1. Computational Methods
CSE 551
Computational Methods
2018/2019 Fall
Chapter 2
MATLAB Basics

2. Outline
Introduction
MATLAB as a Calculator
Variables and Assignment Statement
Arrays
Subarrays
Special Values
Displaying Output Data
Data Files
Arrays and Matrix Operations
Build-in MATLAB Functions
Debugging MATLAB Programs

3. Introduction
MATLAB (short for MATrix LABoratory)
special-purpose computer program
optimized to perform engineering and scientific calculations.
started to perform matrix mathematics
over the years - a flexible computing system The
The MATLAB program implements the MATLAB programming language
a very extensive library of predefined functions to make technical programming tasks easier and more efficient

4. References
Chapman S. J., MATLAB Programming for Engineers, 5th ed., CENGAGE Learning
Attaway S., MATLAB:A Practical Introduction to Programming and Problem Solving, 4th ed., ELSEVIER BH

5. Advantages and Disadvantages of MATLAB
Ease of Use
Platform Independence
Predefined Functions
Device-Independent Plotting
Graphical User Interface
MATLAB Compiler: divice independent p-code
Disadvantages:
an interpreated language
cost

6. The MATLAB Environment
several types of windows:
Command Windows: commands may be entered; Figure Windows: display plots and graphs
Edit Windows: permit a user to create and modify MATLAB programs.

7. The default MATLAB desktop
The default MATLAB desktop. The exact appearance of the desktop may differ slightly on different types of computers.

8. Tools within MATLAB Dasktop
The major tools within or accessible from the MATLAB desktop are
The Command Window
The Command History Window
The Start Button
The Documents Window, including the Editor/Debugger and the Array Editor
Figure Windows
Workspace Browser
Help Browser
Path Browser

9. MATLAB as a Calculator
MATLAB can be used as a calculator to perform mathematical calculations
typed directly into the Command Window
using the symbols +, –, *, /, and ^ (exponentiation)
e.g., to calculate the volume of a cylinder of radius r and
length l
The area of the circle at the base of the cylinder is given by
A = r 2
and the total volume of the cylinder:
V = Al
If the radius is 0.1 m and the length is 0.5 m,

10. » A = pi * 0.1^2
A =
0.0314
» V = A * 0.5
V =
0.0157
Note that pi is predefined to be the value

11. Variables and Assignment Statement
variable - store a value
assignment statement
variablename = expression
>> mynum = 6
mynum = 6
>>

12. suppressing the output
Putting a semicolon at the end of a statement suppresses the output.
>> res = 9 – 2;
>>
it just doesn’t show that result. Instead, another prompt appears immediately.
However, at this point in the Workspace Window the variables mynum and res can be seen

13. Default variable ans
MATLAB uses a default variable named ans
if an expression is typed at the prompt and not assigned to a variable
e.g., the result of the expression 6+3 is stored in the variable ans:
>> 6 + 3
ans = 9
This default variable is reused any time just an expression is typed at the prompt.

14. Variable Names
variables – identifiers
Rules for identgifier names:
1 - must begin with a letter After that can contain letters, digits, and the underscore character (e.g., value_1),
but it cannot have a space
2 - There is a limit to the length of the name; the built-in function namelengthmax tells how many characters this is
3 – MATLAB is case sensitive

15. Variable Names (cont.)
4 - reserved words that cannot be used as variable
name
5 - Names of built-in functions can, but should not, be used as variable names
6 – Conventions
i – use underscore: exchange_rate
ii – Java, C++ convension: exchangeRate

16. Arrays
fundamental unit of data – array
collection of data values organized into rows and columns and known by a single name
Individual data values within an array may be accessed

17. Vectors ands Matices
Individual data values within an array accessed by
name of the array followed by subscripts in parentheses (row,column)
scalars – arrays - one row and one column
Arrays - vectors or matrices.
“vector” - an array with only one dimension
“matrix” - an array with two or more dimensions.
The size of an array - # of rows and # of columns # of rows mentioned first

18. Examples

20. Vectors
Individual array elements are addressed by the array name followed by the
row and column of the element
for row or column vectors - only one subscript is required
e.g., in the preceding arrays a(2,1) is 3
and c(2) = 2.
A MATLAB variable - region of memory containing an array refered by a user-specified name
contents of the array may be used or modified

21. Common Types
common types of MATLAB variables - double and char.
double - scalars or arrays of 64-bit double-precision floating-point numbers
can hold real, imaginary, or complex values.
in the range to with 15 to 16 significant decimal digits of accuracy.
double - the principal numerical data type.

