Some basic commands you will need:
matlab loads the program matlab into your workspace quit quits matlab, returning you to the operating system exit same as quit who lists all of the variables in your matlab workspace whos list the variables and describes their matrix size.
NOTE - When using the workstations, clicking on UP ARROW will recall previous commands. If you make a mistake, the DELETE key OR the backspace key may be used to correct the error; however, one of
these two keys may be inoperable on particular systems.
'matlab' uses variables that are defined to be matrices. A matrix is a collection of numerical values that are organized into a specific configuration of rows and columns. The number of rows and columns can be any number, for example, 3 rows and 4 columns
define a 3 x 4 matrix which has 12 elements in total. A scalar is represented by a 1 x 1 matrix in matlab. A vector of n dimensions or elements can be represented by a n x 1 matrix, in which case it is called a column vector, or a vector can be represented by a 1 x
n matrix, in which case it is called a row vector of n elements.The matrix name can be any group of letters and numbers up to 19,but always beginning with a letter. Thus 'x1' can be a variable name, but '1x' is illegal. 'supercalafragilesticexpealladotious'
can be a variable name; however, only the first 19 characters will be stored! Understand that 'matlab' is "case sensitive", that is,it treats the name 'C' and 'c' as two different variables.Similarly, 'MID' and 'Mid' are treated as two different variables.
Here are examples of matrices that could be defined in 'matlab'.
Note that the set of numerical values or elements of the matrix are
bounded by brackets ......[ ].
c = 5.66 or c = [5.66] c is a scalar or
a 1 x 1 matrix
x = [ 3.5, 33.22, 24.5 ] x is a row vector or
1 x 3 matrix
x1 = [ 2 x1 is column vector or
5 4 x 1 matrix
3
-1]
A = [ 1 2 4 A is a 4 x 3 matrix
2 -2 2
0 3 5
5 4 9 ]
An individual element of a matrix can be specified with the
notation A(i,j) or Ai,j for the generalized element, or by A(4,1)=5
for a specific element.
When 'matlab' prints a matrix on the monitor, it will be organized
according to the size specification of the matrix, with each row appearing on a unique row of the monitor screen and with each column aligned vertically and right-justified.
The numerical values that are assigned to the individual elements of a matrix can be entered into the variable assignment in a number of ways. The simplest way is by direct keyboard entry; however,
large data sets may be more conveniently entered through the use of stored files or by generating the element values using matlab expressions. First, we will look at the use of the keyboard for direct entry.
KEYBOARD DEFINITION OR ENTRY FOR A MATRIX
A matrix can be defined by a number of matlab expressions. Examples
are listed below for a 1 x 3 row vector, x, whose elements are
x(1) = 2, x(2) = 4 and x(3) = -1.
x = [ 2 4 -1 ] or x=[2 4 -1] or x = [ 2,4,-1 ]
(A keystroke 'enter' follows each of the above matlab statements.)
Notice that brackets must be used to open and close the set of numbers, and notice that commas or blanks may be used as delimiters between the fields defining the elements of the matrix. Blanks used
around the = sign and the brackets are superfluous; however, they sometimes make the statement more readable.
A 2x4 matrix, y, whose elements are y(1,1)=0, y(1,2) = y(1,3) = 2,
y(1,4) = 3, y(2,1) = 5, y(2,2) = -3, y(2,3) = 6 and y(2,4) = 4 can
be defined
y = [ 0 2 2 3
5 -3 6 4 ]
or
y = [ 0 2 2 3 ; 5 -3 6 4 ]
The semicolon ";" is used to differentiate the matrix rows when
they appear on a single line for data entry.
The elements of a matrix can be defined with algebraic expressions
placed at the appropriate location of the element. Thus
a = [ sin(pi/2) sqrt(2) 3+4 6/3 exp(2) ]
defines the matrix
a = [ 1.0000 1.4142 7.0000 2.0000 7.3891 ]
A matrix can be defined by augmenting previously defined matrices.
Recalling the matrix, x, defined earlier
x1 = [ x 5 8 ] creates the result
x1 = [ 2 4 -1 5 8 ]
The expression
x(5) = 8
creates
x = [ 2 4 -1 0 8 ]
Notice that the value "0" is substituted for x(4) which has not
been explicitly defined.
Recalling the definition of matrix, y, above, the expressions
c = [ 4 5 6 3 ]
z = [ y;c ]
creates
z = [ 0 2 2 3
5 -3 6 4
4 5 6 3 ]
Note that every time a matrix is defined and an 'enter' keystroke
is executed, matlab echoes back the result. TO CANCEL THIS ECHO,
THE MATLAB COMMAND LINE CAN INCLUDE A SEMICOLON AT THE END OF THE
LINE BEFORE THE KEYSTROKE 'ENTER'.
z = [ y ; c ] ;
LINE CONTINUATION
Occasionally, a line is so long that it can not be expressed in
the 80 spaces available on a line, in which case a line
continuation is needed. In matlab, the ellipsis defining a line
continuation is three successive periods, as in "...". Thus
4 + 5 + 3 ...
+ 1 + 10 + 2 ...
+ 5
gives the result
ans = 30
Notice that in this simple arithmetic operation, no matrix was
defined. When such an operation is executed in matlab, the result
is assigned to the matrix titled "ans". A subsequent operation
without an assignment to a specific matrix name will replace the
results in 'ans' by the result of the next operation. In the above,
'ans' is a 1x1 matrix, but it need not be so in general.
BEFORE YOU QUIT THIS SESSION !!!!!
If this is your first lesson using matlab, execute the matlab
commands 'who' and whos' before you 'quit'. Note that each of these
commands lists the matrices you have defined in this session on the
computer. The command 'whos' also tells you the properties of each
matrix, including the number of elements, the row and column size
(row x column) and whether the elements are complex or real.
IMPORTANT! If you execute the matlab command 'save' before you
quit, all of the matrices that have been defined will be saved in
a file titled matlab.mat stored in your workspace. Should you
desire to save specific matrices during any session, the command
'save' followed by the name of the matrix can be executed. More
detail on how to save and recall your matrices is discussed in
Lesson 2.
PRACTICE PROBLEMS
Determine the size and result for the following matrices.
Subsequently, carry out the operations on matlab that define the
matrices, and check your results using the 'whos' statement.
1. a = [1,0,0,0,0,1]
2. b = [2;4;6;10]
3. c = [5 3 5; 6 2 -3]
4. d= [3 4
5 7
9 10 ]
5. e = [3 5 10 0; 0 0 ...
0 3; 3 9 9 8 ]
6. t = [4 24 9]
q = [t 0 t]
7. x = [ 3 6 ]
y = [d;x]
z = [x;d]
8. r = [ c; x,5]
9. v = [ c(2,1); b ]
10. a(2,1) = -3 (NOTE: Recall matrix "a" was defined in (1)
above.)
