
build by rows and combine


•R1=10k
•R2=R3=5k
•V=10V
V = [0 10 10]’
Solve I = R \ V

I=R\V
(note: transposed)
1
0

0
1
0
R2  R3 
R
 1  i1   0 
R1  i2   V 
   
0  i3  V 
i V
Dot Product Example

Another example of element-by-element

operations
v1
 

dot product of two vectors

v1  3iˆ  2 ˆj  5kˆ

 
 v1  v2  v1 v2 cos( )
v2

v2  2iˆ  4 ˆj  10kˆ
iˆ, ˆj, kˆ are unit vecto rs for a cartesian coordinate system
their coefficien ts are called ' measure numbers'
 
what is v1  v2?
 
v1 v2  (3)(2)  (2)(4)  (5)(10)  6  8  50  52
Dot Product Function Development

Define the problem


Inputs


v1, v2 (three-element row vectors)
Outputs


Create a function that will take two vectors as arguments and
will return their vector dot product
z (the dot product)
Algorithm



Multiply v1 and v2 element-by-element
Sum the element-by-element products
Return the sum
Try it in Matlab/Oct ave :
v1  [3 2  5]
sum(v1 . * v 2)
v 2  [2  4 10]
Dot Product Function in M/O/F

Write the function
function [z] = dot_prod(v1, v2)
% dot_prod(v1,v2) computes the vector dot product between vectors v1 and v2
% Function dot_prod(v1,v2) computes and returns the vector dot product between
vectors v1 and v2
z = sum(v1.*v2);

Test it out
A = [ 1 2 3 ];
B = [ 4 5 6 ];
% what should A dot B result in?
A_dot_B = dot_prod(A,B)
Review of Functions

Functions

Like script M-files, but several differences:

first line (function declaration) must be of the form:
function [output args] = function_name(input args)




variables generated in the function are local to the
function, whereas for script files, variables are global
must be named, ‘function_name.m’ (same as file name)
Make sure you add comments at the start that
describe what the function does (see example code)
N
Example: root-mean-square function,
2
x
rms.m
i
Given, x  [ x1 , x2 ,..., x N ] RMS  i 1
N

Root Mean-Square Function Development

Functions, cont.

Example: root-meansquare function, cont.

Pseudocode:





Square each element


xs = x .^2
Sum the squares


square each element of
x
sum the squares
divide by N
take the square root
sums = sum(xs)
Divide by N


N = length(x)
ms = sums/N
N
Given, x  [ x1 , x2 ,..., x N ] RMS 

Take the square root


rms = sqrt(ms)
Before you write the
function, make sure the
name you propose is
not already used!

Use: which name
to check

xi2
i 1
N
Root Mean-Square Function Implementation

Functions, cont.

Example: root-mean-square
function, cont.
H1 comment line
(used in lookfor)
Comments that
will be displayed
by help command
function [y] = rms(v)
% RMS(v) root mean square of the elements of the column vector v
% Function rms(v) returns the root mean square of the elements
% of the column vector, v
vs = v.^2; % what does this line do? Also note semicolon.
s = length(v);
y = sqrt(sum(vs)/s);
Let v=sin([0: 0.01*pi: 2*pi]’), one period of a sine wave. The RMS value
of a sine wave is its amplitude*1/sqrt(2)
Does rms() work with a row vector? How about a matrix?
More Robust Root Mean-Square Function
 Functions,
cont.

Make rms
function more
robust

to work with
row or column
vector or
matrix with
column vectors
of data
function [rmsout] = rms2(v)
%RMS2(v) Root mean square of v
% Function rms2(v) returns a row vector, where
% each element is the rms value of values in each
% column of v
vs = v.^2;
s = size(v);
rmsout = sqrt(sum(vs,1)/s(1));
File I/O with M/O/F

Data Input - simplest method

load command

Ex: load(‘data_file.txt’)
reads on a row-by-row basis
 data values separated by spaces or commas and
rows terminated by new line
 columns must have the same number of elements
 data is stored in workspace in an array with same
name as the argument used in the load function
 Ex. Portland International Airport monthly rainfall


load (‘PDXprecip.dat’) % must be in search path!
File I/O with M/O/F, cont.

Data Output - simplest method

Save command

Ex: save(‘data_file_name’)


Saves all the variables into a .mat file named
‘data_file_name’
Many other commands are available for special
purpose file I/O
File I/O and Plotting Example
% read data into PDXprecip matrix
load('PDXprecip.dat');
% copy first column of PDXprecip into month
month = PDXprecip(:,1);
% and second column into precip
precip = PDXprecip(:,2);
% plot precip vs. month with circles
plot(month,precip,'o');
% add axis labels and plot title
xlabel('month of the year');
ylabel('mean precipitation (inches)');
title('Mean monthly precipitation at Portland International Airport');
file_io_example.m
Adapted from: http://web.cecs.pdx.edu/~gerry/MATLAB/plotting/loadingPlotData.html
visited 15NOV2009
More on Plotting




Add a red line through the
data
Plot multiple sets of data
on a single graph and add
a legend
grid on
Sub-plots

Format:



subplot (m,n,p)
Figure window divided
into m x n matrix of
plotting areas
Procedure:


Pick the sub-plot
window
Execute plot
commands for that
sub-plot
General Format:
plot (x, y, fmt, ...)
% plot precip vs. month with circles
plot(month,precip,'o',month,precip,'-r');
% copy first column of PDXtemperature into month
month = PDXtemperature(:,1);
% and second column into high_temp
high_temp = PDXtemperature(:,2);
% and third column into low temp
low_temp = PDXtemperature(:,3);
% and fourth column into avg temp
avg = PDXtemperature(:,4);
% generate the plot
plot(month,high_temp,'ko',month,low_temp,'k+',month,avg,‘r-');
% add axis labels and plot title
xlabel('Month');
ylabel('temperature (degrees F)');
title('Monthly average temperature for PDX');
% add a plot legend using labels read from the file
legend('High','Low','Avg');
multi_plot.m
Vector Dot Product Example
Find the X and Y components of the vector, V


 

that is, find v x and v y , so that v x  v y  v

v

v
Y
ĵ


iˆ

vy

vx
X
    ˆ  v iˆ cos( )  v (1) cos( )  v cos( )
vx v i
x
x
x





v y  v  ˆj  v y ˆj cos(90   )  v y (1) cos(90   )  v y sin( )
Back
Review
References

Matlab. (2009, November 6). In Wikipedia, the free encyclopedia.
Retrieved November 6, 2009, from
http://en.wikipedia.org/wiki/Matlab

Matlab tutorials:
http://www.mathworks.com/academia/student_center/tutorials/launchpad.html

GNU Octave. (2009, October 31). In Wikipedia, the free
encyclopedia. Retrieved November 6, 2009, from
http://en.wikipedia.org/wiki/GNU_Octave

Octave main page: http://www.gnu.org/software/octave/
(http://octave.sourceforge.net/ access to pre-built installers)

Octave tutorials: http://homepages.nyu.edu/~kpl2/dsts6/octaveTutorial.html,
http://smilodon.berkeley.edu/octavetut.pdf


FreeMat. http://freemat.sourceforge.net/index.html
ftp://www.chabotcollege.edu/faculty/bmayer/ChabotEngineeringCour
ses/ENGR-25.htm