22. imaginary number - appending the letter i or j to a number 10i and –4j
double Type
variable type double automatically created whenever a numerical value is assigned to a variable name
real, imaginary, or complex
var = 10.5
imaginary number - appending the letter i or j to a number
10i and –4j
var = 4i
complex value
var = i

23. char Type
Variables of type char consist of scalars or arrays of 16-bit values, each
representing a single character.
Arrays of this type are used to hold character strings.
e.g.,
comment will be a character array.
comment = 'This is a character string'

24. Strongly and Weakly typed languages
In a strongly typed language, the type of every variable must be explicitly declared before it is used.
MATLAB - weakly typed language
Variables may be created at any time by simply assigning values to them,
the type of data assigned to the variable determines the type of variable

25. MATLAB variables are automatically created when they are initialized. :
1. Assign data to the variable in an assignment statement.
2. Input data into the variable from the keyboard.
3. Read data from a file.

26. assign it one or more values assignment statement. var = expression;
scalar constant
an array
combination of constants, other variables, and mathematical operations
var = 40i;
var2 = var/5;
x = 1; y = 2;
array = [ ];

27. variables can be initialized with arrays of data.
constructed using brackets ([]) and semicolons.
All of the elements
of an array are listed in row order.
the values in each row are listed
from left to right, with the topmost row first and the bottommost row last. Individual
values within a row - separated by blank spaces or commas
the rows - separated by semicolons or new lines.

28. Clearing Variables

29. The number of elements in every row must be the same
the number of elements in every column must be the same.
An expression such as
[1 2 3; 4 5];
is illegal

30. The expressions used to initialize arrays can include algebraic operations and all of or portions of previously defined arrays
e.g.,
a = [0 1+7];
b = [a(2) 7 a];
will define an array a [0 8]
b [ ].

31. not all of the elements in an array must be defined when it is created.
If a specific array element is defined and one or more of the elements before it are not
then the earlier elements will automatically be created and initialized to zero
e.g.,
c(2,3) = 5;
will produce the matrix c =

32. an array can be extended by specifying a value for an element beyond the currently defined size.
suppose that array d [1 2]
the statement
d(4) = 4;
will produce the array d [ ]

33. Initilizing with Shortcut Expressions
for large arrays:
the colon operator. specifies a whole series of values
The general form of a colon operator is
first:incr:last
first: first value in the series
incr: stepping increment
last: last value in the series.
If the increment is one, it may be omitted
generate an array containing the values
first, first+incr, first+2*incr, first+3*incr, and so forth as long as the values are less than or equal to last.
The list stops when the next value in the series is greater than the value of last.

34. An Example
the expression 1:2:10 - shortcut for a 1 x 5 row vector containing the values 1, 3, 5, 7, and 9.
The next value in the series would be 11, which
is greater than 10, so the series terminates at 9.
» x = 1:2:10
x =

35. Shortcut expressions can be combined with the transpose operator (')
to initialize column vectors and more complex matrices.
The transpose operator
swaps the row and columns of any array that it is applied to.
Thus the expression
f = [1:4]';
generates a 4-element row vector [ ]
then transposes it into the
4-element column vector f.

36. Example
the expressions
g = 1:4;
h = [g' g'];
will produce the matrix h

37. Initilizing with Build-in Functions
zeros function - create an all-zero array of any size.
single scalar argument - a square array
two scalar arguments –
first: # of rows, second: # of columns
size function returns two values:
# of rows and columns in an array
combined with the zeros function
ones function - arrays containing all ones
eye function - arrays containing identity matrices

38. Examples
a = zeros(2);
b = zeros(2,3);
c = [1 2; 3 4];
d = zeros(size(c));
These statements generate the following arrays:

39. list of common MATLAB functions for initializing variables.

40. Initilizing Variables with Keybord Input
input function
displays a prompt string in the Command Window
and then waits for the user to type in a response.
my_val = input('Enter an input value:');
If enters a single number - be typed in
If enters an array – enclosed in brackets
If only the return key is entered - an empty
matrix

41. Multidimensional Arrays
One dimensional arrays - series of values laid out
row or column –
single subscript - select individual array elements
MATLAB - arrays with as many dimensions as necessary
one subscript for each dimension,

43. Example » c(:,:,1)=[1 2 3; 4 5 6]; » c(:,:,2)=[7 8 9; 10 11 12];
» whos c
Name Size Bytes Class Attributes
c 2x3x2 96 double
» c
c(:,:,1) =
1 2 3
4 5 6
c(:,:,2) =
7 8 9