New commands in this lesson:
save saves all the matrices defined in the current
session into the file, matlab.mat, located in
the directory from which you executed matlab.
load loads contents of matlab.mat into current workspace
save filename x y z
save the matrices x, y and z into the file titled
filename.mat.
load filename loads the contents of filename into current
workspace; the file can be a binary (.mat) file
or an ASCII file.
clear x erases the matrix 'x' from your workspace
clear erases ALL matrices from your workspace
NOTE - When using PROMATLAB on a workstation, files are stored in
the directory from which you invoked the 'matlab' command. When
using the workstation, create a matlab subdirectory titled 'matlab'
or some similar name. Thereafter, store all files and conduct all
matlab sessions in that subdirectory.
ASCII FILES
An ASCII file is a file containing characters in ASCII format, a
format that is independent of 'matlab' or any other executable
program. Using ASCII format, you can build a file using a screen
editor or a wordprocessing program, for example, that can be read
and understood by 'matlab'. The file can be read into 'matlab'
using the "load" command described above.
Using a text editor or using a wordprocessor that is capable of
writing a file in ASCII format, you simply prepare a matrix file
for which EACH ROW OF THE MATRIX IS A UNIQUE LINE IN THE FILE, with
the elements in the row separated by blanks. An example is the 3 x
3 matrix
2 3 6
3 -1 0
7 0 -2
If these elements are formatted as described above and stored in a
filename titled, for example, x.dat, the 3 x 3 matrix 'x' can be
loaded into 'matlab' by the command
load x.dat
Open the Text Editor window on your workstation. Build a 3x3
matrix in the editor that follows the format explained above.
Store this in your matlab directory using the command
save ~/matlab/x.dat
The suffix, .dat, is not necessary; however, it is strongly
recommended here to distinguish the file from other data files,
such as the .mat files which will be described below. Of course if
you define the file with the .dat or any other suffix, you must use
that suffix when downloading the file to 'matlab' with the 'load'
command. Remember, the file must be stored in the directory in
which you are executing matlab. In the above example, it is
assumed that this directory is titled 'matlab'.
Now go to a window in which matlab is opened. We desire to
load the matrix x into matlab. We might wonder if the file x.dat
is actually stored in our matlab directory. To review what files
are stored therein, the unix command 'ls' can be evoked in matlab
by preceding the command with an exclamation mark, ! . Type the
command
! ls
Matlab will list all of the files in your directory. You should
find one titled "x.dat".
Now, type the command
load x.dat
The 3x3 matrix defined above is now down-loaded into your working
space in matlab. To ensure that is so, check your variables by
typing 'who' or 'whos'.
FILES BUILT BY MATLAB....THE .mat FILE
If at any time during a matlab session you wish to store a matrix
that has been defined in the workspace, the command
save filename
will create a file in your directory that is titled filename.mat.
The file will be in binary format, understandable only by the
'matlab' program. Note that 'matlab' appends the .mat suffix and
this can be omitted by the user when describing the filename.
If the matrix 'x' has been defined in the workspace, it can be
stored in your directory in binary matlab format with the command
save x
The filename need not be the same as the matrix name. The file
named, for example, 'stuff' can be a file that contains the matrix
'x'. Any number of matrices can be stored under the filename. Thus
the command string
save stuff x y z
will store the matrices x, y and z in the file titled 'stuff'. The
file will automatically have the suffix .mat added to the
filename. If, during a session, you type the command 'save' with
no other string, all of the current matrices defined in your
workspace will be stored in the file 'matlab.mat'.
The load command when applied to .mat files follows the same
format as discussed above for ASCII files; however, you can omit
the file suffix '.mat' when loading .mat files to your workspace.
COLON OPERATOR
The colon operator ' : ' is understood by 'matlab' to perform
special and useful operations. If two integer numbers are separated
by a colon, 'matlab' will generate all of the integers between
these two integers.
a = 1:8
generates the row vector, a = [ 1 2 3 4 5 6 7 8 ].
If three numbers, integer or non-integer, are separated by two
colons, the middle number is interpreted to be a "range" and the
first and third are interpreted to be "limits". Thus
b = 0.0 : .2 : 1.0
generates the row vector b = [ 0.0 .2 .4 .6 .8 1.0 ]
The colon operator can be used to create a vector from a matrix.
Thus if
x = [ 2 6 8
0 1 7
-2 5 -6 ]
The command
y = x(:,1)
creates the column vector
y = [ 2
0
-2 ]
and
yy = x(:,2)
creates
yy = [ 6
1
5 ]
The command
z = x(1,:)
creates the row vector
z = [ 2 6 8 ]
The colon operator is useful in extracting smaller matrices from
larger matrices. If the 4 x 3 matrix c is defined by
c = [ -1 0 0
1 1 0
1 -1 0
0 0 2 ]
Then
d1 = c(:,2:3)
creates a matrix for which all elements of the rows from the 2nd
and third columns are used. The result is a 4 x 2 matrix
d1 = [ 0 0
1 0
-1 0
0 2 ]
The command
d2 = c(3:4,1:2)
creates a 2 x 2 matrix in which the rows are defined by the 3rd and
4th row of c and the columns are defined by the 1st and 2nd columns
of the matrix, c.
d2 = [ 1 -1
0 0 ]
USING THE CLEAR COMMANDS
Before quitting this session of 'matlab', note the use of the
commands 'clear' and 'clc'. Note that 'clc' simply clears the
screen, but does not clear or erase any matrices that have been
defined. The command 'clear' erase or removes matrices from your
workspace. Use this command with care. NOTE THE COMMAND 'CLEAR'
WITHOUT ANY MODIFYING STRING WILL ERASE ALL MATRICES IN YOUR
WORKSPACE.
PROBLEMS
Define the 5 x 4 matrix, g.
g = [ 0.6 1.5 2.3 -0.5
8.2 0.5 -0.1 -2.0
5.7 8.2 9.0 1.5
0.5 0.5 2.4 0.5
1.2 -2.3 -4.5 0.5 ]
Determine the content and size of the following matrices and check
your results for content and size using 'matlab'.
1. a = g(:,2)
2. b = g(4,:)
3. c = [10:15]
4. d = [4:9;1:6]
5. e = [-5,5]
6. f= [1.0:-.2:0.0]
7. t1 = g(4:5,1:3)
8. t2 = g(1:2:5,:)
New commands in this lesson:
plot(x,y) creates a Cartesian plot of the vectors x & y
plot(y) creates a plot of y vs. the numerical values of the
elements in the y-vector.
semilogx(x,y) plots log(x) vs y
semilogy(x,y) plots x vs log(y)
loglog(x,y) plots log(x) vs log(y)
grid creates a grid on the graphics plot
title('text') places a title at top of graphics plot
xlabel('text') writes 'text' beneath the x-axis of a plot
ylabel('text') writes 'text' beside the y-axis of a plot
text(x,y,'text') writes 'text' at the location (x,y)
text(x,y,'text','sc') writes 'text' at point x,y assuming
lower left corner is (0,0) and upper
right corner is (1,1).
polar(theta,r) creates a polar plot of the vectors r & theta
where theta is in radians.
bar(x) creates a bar graph of the vector x. (Note also
the command stairs(x).)
bar(x,y) creates a bar-graph of the elements of the vector y,
locating the bars according to the vector elements
of 'x'. (Note also the command stairs(x,y).)