44. Storing Arrays in Memory
A two-dimensional array m rows, n columns
m x n successive locations in the computer’s
memory
MATLAB always allocates array elements in column major order.
allocates the first column in memory
then the second, then the third, and so on.,
e.g.,
element a(1,2) - fifth element allocated in memory
single-subscript addressing
low-level I/O functions

45. Data values for array a. (b) Layout of values in memory for array a.

46. Storing Arrays with More Then TWo Dimensions
This same allocation scheme applies to arrays with more than two dimensions.
The first array subscript is incremented most rapidly,
the second subscript is incremented less rapidly, and so on, and the last subscript in incremented
most slowly.
e.g., in a 2 x 2 x 2 array, the elements would be
allocated in the following order: (1,1,1), (2,1,1), (1,2,1), (2,2,1), (1,1,2), (2,1,2),
(1,2,2), (2,2,2)

47. Accessing Multidimensional Arrays with One Dimension
MATLAB permit treat
a multidimensional array - a one-dimensional array
length = # of elements in the multidimensional array.
If a MDA addressed with a single dimension, then the elements accessed:
in the order in which they were allocated in memory.

48. Example
a :4 x 3
» a = [1 2 3; 4 5 6; 7 8 9; ]
a =
value of a(5) is 2 - value of element a(1,2),
a(1,2) allocated fifth in memory.

49. Subarrays possible to select and use
subsets of MATLAB arrays as if separate arrays
To select a portion of an array
just include a list of all of the elements
to be selected in the parentheses after the array name.
e.g., array arr1:
arr1 = [ ];
arr1(3): 3.3
arr1([1 4]): array [ ]
arr1(1:2:5): array [ ].

50. Examples
For a two-dimensional array
a colon can be used in a subscript to select all
of the values of that subscript.
e.g.,
arr2 = [1 2 3; ; 3 4 5];
arr2(1,:): [1 2 3]
arr2(:,1:2:3):

51. The end Function
end functrion:creating array subscripts.
in an array subscript - end returns the highest value taken on by that subscript
e.g., arr3:
arr3 = [ ];
arr3(5:end): [ ]
array(end): value 8.

52. The value returned by end - the highest value of a given subscript.
Example
The value returned by end - the highest value of a given subscript.
If end appears in different subscripts, it can return different values
e.g., arr4:
arr4 = [ ; ; ];
arr4(2:end,2:end):
the first end returned the value 3
the second end returned the value 4

53. Using Subarrays on the Left-Hand Side of an Assignment Statement
It is possible to use subarrays on the left-hand side of an assignment statement
to update only some of the values in an array,
as long as the shape ( # of rows and columns) of the values being assigned matches the shape of the subarray.
If the shapes do not match,
an error will occur.

54. Example » arr4 = [1 2 3 4; 5 6 7 8; 9 10 11 12] arr4 = 1 2 3 4 5 6 7 8
following assignment statement is legal, since the expressions on both sides of the equal sign have the same shape :
» arr4(1:2,[1 4]) = [20 21; 22 23]
the array elements (1,1), (1,4), (2,1), and (2,4) were updated

55. Example (cont.)
In contrast, the following expression is illegal
because the two sides do not have the same shape.
» arr5(1:2,1:2) = [3 4]
??? In an assignment A(matrix,matrix) = B, the number of rows in B and the number of elements in the A row index matrix must be the same.

56. Difference Between Arrays and Subarrays
major difference in MATLAB between
assigning values to a subarray and
assigning values to an array.
If values are assigned to a subarray, only
those values are updated
while all other values in the array remain unchanged.
if values are assigned to an array, the entire contents of thearray are deleted and replaced by the new values.

57. Example arr4 » arr4 = [1 2 3 4; 5 6 7 8; 9 10 11 12] arr4 = 1 2 3 4
the following assignment statement replaces the specified elements of arr4:
» arr4(1:2,[1 4]) = [20 21; 22 23]
the following assignment statement replaces the entire contents of
arr4 with a array:
» arr4 = [20 21; 22 23]
20 21
22 23:

58. Assigning a Scaler to a Subarray
A scalar value on the right-hand side of an assignment statement
always matches the shape specified on the left-hand side.
The scalar value - copied into every element
specified on the left-hand side of the statement.
e.g., arr4 :
arr4 = [ ; ; ];
one is assigned to four elements of the array.
» arr4(1:2,1:2) = 1
arr4 =

59. Special Values predefined special values.
may be used at any time in MATLAB without initializing them first
stored in ordinary variables,
so they can be
overwritten or modified
e.g.,:
circ1 = 2 * pi * 10
pi = 3;
circ2 = 2 * pi * 10
In the first statement, pi has its default value of , so circ1: 62.8
The second statement redefines pi as 3,
so in the third statement circ2 is 60

60. Predefined Special Values

61. Displaying Output Data
several ways to display output data:
simplest way:
leave the semicolon off of the end of a statement
and it will be echoed to the Command Window
a few other ways to display data:
Changing the Default Format
The disp Function
The fprintf Function

62. Changing the Default Format
When data is echoed in the Command Window,
integer values displayed as integers
character values as strings
other values are printed using a default format
default format
four digits after the decimal point
it may be displayed in scientific notation with an exponent
if the number is too large or too small

63. Example
x =
y =
z =
output:
x =
y =
1.0011e+003
z =
1.0011e-004

64. Changing Default Format
default format - changed two ways:
from the main MATLAB Window menu:
by selecting the “File / Preferences” menu option. pop up the preferences window
the format can be selected from the Command Window item in the preferences list.
using the format command

65. changes the default format
format Command
changes the default format
to display more significant digits of data
to force the display to be in scientific notation; to display data to two decimal digits
to eliminate extra line feeds to get more data visible in the Command Window at a single time.

66. Output Display Formats

67. The disp Function disp function: accepts
an array argument and displays the value of the array in the Command Window
array type – char:
the character string contained in the array is printed out.
often combined with the functions
num2str (convert a number to a string)
int2str (convert an integer to a string)
e.g., the following MATLAB statements will display
“The value of pi ”
first statement creates a string array containing the message
second statement displays the message.
str = ['The value of pi = ' num2str(pi)];
disp (str);

68. Formated Output with the fprintf Function
fprintf function: more flexible way to display data
displays one or more values together with related text
the programmer can control the way that the displayed value appears
general form - print to the Command Window:
fprintf(format,data)
where
format: a character string describing the way the data is to be printed
containing text to be printed plus special characters describing the format of the data
data: one or more scalars or arrays to be printed

69. fprintf('The value of pi is %f \n',pi) prints out
An Example
fprintf('The value of pi is %f \n',pi)
prints out
The value of pi is
followed by a line feed
%f - conversion characters
a value in the data list should be printed out in floating-point format at that location in the format string.
\n - escape characters
a line feed should be issued A

70. fprintf('The value of pi is %6.2f \n',pi) print out:
specify
width of the field - a number will be displayed
# of decimal places to display
specifying
the width and precision after the % sign and before the f.
e.g.,
fprintf('The value of pi is %6.2f \n',pi)
print out:
The value of pi is 3.14
followed by a line feed
The conversion characters %6.2f
the first data item in the function - in floating-point format
in a field six characters wide
including two digits after the decimal point.

71. Limitation of fprintf Function
displays only the real portion of a complex value
use the disp function
e.g., following statements calculate a complex value x and display it using both fprintf and disp.
x = 2 * ( 1 - 2*i )^3;
str = ['disp: x = ' num2str(x)];
disp(str);
fprintf('fprintf: x = %8.4f\n',x);
output:
disp: x = -22+4i
fprintf: x =

72. Common Special Characters in fprintf Format Strings

73. Data Files
many ways to load and save data files in MATLAB
the simplest ones to use
load and save commands

74. saves data from the current MATLAB workspace into a disk file
The save Command
saves data from the current MATLAB workspace into a disk file
The most common form:
save filename var1 var2 var3
filename: name of the file where the variables are saved
var1, var2, etc: variables to be saved in the file
By default, the file name extension “mat,” – MAT-files
If no variables are specified
the entire contents of the workspace are saved

75. MATLAB saves MAT-files in a special compact format that preserves
many details, including the name and type of each variable, the size of each array, and all data values
A MAT-file created on any platform (PC, Mac, Unix, or Linux) can be read on any other platform,
so MAT-files are a good way to exchange data between computers if both computers run MATLAB.
Unfortunately, the MAT-file is in a format that cannot be read by other programs.
If data must be shared with other programs, then the -ascii option
should be specified, and the data values will be written to the file as ASCII
character strings separated by spaces. However, the special information such as
variable names and types are lost when the data is saved in ASCII format, and
the resulting data file will be much larger

76. Example For example, suppose the array x is defined as
the command: “save -ascii x.dat x” will produce a file named x.dat
containing the following data:
e e e+000
e e e+000
This data is in a format that can be read by spreadsheets or by programs written
in other computer languages, so it makes it easy to share data between MATLAB
programs and other applications.

77. MATLAB doesn’t care what file extension is used for ASCII files
MATLAB doesn’t care what file extension is used for ASCII files. However,
it is better for the user if a consistent naming convention is used, and an extension
of “dat” is a common choice for ASCII files.