CARTESIAN OR X-Y PLOTS
One of 'matlab' most powerful features is the ability to
create graphic plots. Here we introduce the elementary ideas for
simply presenting a graphic plot of two vectors. More complicated
and powerful ideas with graphics can be found in the 'matlab'
documentation.
A Cartesian or orthogonal x,y plot is based on plotting the
x,y data pairs from the specified vectors. Clearly, the vectors x
and y must have the same number of elements. Imagine that you wish
to plot the function ex for values of x from 0 to 2.
x = 0:.1:2;
y = exp(x);
plot(x,y)
NOTE: The use of selected functions such as exp() and sin() will be
used in these tutorials without explanation if they take on an
unambiguous meaning consistent with past experience. Here it is
observed that operating on a matrix with these functions simply
creates a matrix in which the elements are based on applying the
function to each corresponding element of the argument.
Of course the symbols x,y are arbitrary. If we wanted to plot
temperature on the ordinate and time on the abscissa, and vectors
for temperature and time were loaded in 'matlab', the command would
be
plot(time,temperature)
Notice that the command plot(x,y) opens a graphics window. If you
now execute the command 'grid', the graphics window is redrawn.
(Note you move the cursor to the command window before typing new
commands.) To avoid redrawing the window, you may use the line
continuation ellipsis. Consider
plot(x,y),...
grid,...
title('Exponential Function'),...
xlabel('x'),...
ylabel('exp(x)'),
text(.6,.4,' y = exp(x)','sc')
Note that if you make a mistake in typing a series of lines of
code, such as the above, the use of line continuation can be
frustrating. In a future lesson, we will learn how to create batch
files (called '.m' files) for executing a series of 'matlab'
commands. This approach gives you better opportunity to return to
the command string and edit errors that have been created.
Having defined the vectors x and y, construct semilog and log-
log plots using these or other vectors you may wish to create. Note
in particular what occurs when a logarithmic scale is selected for
a vector that has negative or zero elements. Here, the vector x has
a zero element (x(1)=0). The logarithm of zero or any negative
number is undefined and the x,y data pair for which a zero or
negative number occurs is discarded and a warning message provided.
Try the commands plot(x) and plot(y) to ensure you understand
how they differ from plot(x,y).
Notice that 'matlab' draws a straight line between the data-
pairs. If you desire to see a relatively smooth curve drawn for a
rapidly changing function, you must include a number of data
points. For example, consider the trigonometric function, sin(x1)
(x1 is selected as the argument here to distinguish it from the
vector 'x' that was defined earlier. )
Plot sin(x1) over the limits 0 <= x1 <= pi. First define x1
with 5 elements,
x1 = 0 : pi/4 : pi
y1 = sin(x1)
plot(x1,y1)
Now, repeat this after defining a new vector for x1 and y1 with 21
elements. (Note the use of the semicolon here to prevent the
printing and scrolling on the monitor screen of a vector or matrix
with a large number of elements!)
x1 = 0 : .05*pi : pi ;
y1 =sin(x1);
plot(x1,y1)
POLAR PLOTS
Polar plots are constructed much the same way as are Cartesian
x-y plots; however, the arguments are the angle 'theta' in radians
measured CCW from the horizontal (positive-x) axis, and the length
of the radius vector extended from the origin along this angle.
Polar plots do not allow the labeling of axes; however, notice that
the scale for the radius vector is presented along the vertical and
when the 'grid' command is used the angles-grid is presented in 15-
degree segments. The 'title' command and the 'text' commands are
functional with polar plots.
angle = 0:.1*pi:3*pi;
radius = exp(angle/20);
polar(angle,radius),...
title('An Example Polar Plot'),...
grid
Note that the angles may exceed one revolution of 2*pi.
BAR GRAPHS
To observe how 'matlab' creates a bargraph, return to the
vectors x,y that were defined earlier. Create bar and stair graphs
using these or other vectors you may define. Of course the 'title'
and 'text' commands can be used with bar and stair graphs.
bar(x,y) and bar(y)
stair (x,y) and stair(y)
MULTIPLE PLOTS
More than a single graph can be presented on one graphic plot.
One common way to accomplish this is hold the graphics window open
with the 'hold' command and execute a subsequent plotting command.
x1=0:.05*pi:pi;
y1=sin(x1);
plot(x1,y1)
hold
y2=cos(x1);
plot(x1,y2)
The "hold" command will remain active until you turn it off with
the command 'hold off'.
You can create multiple graphs by using multiple arguments.
In addition to the vectors x,y created earlier, create the vectors
a,b and plot both vector sets simultaneously as follows.
a = 1 : .1 : 3;
b = 10*exp(-a);
plot(x,y,a,b)
Multiple plots can be accomplished also by using matrices
rather than simple vectors in the argument. If the arguments of
the 'plot' command are matrices, the COLUMNS of y are plotted on
the ordinate against the COLUMNS of x on the abscissa. Note that x
and y must be of the same order! If y is a matrix and x is a
vector, the rows or columns of y are plotted against the elements
of x. In this instance, the number of rows OR columns in the matrix
must correspond to the number of elements in 'x'. The matrix 'x'
can be a row or a column vector!
Recall the row vectors 'x' and 'y' defined earlier. Augment
the row vector 'y' to create the 2-row matrix, yy.
yy=[y;exp(1.2*x)];
plot(x,yy)
PLOTTING DATA POINTS & OTHER FANCY STUFF
Matlab connects a straight line between the data pairs
described by the vectors used in the 'print' command. You may wish
to present data points and omit any connecting lines between these
points. Data points can be described by a variety of characters
( . , + , * , o and x .) The following command plots the x,y data
as a "curve" of connected straight lines and in addition places an
'o' character at each of the x1,y1 data pairs.
plot(x,y,x1,y1,'o')
Lines can be colored and they can be broken to make
distinctions among more than one line. Colored lines are effective
on the color monitor and color printers or plotters. Colors are
useless on the common printers used on this network. Colors are
denoted in 'matlab' by the symbols r(red), g(green), b(blue),
w(white) and i(invisible). The following command plots the x,y
data in red solid line and the r,s data in broken green line.
plot(x,y,'r',r,s,'--g')
PRINTING GRAPHIC PLOTS
Executing the 'print' command will send the contents of the
current graphics widow to the local printer.
You may wish to save in a graphics file, a plot you have
created. To do so, simply append to the 'print' command the name
of the file. The command
print filename
will store the contents of the graphics window in the file titled
'filename.ps' in a format called postscript. You need not include
the .ps suffix, as matlab will do this. When you list the files
in your matlab directory, it is convenient to identify any graphics
files by simply looking for the files with the .ps suffix. If
you desire to print one of these postscript files, you can
conveniently "drag and drop" the file from the file-manager window
into the printer icon.