78. The load Command
loads data from a disk file into the current MATLworkspace
most common form:
load filename
for MAT-file:
all of the variables in the file will be restored, with the names and types
If a list of variables is included
only those variables will be restored
If filename has
no extent or
extent is .mat –
treat the file as a MAT-file

79. can load data created by other programs in
comma- or spaceseparated ASCII format
If the filename extension - other than .mat
treat the file - ASCII file
The contents of an ASCII file - converted into a MATLAB array - same name as the file (without the file extension)
e.g.,
an ASCII data file named x.dat -
load x.dat
create a array - x - in the current workspace, containing these data values.

80. forced to treat a file as a MAT-file load –mat x.dat
by specifying the –mat option
load –mat x.dat
treat file x.dat as a MAT-file even though its file extent is not .mat.
can be forced to treat a file as an ASCII file
by specifying the –ascii option.

81. Scaler and Array Operations
an assignment statement:
variable_name = expression;
the equal sign - the assignment operator
scaler operations
array and matrix operations

82. Scaler Operations
expression - any valid combination of
scalars, array elements, parentheses, and arithmetic operators
2 ^ ((8+2)/5)
is evaluated as
= 2 ^ (10/5)
= 2 ^ 2
= 4

83. Arithmetic Operations Between Two Scalars

84. Array and Matrix Operations
two types of operations between arrays:
array operations and matrix operations.
Array operations: performed between arrays on an element-by-element basis.
on corresponding elements in the two arrays
e.g.,
# of rows and columns in both arrays must be the same.
If not, MATLAB will generate an error message.

85. Array operations - between an array and a scalar. e.g.,
value of the scalar is applied to every element of the array.
e.g.,

86. Matrix Operations
follow the normal rules of linear algebra,
matrix multiplication:
c = a x b:
# of columns in matrix a must be equal to # of rows in matrix b.

87. a special symbol - distinguish array operations from matrix operations
a period before the symbol to indicate an array operation
e.g., .*
square matrices. of the same size - both
array multiplication and matrix multiplication - legal
give totally different answers

88. Example
What is the result of each of the following expressions?
(a) a + b (e) a + c
(b) a .* b (f ) a + d
(c) a * b (g) a .* d
(d) a * c (h) a * d

89. Solution

90. Solution (cont.)

91. Common Array and Matrix Operations

92. Common Array and Matrix Operations (cont.)

93. Matrix Left Division Pperator
matrix left division operation A 3 x 3 set of simultaneous linear equations:
expressed as
can be solved for x = A-1b

94. Hierarchy of Operations
To make the evaluation of expressions unambiguous
a series of rules governing the hierarchy or order in which operations are evaluated
generally follow the normal rules of algebra.

95. Hierarchy of Arithmetic Operations

96. Built-in MATLAB Functions
In mathematics, a function is an expression that accepts one or more input values and calculates a single result from them
Optional Results:
MATLAB functions can return more than one result.
e.g., function max.
normally returns the maximum value of an input vector, it can
also return a second argument: location in the input vector where the maximum value was found

97. An Example
maxval = max ([1 –5 6 –3])
returns maxval = 6
if two variables are provided to store results in,
returns both the maximum value and the location of the maximum value.
[maxval index] = max ([1 –5 6 –3])
maxval = 6 , index = 3

98. Using MATLAB Functions with Array Inputs
Many MATLAB functions - one or more scalar inputs
produce a scalar output
y = sin(x)
stores the result in y
If receive an array of input values
calculate an array of output values on an element-by-element basis
if x = [ 0 pi/2 pi 3*pi/2 2*pi]
the result: y = [0 1 0 –1 0].

99. Common MATLAB Functions
many MATLAB functions work correctly for both real and complex inputs
sqrt(-2)
MATLAB imaginary answer:
ans = i

103. Debugging MATLAB Programs
Three types of error:
syntax error
run-time error
logical error

104. Syntax Errors errors in the MATLAB statements
spelling errors or punctuation errors
detected by the MATLAB compiler
x = (y + 3) / 2);
in an M-file - test.m when executed:
» test
??? x = (y + 3) / 2) |
Missing operator, comma, or semi-colon.
Error in ==> d:\book\matlab\chap1\test.m
On line 2 ==>

105. cause the program to return Inf or NaN
Run Time Errors
occurs when an illegal mathematical operation is attempted during program execution
e.g., attempting to divide by 0
cause the program to return Inf or NaN
then used in further calculations
The results of a program that contains calculations using Inf or NaN are usually invalid

106. Logical Errors
occur when the program compiles and runs successfully but produces the wrong answer