Commands introduced in this lesson:
max(x) returns the maximum value of the elements in a
vector or if x is a matrix, returns a row vector
whose elements are the maximum values from each
respective column of the matrix.
min (x) returns the minimum of x (see max(x) for details).
mean(x) returns the mean value of the elements of a vector
or if x is a matrix, returns a row vector whose
elements are the mean value of the elements from
each column of the matrix.
median(x) same as mean(x), only returns the median value.
sum(x) returns the sum of the elements of a vector or if x
is a matrix, returns the sum of the elements from
each respective column of the matrix.
prod(x) same as sum(x), only returns the product of
elements.
std(x) returns the standard deviation of the elements of a
vector or if x is a matrix, a row vector whose
elements are the standard deviations of each
column of the matrix.
sort(x) sorts the values in the vector x or the columns of
a matrix and places them in ascending order. Note
that this command will destroy any association that
may exist between the elements in a row of matrix x.
hist(x) plots a histogram of the elements of vector, x. Ten
bins are scaled based on the max and min values.
hist(x,n) plots a histogram with 'n' bins scaled between the
max and min values of the elements.
hist((x(:,2)) plots a histogram of the elements of the 2nd
column from the matrix x.
The 'matlab' commands introduced in this lesson perform
simple statistical calculations that are mostly self-explanatory.
A simple series of examples are illustrated below. Note that
'matlab' treats each COLUMN OF A MATRIX as a unique set of "data",
however, vectors can be row or column format. Begin the exercise
by creating a 12x3 matrix that represents a time history of two
stochastic temperature measurements.
Load these data into a matrix called 'timtemp.dat' using an
editor to build an ASCII file.
Time(sec) Temp-T1(K) Temp-T2(K)
0.0 306 125
1.0 305 121
2.0 312 123
3.0 309 122
4.0 308 124
5.0 299 129
6.0 311 122
7.0 303 122
8.0 306 123
9.0 303 127
10.0 306 124
11.0 304 123
Execute the commands at the beginning of the lesson above and
observe the results. Note that when the argument is 'timtemp', a
matrix, the result is a 1x3 vector. For example, the command
M = max(timtemp)
gives
M = 11 312 129
If the argument is a single column from the matrix, the
command identifies the particular column desired. The command
M2 = max(timtemp(:,2))
gives
M2 = 312
If the variance of a set of data in a column of the matrix is
desired, the standard deviation is squared. The command
T1_var = (std(timtemp(:,2)))^2
gives
T1_var = 13.2727
If the standard deviation of the matrix of data is found using
STDDEV = std(timtemp)
gives
STDDEV = 3.6056 3.6432 2.3012
Note that the command
VAR = STDDEV^2
is not acceptable; however, the command
VAR = STDDEV.^2
is acceptable, creating the results,
VAR = 13.0000 13.2727 5.2955
USING m-FILES - SCRATCH and FUNCTION FILES
Sometimes it is convenient to write a number of lines of
'matlab' code before executing the commands. We have seen how this
can be accomplished using the line continuation ellipsis; however,
it was noted in this approach that any mistake in the code required
the entire string to be entered again. Using an m-file we can
write a number of lines of 'matlab' code and store it in a file
whose name we select with the suffix '.m' added. Subsequently the
command string or file content can be executed by invoking the name
of the file while in 'matlab'. This application is called a
SCRATCH FILE.
On occasion it is convenient to express a function in 'matlab'
rather than calculating or defining a particular matrix. The many
'matlab' stored functions such as sin(x) and log(x) are examples of
functions. We can write functions of our definition to evaluate
parameters of our particular interest. This application of m-files
is called FUNCTION FILES.
SCRATCH FILES
A scratch file must be prepared in a text editor in ASCII
format and stored in the directory from which you invoked the
command to download 'matlab'. The name can be any legitimate file
name with the '.m' suffix.
As an example, suppose we wish to prepare a x-y plot of the
function
y = e-x/10 sin(x) 0 x ø 10 .
To accomplish this using a scratch ".m-file", we will call the file
'explot.m'. Open the file in a text editor and type the code
below. (Note the use of the '%' in line 1 below to create a
comment. Any text or commands typed on a line after the'%' will be
treated as a comment and ignored in the executable code.)
% A scratch m-file to plot exp(-x/10)sin(x)
x = [ 0:.2:10 ];
y = exp(-x/10) .* sin(x);
plot(x,y),...
title('EXPONENTIAL DAMPED SINE FUNCTION'),...
xlabel('x'),...
ylabel('y'),...
text(.6,.7,'y = exp(-x/10)*sin(x)','sc')
Store this file under the name 'explot.m in your 'matlab'
directory. Now when you are in 'matlab', any time you type the
command 'explot' the x-y plot of the damped exponential sine
function will appear.
FUNCTION FILES
Suppose that you wish to have in your 'matlab' workspace a
function that calculates the sine of the angle when the argument is
in degrees. The matlab function sin(x) requires that x be in
radians. A function can be written that use the 'matlab' sin(x)
function, where x is in radians, to accomplish the objective of
calculating the sine of an argument expressed in degrees. Again,
as in the case of the scratch file, we must prepare the file in
ASCII format using a text editor, and we must add the suffix '.m'
to the file. HERE THE NAME OF THE FILE MUST BE THE NAME OF THE
FUNCTION.
The following code will accomplish our objective. Note that the
first line must always begin with the word "function" followed by
the name of the function expressed as " y = function-name". Here
the function name is "sind(x)". This is the name by which we will
call for the function once it is stored.
function y = sind(x)
% This function calculates the sine when the argument is degrees
% Note that array multiplication and division allows this to
% operate on scalars, vectors and matrices.
y = sin( x .* pi ./ 180 )
Again, note the rule in writing the function is to express in the
first line of the file the word 'function', followed by
y = function call,
here 'sind' with the function argument in parentheses, 'x' is the
function call. Thus,
function y=sind(x)
Now every time we type sind(x) in 'matlab', it will return the
value of the sine function calculated under the assumption that x
is in degrees. Of course any matrix or expression involving a
matrix can be written in the argument.
ALGEBRAIC OPERATIONS IN MATLAB
Scalar Calculations. The common arithmetic operators used in
spreadsheets and programming languages such as BASIC are used in
'matlab'. In addition a distinction is made between right and left
division. The arithmetic operators are
+ addition
- subtraction
* multiplication
/ right division (a/b means a ÷ b)
\ left division (a\b means b ÷ a)
^ exponentiation
When a single line of code includes more than one of these
operators the precedence or order of the calculations follows the
below order:
Precedence Operation
1 parentheses
2 exponentiation, left to right
3 multiplication and division, left right
4 addition and subtraction, left right
These rules are applied to scalar quantities (i.e., 1x1 matrices)
in the ordinary manner. (Below we will discover that nonscalar
matrices require additional rules for the application of these
operators!!) For example,
3*4 executed in 'matlab' gives
ans=12
4/5 executed in 'matlab' gives
ans=.8000
4\5 executed in 'matlab' gives
ans=1.2500
x = pi/2; y = sin(x) executed in 'matlab' gives
y = 1
z = 0; w = exp(4*z)/5 executed in 'matlab" gives
z= .2000
Note that many programmers will prefer to write the expression for
w above in the format
w = (exp(4*x))/5
which gives the same result and is sometimes less confusing when
the string of arithmetic operations is long. Using 'matlab', carry
out some simple arithmetic operations with scalars you define. In
doing so utilize the 'matlab' functions sqrt(x), abs(s), sin(x),
asin(x), atan(x), atan2(x), log(x), and log10(x) as well as exp(x).
Many other functions are available in 'matlab' and can be found in
the documentation.
Matrix Calculations. Because matrices are made up of a number
of elements and not a single number (except for the 1x1 scalar
matrix), the ordinary rules of commutative, associative and
distributive operations in arithmetic do not always follow.
Moreover, a number of important but common-sense rules will prevail
in matrix algebra and 'matlab' when dealing with nonscalar
quantities.
Addition and Subtraction of Matrices. Only matrices of the SAME
ORDER can be added or subtracted. When two matrices of the same
order are added or subtracted in matrix algebra, the individual
elements are added or subtracted. Thus the distributive rule
applies.
A + B = B + A and A - B = B - A
If C = A + B then each element Cij = Aij + Bij.
Define A and B as follows:
A=[1 2 3; 3 3 3; 5 3 1]
B=[2 -3 4;2 -2 2; 0 4 0]
Then note that
C = A + B and C = B + A
gives
C =
3 -1 7
5 1 5
5 7 1
Now define the row vector
x= [3 5 7]
and the column vector
y = [4; -1; -3]
Note that the operation
z = x + y
is not valid! It is not valid because the two matrices do not have
the same order. (x is a 1x3 matrix and y is a 3x1 matrix.) Adding
any number of 1x1 matrices or scalars is permissible and follows
the ordinary rules of arithmetic because the 1x1 matrix is a
scalar. Adding two vectors is permissible so long as each is a row
vector (1xn matrix) or column vector (nx1 matrix). Of course any
number of vectors can be added or subtracted, the result being the
arithmetic sum of the individual elements in the column or row
vectors. Square matrices can always be added or subtracted so long
as they are of the same order. A 4x4 square matrix can not be added
to a 3x3 square matrix, because they are not of the same order,
although both matrices are square.
Multiplication of Matrices. Matrix multiplication, though straight-
forward in definition, is more complex than arithmetic
multiplication because each matrix contains a number of elements.
Recall that with vector multiplication, the existence of a number
of elements in the vector resulted in two concepts of
multiplication, the scalar product and the vector product. Matrix
multiplication has its array of special rules as well.
In matrix multiplication, the elements of the product, C, of
two matrices A*B are calculated from
Cij = ‘ Aik * Bkj
To form this sum, the number of columns of the first or left matrix
(A) must be equal to the number of rows in the second or right
matrix (B). The resulting product, matrix C, has an order for which
the number of rows equals the number of rows of the first (left)
matrix (A) and the product (C) has a number of columns equal to the
number of columns in the second (right) matrix (B). It is clear
that A*B IS NOT NECESSARILY EQUAL TO B*A! It is also clear that
A*B and B*A only exist for square matrices!
Consider the simple product of two square 2x2 matrices.
a = [ 1 2; 3 4];
b = [ 8 7; 6 5];
Calling the product c = a*b
c11 = a11*b11 + a12*b21
c12 = a11*b12 + a12*b22
c21 = a21*b11 + a22*b21
c22 = a21*b12 + a22*b22
Carry out the calculations by hand and verify the result using
'matlab'. Next, consider the following matrix product of a 3x2
matrix X and a 2x4 matrix Y.
X = [2 3 ; 4 -1 ; 0 7];
Y = [5 -6 7 2 ; 1 2 3 6];
First, note that the matrix product X*Y exists because X has the
same number of columns (2) as Y has rows (2). (Note that Y*X does
NOT exist!) If the product of X*Y is called C, the matrix C must
be a 3x4 matrix. Again, carry out the calculations by hand and
verify the result using 'matlab'.
Note that the PRODUCT OF A SCALAR AND A MATRIX is a matrix in
which every element of the matrix has been multiplied by the
scalar. Verify this by recalling the matrix X defined above, and
carry out the product 3*X on 'matlab'. (Note that this can be
written X*3 as well as 3*X, because quantity '3' is scalar.)
ARRAY PRODUCTS
Recall that addition and subtraction of matrices involved
addition or subtraction of the individual elements of the matrices.
Sometimes it is desired to simply multiply or divide each element
of an matrix by the corresponding element of another matrix. These
are called 'array operations" in 'matlab'. Array or element-by-
element operations are executed when the operator is preceded by a
'.' (period). Thus
a .* b multiplies each element of a by the respective
element of b
a ./ b divides each element of a by the respective element
of b
a .\ b divides each element of b by the respective element
of a
a .^ b raise each element of a by the respective b element
For example, if matrices G and H are defined
G = [ 1 3 5; 2 4 6];
H = [-4 0 3; 1 9 8];
G .* H = [ -4 0 15
2 36 48 ]
TRANSPOSE OF A MATRIX
The transpose of a matrix is obtained by interchanging the
rows and columns. The 'matlab' operator that creates the transpose
is the single quotation mark, '. Recalling the matrix G
G' = [ 1 2
3 4
5 6 ]
Note that the transpose of a m x n matrix creates a n x m matrix.
One of the most useful operations involving the transpose is that
of creating a column vector from a row vector, or a row vector from
a column vector.
INNER (SCALAR) PRODUCT OF TWO VECTORS
The scalar or inner product of two row vectors, G1 and G2 is
found as follows. Create the row vectors for this example by
decomposing the matrix G defined above.
G1 = G(1,:)
G2 = G(2,:)
Then the inner product of the 1x3 row vector G1 and the 1x3 row
vector G2 is
G1 * G2' = 44
Verify this result by carrying out the operations on 'matlab'.
If the two vectors are each column vectors, then the inner
product must be formed by the matrix product of the transpose of a
column vector times a column vector, thus creating an operation in
which a 1 x n matrix is multiplied with a n x 1 matrix.
In summary, note that the inner product must always be the
product of a row vector times a column vector.
OUTER PRODUCT OF TWO VECTORS
If two row vectors exist, G1 and G2 as defined above, the
outer product is simply
G1' * G2 {Note G1' is 3x1 and G2 is 1x3}
and the result is a square matrix in contrast to the scalar result
for the inner product. DON'T CONFUSE THE OUTER PRODUCT WITH THE
VECTOR PRODUCT IN MECHANICS! If the two vectors are column
vectors, the outer product must be formed by the product of one
vector times the transpose of the second!
OTHER OPERATIONS ON MATRICES
As noted above, many functions that exist can be applied
directly to a matrix by simply allowing the function to operate on
each element of the matrix array. This is true for the
trigonometric functions and their inverse. It is true also for the
exponential function, ex, exp(), and the logarithmic functions,
log() and log10().
Exponential operations (using ^) on matrices are quite
different from their use with scalars. If W is a square matrix
W^2 implies W*W which is quite different from W.*W. Be certain of
your intent before raising a matrix to an exponential power. Of
course W^2 only exists if W is a square matrix.
SPECIAL MATRICES
A number of special functions exist in 'matlab' to create
special or unusual matrices. For example, matrices with element
values of zero or 1 are sometimes useful.
zeros(4)
creates a square matrix (4 x 4) whose elements are zero.
zeros(3,2)
creates a 3 row, 2 column matrix of zeros.
Similarly the commands ones(4) creates a 4 x 4 square matrix
whose elements are 1's and 'ones(3,2)' creates a 3x2 matrix whose
elements are 1's.
If a m x n matrix, say A, already exists, the command
'ones(A)' creates an m x n matrix of 1's. The command 'zeros(A)'
creates an mxn matrix of 0's. Note that these commands do not
alter the previously defined matrix A, which is only a basis for
the size of the matrix created by commands zeros() and ones().
The identity matrix is a square matrix with 1's on the
diagonal and 0's off the diagonal. Thus a 4x4 identity matrix, a,
is
a=[1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1]
This matrix can be created with the 'eye' function. Thus,
a = eye(4)
The 'eye' function will create non-square matrices. Thus
eye(3,2)
creates
1 0
0 1
0 0
and
eye(2,3)
creates
1 0 0
0 1 0
PRACTICE
1. The first four terms of the fourier series for the square wave
whose amplitude is 5 and whose period is 2ƒ is
y = (20/ƒ)[sinx + (1/3)sin3x + (1/5)sin5x + (1/7)sin7x)]
Calculate this series, term by term, and plot the results for each
partial sum.
SOLVING SIMULTANEOUS LINEAR EQUATIONS
One of the most common applications of matrix algebra occurs
in the solution of linear simultaneous equations. Consider a set of
n equations for which the unknowns are x1, x2, ....... xn.
a11x1 + a12x2 + a13x3 + ............... a1nxn = b1
a21x1 + a22x2 + a23x3 + ............... a2nxn = b2
a31x1 + a32x2 + a33x3 + ............... a3nxn = b3
. . . . .
. . . . .
. . . . .
. . . . .
an1x1 + an2x2 + an3x3 + ............... annxn = bn
The matrix format for these equations is
[a][x] = [b]
where
a11 a12 a13 .... a1n x1 b1
a21 a22 a23 .....a2n x2 b2
[a] = a31 a32 a33 .... a3n [x] = x3 [b] = b3
. . . . . .
. . . . . .
. . . . . .
. . . . . .
an1 an2 an3 .... ann xn bn
Note that only when the equations are linear in the unknown xi's is
the matrix formulation possible.
The classical matrix solution to this equation is based on the
definition of the inverse of a matrix. The inverse of a matrix is
that matrix which when multiplied by the original matrix, results
in the identity matrix. Then
a-1 a = I
where a-1 denotes the 'inverse of matrix a' and I denotes the
identity matrix of the same order as matrix 'a'. If 'a' is a known
coefficient matrix and if 'b' is a column vector of known terms,
the problem is one of finding the n vales of x1, x2, .... xn that
satisfy these simultaneous equations. Pre-multiplying both sides of
the equation by the inverse of 'a' gives
[a-1][a][x] = [a-1][b]
or
[x] = [a-1][b]
Thus if we find the inverse of the coefficient matrix, the product
of the inverse times the matrix of non-homogeneous terms, b, yields
the unknown matrix, x. To observe the simple property of the
inverse, define the 4x4 matrix, C
C = [1 -4 3 2; 3 1 -2 1; 2 1 1 -1; 2 -1 3 1]
calculate the inverse using the 'matlab' command, inv().
C_inverse = inv(C)
and note that
C_inverse * C = [1 0 0 0;0 1 0 0;0 0 1 0;0 0 0 1]
where the right hand side is the 4x4 identity matrix.
We leave to a study of linear algebra, the method and steps
required to calculate the inverse of a matrix. Suffice it to say
here that for systems of equations numbering in the hundreds or
thousands, which often occur in complex engineering problems, the
calculation of the matrix inverse is a time consuming operation
even for modern computers.
As an example of solving a system of equations using the
matrix inverse, consider the following system of three equations.
x1 - 4x2 + 3x3 = -7
3x1 + x2 - 2x3 = 14
2x1 + x2 + x3 = 5
Using 'matlab', define the two matrices for [a] and [b].
a = [ 1 -4 3; 3 1 -2; 2 1 1];
b = [ -7; 14; 5];
Find the solution vector, x, by
x = inv(a)*b
giving
x = [ 3
1
-2 ]
LEFT AND RIGHT MATRIX 'DIVISION' IN MATLAB
In contrast to matrix inversion, the preferred methods of
solution for large-scale systems of simultaneous equations are
methods of back substitution whereby the equations can be
rearranged so that first xn is solved, then xn-1 and so on until
finally x1 is obtained in order. Thus in back substitution we never
solve the equations simultaneously, rather we solve them
sequentially. For example, the above set of equations can be
rearranged through appropriate combination into the below
equations. (You should carry out this algebra to ensure you
understand the steps matlab takes in solving with back
substitution. In the general case, the steps or algorithm is much
more complex than that required for 3 equations.)
x1 - 4x2 + 3x3 = -7
13x2 - 11x3 = 35
(34/13)x3 = (68/13)
In this format, we see from the third equation the solution for x3
= 2 and knowing this, the second equation yields x2 = 1, and
knowing both x3 and x2, the first equation gives x1 = 1. This is an
application of back substitution, whereby each unknown is obtained
by simple sequential solution to a single equation. Methods of back
substitution are employed by 'matlab' when we invoke the 'right
division' and 'left division' commands.
\ left division
/ right division
Understand that matrices cannot be divided. The operation is not
defined. The syntax of 'right division' and 'left division' simply
invoke back substitution methods for obtaining a solution vector
from a system of simultaneous equations. Whether 'right division'
(/) or 'left division' (\) is appropriate depend on how the matrix
equation is posed.
Left Division. If the matrix equation is
[a][x] = [b]
then we have a nxn matrix [a] pre-multiplying a nx1 matrix [x],
resulting in a nx1 matrix [b]. The solution for x is obtained by
using the left division operation.
[x] = [a]\[b]
If the matrix equation is cast in this format, the use of right
division to obtain the solution vector x will result in an error
message.
Returning to 'matlab', recall we have defined the matrices [a]
and [b] in the format of the matrix equation appropriate for left
division.
a = [ 1 -4 3; 3 1 -2; 2 1 1];
b = [ -7; 14; 5];
Now enter the command
x1 = a\b
and obtain the result
x1 = [ 3
1
-2 ]
which i s the same result found earlier for x when solving by
matrix inversion.
Right Division. The matrix format in which right division is
employed is less common but no less successful in solving the
problem of simultaneous equations. Right Division is invoked when
the equations are written in the form
[x][A] = [B]
Note that in this form [x] and [B] are row vectors rather than
column vectors as above. This form represent a 1xn matrix (x) pre-
multiplying a nxn matrix (A), resulting an a 1xn matrix (B). Again,
recalling the set of equations,
x1 - 4x2 + 3x3 = -7
3x1 + x2 - 2x3 = 14
2x1 + x2 + x3 = 5
These equations can be written in matrix format
[x][A] =[B]
if
x = [x1 x2 x3] B = [ -7 14 5]
and
1 3 2
[A] = -4 1 1
3 -2 1
Note that [A] is equal to the transpose of [a].
A = a'
Having so defined A and B, the solution for x can be obtained by
right division.
x = B/A
results in
x = [ 3
1
-2 ]
PRACTICE PROBLEMS
1. Find the solution to
r + s + t + w = 4
2r - s + w = 2
3r + s - t - w = 2
r - 2s - 3t + w = -3
using the matrix inverse and left and right division.
2. Find the solution to
2x1 + x2 - 4x3 + 6x4 + 3x5 - 2x6 = 16
-x1 + 2x2 + 3x3 + 5x4 - 2x5 = -7
x1 - 2x2 - 5x3 + 3x4 + 2x5 + x6 = 1
4x1 + 3x2 - 2x3 + 2x4 + x6 = -1
3x1 + x2 - x3 + 4x4 + 3x5 + 6x6 = -11
5x1 + 2x2 - 2x3 + 3x4 + x5 + x6 = 5
using the matrix inverse and left and right division.
3. Back substitution is facilitated by recognizing that any
nonsingular symmetric matrix can be decomposed or factored into the
product of two matrices, called here L and U.
A = L U
This is called LU factorization. U is an upper triangular matrix
(all 0's below the diagonal). L is called a permutation of a lower
triangular matrix in which 1's occur at every element which is a
permutation of the two numbers defining the size of the matrix.
These matrices can be obtained in 'matlab' using the command
[L,U] = lu(A)
Using the matrix A defined in Prob 2 above, find the upper and
lower triangular factor matrices for the matrix of coefficients, A.
If these matrices are called AU and AL, show that
AL * AU * x = B
where
x [ 2 -1 1 0 3 -4 ]' and B = [16 -7 1 -1 -11 5]'
EQUATION FITTING
Many occasions arise when we have a set of (x,y) pairs and we
desire to find an equation or function that "fits" these data. The
procedure we follow can be generally classified in one of two
categories, interpolating functions or least squares functions.
The least-squares function is one that obtains the best fit,
where the meaning here of "best" is based on minimizing the sum of
the squares of the differences between the function and the data
pairs. Clearly the idea of "least squares" is to achieve some sort
of average or mean value function whose graphical curve does not
typically pass through any of the (x,y) data pairs. It is
appropriate for experimental data in which every data pair is
obtained under conditions of uncertainty and for which any
collected data, as a consequence of experimental error, may be
greater or less than the true value.
INTERPOLATING POLYNOMIAL
The interpolating function is one that satisfies every data
point exactly. When viewed in a graphical plot, the interpolating
function passes through each data point. If for convenience we
select the polynomial form as the function, a polynomial of degree
'n' can be found for a set of 'n+1' (x,y) data pairs. The general
form of the polynomial can be written
y = a0 + a1x + a2x2 + a3x3 + ....... + anxn
To evaluate the interpolating polynomial of degree 'n' for any data
set, we must evaluate 'n+1' coefficients, a0, a1, a2, a3, .....an.
Given the 'n+1' (x,y) data pairs, we can form 'n+1' equations
y1 = a0 + a1x1 + a2x12 + a3x13 + ...... + anx1n
y2 = a0 + a1x2 + a2x22 + a3x23 + ...... + anx2n
y3 = a0 + a1x3 + a2x32 + a3x33 + ...... + anx3n
. . . . . .
. . . . . .
. . . . . .
yn+1 = a0 + a1xn+1 + a2xn+12 + a3xn+13 + ... + anxn+1n
This is a solvable set of 'n+1' equations with 'n+1' unknowns. The
unknowns are the coefficients a0, a1,......an. If there exists only
two or three equations, the solution procedure for the unknown
coefficients can conveniently follow Cramer's Rule. In general,
however, it is common to rely on matrix algebra to manipulate these
equations into a format for which n operations of back substitution
will obtain the unknown coefficients. Here we will simply rely on
the methods of right (or left) division in 'matlab' to obtain the
unknown coefficients, leaving the details of the numerical method
for back substitution for later discussion.
Note that the set of equations can be conveniently expressed
by the matrix equation
[y] = [X][a]
where
y1 1 x1 x12.....x1n+1 a1
y2 1 x2 x22.....x2n+1 a2
[y] = [X] = [a] =
y3 1 x3 x32.....x3n+1 a3
. . . . . .
. . . . . .
. . . . . .
yn+1 1 xn+1 xn+12... xn+1n+1 an+1
Note that the right side of the matrix equation must be the product
of a 'n+1' x 'n+1' square matrix times a 'n+1' x 1 column matrix!
The solution using Gauss elimination calls for left division in
'matlab' or
[a] = [X]\[y]
Using this method, we can find the coefficients for a n-degree
polynomial that passes exactly through 'n+1' data points.
If we have a large data set, each data pair including some
experimental error, the n-degree polynomial is NOT A GOOD CHOICE.
Polynomials of degree larger than five or six often have terribly
unrealistic behavior BETWEEN the data points even though the
polynomial curve passes through every data point ! As an example,
consider these data.
x y
2 4
3 3
4 5
5 4
6 7
7 5
8 7
9 10
10 9
There are 9 data points in this set. It is clear that as x
increases, so also does y increase; however, it appears to be doing
so in a nonlinear way. Let's see what the data looks like. In
'matlab', create two vectors for these data.
x9 = [2:1:10];
y9 = [ 4 3 5 4 7 5 7 10 9 ];
To observe these data plotted as points, execute
plot(x9,y9,'o')
Because we have nine data pairs, it is possible to construct an
eighth-degree interpolating polynomial.
y = a0 + a1x + a2x2 + a3x3 + ....... + a8x8
To find the unknown coefficients, define the column vector y
y = y9'
and the matrix X
X = [ones(1,9);x9;x9.^2;x9.^3;x9.^4;x9.^5;x9.^6;x9.^7;x9.^8]'
Note that X is defined using the transpose, the ones() function and
the array operator ' .^ '. With X and y so defined, they satisfy
the equation
[X][a] = [y]
Solve for the coefficients, matrix a in the above equation, by
entering the command
a = X\y
which results in
a = [ 1.0e+003*
3.8140
-6.6204
4.7831
-1.8859
.4457
- .0649
.0057
- .0003
.0000 ]
Note that the ninth coefficient a(8) appears to be zero. Actually,
it is finite, it only appears to be zero because 'matlab' is
printing only 4 significant figures to the left of the decimal
point. To observe the coefficients with more significant figures,
enter the commands
format long
a
and the format is changed to one with 15 significant digits.
Clearly a(8) is not zero, it is a small number because it is
multiplying a number, x, raised to the eight power.
Now that we have the coefficients, let's generate a
sufficient number of points to create a smooth curve. For x, form
a vector over the range 2 <= x <= 10 in increments of 0.1.
x = [ 2:.1:10 ];
For y, calculate the value of the eighth degree polynomial for each
x.
y =a(1)+ a(2).*x + a(3).*x.^2 + a(4).*x.^3 + a(5).*x.^4...
+ a(6).*x.^5 + a(7).*x.^6 + a(8).*x.^7 + a(9).*x.^8;
Now plot (x,y) and the data points (x9,y9).
plot(x,y,x9,y9,'o')
The polynomial results appear to pass exactly through every data
point, but clearly the polynomial is useless for representing our
impression of the data at any other point within the range of x!
This is a common behavior for high-order interpolating polynomials.
For this reason, one should never attempt to use high order
interpolating polynomials to represent experimental data.
LEAST SQUARES
A lower order polynomial, possibly a cubic polynomial might be
a sufficient choice; however, a cubic polynomial would have 4
unknown coefficients and we have 9 data points. Which four of the
nine do we select? The method of least squares responds to this
problem by taking an "average" or best curve that includes
information from every one of the data points. The method is well
explained in Chapter 3 of the text, EXPERIMENTAL METHODS FOR
ENGINEERS by J. P. Holman. The details will not be repeated here;
however, the text specializes the result to a first and second
order polynomials. Here we would like to retain the polynomial
format and express the result for the third degree polynomial. If
we desire a quadratic or linear equation, it is easy to select the
reduced matrices that create the desired coefficients. Thus we
find that for the equation
y = a0 + a1x + a2x2 + a3x3
the four unknown coefficients are evaluated from
a = X \ B
where
a0 m ‘xi ‘xi2 ‘xi3 ‘yi
a1 ‘xi ‘xi2 ‘xi3 ‘xi4 ‘xiyi
[a] = [X] = [B] =
a2 ‘xi2 ‘xi3 ‘xi4 ‘xi5 ‘xi2yi
a3 ‘xi3 ‘xi4 ‘xi5 ‘xi6 ‘xi3yi
The summation is over the range 1 to m, where m is the number of
(x,y) data pairs. Note that for any least-squares equation fit,
you must have at least one more data point than unknown
coefficients, or m > n+1.
MATLAB FUNCTION FOR LEAST SQUARES
There is a convenient function in 'matlab' for polynomial
equation fitting with the method of least squares. The command is
polyfit(x,y,n)
where x,y are the data vectors and 'n' is the order of the
polynomial for which the least-squares fit is desired. The command
'polyfit' returns a vector whose elements are the coefficients of
the polynomial. THE ELEMENTS OF THE VECTOR ARE IN REVERSE ORDER
FROM WHAT YOU MIGHT ANTICIPATE, that is, the first element is the
coefficient of the highest order of x and the last element is the
coefficient of the lowest order (always order '0' or x0).
NON-POLYNOMIAL FORMS FOR EQUATION FITTING
The number of equation formats that can be employed in
equation fitting is limited only by the imagination of the analyst.
A few examples deserve special comment.
EXPONENTIAL FORM. The equation
y = a*ebx
is an important form. One basis for it popularity lies in the fact
that this nonlinear equation appears to be a straight line when
plotted on ln(y) vs x coordinates (semi-log coordinates). This
can be best observed by taking the logarithm of the above equation.
ln(y) = ln(a) + bx
With a simple transformation , y1 = ln(y) and ln(a) = A, the above
equation can be written
y1 = A + bx
Thus if we plot the logarithm of y vs x, the intercept is ln(a) and
the slope is the coefficient 'b'.
POWER FORM. The equation
y = a*xn
appears to be linear when plotted in log-log coordinates. Again we
observe this by taking the logarithm of the equation, giving
ln(y) = ln(a) + n*ln(x)
Again with a simple and obvious transformation, this equation can
be written in the format of the linear equation
y1 = A + n*x1
Of course it is convenient to obtain least squares curves of these
forms by first applying the transformation to the data, and then
fitting the linear equation to the transformed data.
HYPERBOLIC FORM. The hyperbolic equation
y = a + b/x
appears to be linear when y is plotted versus 1/x rather than x.
This equation is sometimes useful when data exhibits a decreasing
value of y as x increases.
ASYMPTOTES. The hyperbolic form above exhibits an asymptote.
As x Π, y a . Asymptotes may exist for either variable, x or
y. If an asymptote can be recognized from knowledge of the physics
in the problem, it will facilitate the equation fitting process if
a transformation is introduced. For example, in the hyperbolic
equation
y = a + b/x
the transformation z = y - a results in
z = b/x
which eliminates one of the unknown parameters in the equation
fitting problem.
PRACTICE PROBLEMS
SATURATION PROPERTIES OF WATER
TEMP(C) PRESSURE(kPa) SPEC.VOL(m3/kg) ENTHALPY-hg(kJ/kg)
0 .6108 206.3 2501.6
10 1.227 106.4 2519.9
20 2.337 57.84 2538.2
30 4.241 32.93 2556.4
40 7.375 19.55 2574.4
50 12.335 12.05 2592.2
60 19.92 7.679 2609.7
70 31.16 5.046 2626.9
80 47.36 3.409 2643.8
90 70.11 2.361 2660.1
100 101.33 1.673 2676.0
1. Using the above data and linear interpolation, calculate the
saturation pressure at 65 C. Next, find the interpolating
polynomial that satisfies the above data for saturation temperature
and pressure for 50, 60 and 70 C. Using this polynomial, compute
the saturation pressure at 65 C and compare the result with your
linear interpolation.
2. Find the interpolating polynomial that satisfies exactly the
above data for saturation temperature and specific volume for the
temperatures 10, 20 and 30 C. Using this polynomial, compute the
specific volume at a temperature of 15 C and compare the result
with linear interpolation from the table.
3. A function is expected to be of the form y = axn. Two data
points are available to evaluate the unknown coefficient a, and
exponent n. The data are (100,50) and (1000,10). Find the
parameters a and n.
4. (a) Find the linear equation using least-squares that represents
the temperature-enthalpy data in the above table. ( h = h(T) )
(b) Find a second-degree polynomial that represents the T-
enthalpy data in the above table using the method of least squares.
(c) Plot the (h,T) data and the above two equations.
5. Find the unknown coefficients in the below equation that
represents the T-p data in the above table.
ln(p) = A + B/T
Plot the data and the equation.
6. (a) Find the 1st, 2nd and 3rd order polynomials that satisfy the
p vs. T data above, and plot them on a single graphics for
comparison.
(b) Prepare a graphic plot that compares the result of (5.) above
with the 3rd order equation found in 6(a.).
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