90

But, Our MATLAB Tutorial is Not Over!MATLAB Has Many More Useful Features.

• Built-in functions

• Data plotting

• Logical expressions

• Branches and loops

• User-defined functions

91

MATLAB’s Built-In Functions

MATLAB comes with a very large variety of built-in mathfunctions that are ready for use. Common ones include:

sin(x) cos(x) tan(x)

asin(x) acos(x) atan(x) atan2(y,x)

exp(x) log(x) sqrt(x)

abs(x) angle(x)

Less common ones include hyperbolics and their inverses,Bessel functions of various flavors, and other functionsused in advanced engineering and scientific analyses.This is one of MATLAB’s greatest strengths.

92

Function Operations on Arrays

Upon receiving an array of input numbers, many MATLABfunctions calculate an array of output values on anelement-by-element basis. For example, if

x = [0 pi/2 pi 3*pi/2 2*pi]

then the statement

y = sin(x)

produces the result y = [0 1 0 -1 0] .

93

Complex-Valued Inputs

Unlike many languages such as C and Fortran, manyMATLAB functions work correctly for both real-valuedand complex-valued inputs. For example:

>> sqrt(-2) ans = 0 + 1.4142i

For information about the admissible inputs for a specificfunction, use the MATLAB Help Browser.

94

Introduction to Data Plotting

To plot a data set in MATLAB, simply create two vectorscontaining the x and y values to be plotted and use theplot function. Any pair of vectors can be plotted versuseach other as long as both have the same length.

For example, say we want to plot f(x) = sin(2x) for0 ≤ x ≤ 2π in increments of π/100 radian.

x = 0:pi/100:2*pi;y = sin(2*x);plot(x,y);

When the plot function is executed,MATLAB opens a Figure Window anddisplays the plot in that window.

95

Add Title, Axis Labels, and Grid Lines

Titles and axis labels can be added with the title ,xlabel , and ylabel functions. Grid lines can be addedwith grid on and removed with grid off . Example:

x = 0:pi/100:2*pi; y = sin(2*x); plot(x,y);

title( ' Plot of f(x) = sin(2x) ' );

xlabel( ' x ' );

ylabel( ' y ' ); grid on;

Addedcode

96

Plot Multiple Functions on the Same Graph

Example: Plot f(x) = sin(2x) and its derivativeg(x) = 2cos(2x) on the same graph.

x = 0:pi/100:2*pi;

y1 = sin(2*x);

y2 = 2*cos(2*x);

plot(x,y1,x,y2);

97

Specify Line Colors and Styles

Example: Plot f(x) = sin(2x) and its derivative

g(x) = 2cos(2x) on the same graph. Use a solid red

line for f(x) and a dashed green line for g(x).

x = 0:pi/100:2*pi;

y1 = sin(2*x);

y2 = 2*cos(2*x);

plot(x,y1,'r-',x,y2,'g--');

denotes a

red solid

line

denotes a

green dashed

line

98

Add an Explanatory Legend

Example: Add an explanatory legend to the previous

graph.

x = 0:pi/100:2*pi;

y1 = sin(2*x);

y2 = 2*cos(2*x);

plot(x,y1,'r-',x,y2,'g--');

legend('f(x)','d/dx f(x)',1);

Legend position parameter: 0 for automatic

best placement (least conflict with the data);

1 for upper right-hand corner; etc.

99

Typical Complete MATLAB Plot

Title

Legend

xlabel

ylabel

grid lines

100

Implement Logarithmic Axes, If Needed

Logarithmic x and / or y axes can be very useful whenthe data to be displayed varies over ranges of about100:1 or more. This feature can be implemented byusing the following variations of the plot function:

plot Both x and y data are plotted on linearaxes.

semilogx x data are plotted on a log axis and y dataare plotted on a linear axis.

semilogy x data are plotted on a linear axis andy data are plotted on a log axis.

loglog Both x and y data are plotted on log axes.

101

Choice of Linear and/or Log Scales

102

Relational Operators in MATLAB

• Allow us to test whether two values (floating point,integer, or character) are the same, or whether one isgreater than or less than the other.

• If the test yields a “True” condition, the number 1 isreturned. If the test yields a “False” condition, thenumber 0 is returned.

• We can then act on the “True” or “False” conditionby subsequently choosing different paths within theprogram.

103

Cautionary Remark

• Unlike most programming languages, MATLAB doesnot have a logical type. In MATLAB, the “True” and“False” conditions are represented as 1 and 0 ,respectively.

• Therefore, we must be careful to distinguish betweenthe numbers 1 and 0 produced during normalcalculations and the numbers 1 and 0 produced as aresult of “True” and “False” tests. The context is thekey!

104

List of Relational Operators

== equal to

!= not equal to

> greater than

>= greater than or equal to

< less than

<= less than or equal to

105

Sample Results of Relational Operators

Operation Result

5 == 6 0

5 < 6 1

5 <= 5 1

5 ~= 4 1

[5 6] < [7 4] [1 0]

[5 6] < 6 [1 0]

106

More Sample Results of

Relational Operators

Operation Result

97 == 'a' 1

65 == 'A' 1

'B' == 66 1

32 == ' ' 1

41 == ')' 1

'a' == 'A' 0

'a' < 'B' 0

')' < 'A' 1

ASCII

values

(see

Slide

328)

Interesting side note:

ASCII values can be

used in arithmetic

expressions such as

>> 2 * ')'

ans =

82

107

Two Cautions About theUse of the == Operator

1. Do not confuse the equality relational operator ==

with the arithmetic assignment operator = .

Here is an example where both are used in the

same statement:

>> a = 'fate';

>> b = 'cake';

>> result = a == b

result =

0 1 0 1

108

Two Cautions About theUse of the == Operator

2. Beware of checking for the equality of two floating-

point numbers. Roundoff error during calculations

can cause two theoretically equal numbers to differ

enough so that the equality test fails. Example:

>> a = 0;

>> b = sin(pi);

>> a == b

ans =

0

MATLAB yields 1.2246e-16

instead of exactly 0!

109

Testing for the “Equality” of Two Values

Instead of comparing two floating-point values forexact equality, set up a test to determine if thenumbers are nearly equal to each other within somemargin that accounts for the amount of roundoff errorexpected during the calculation. Example:

>> a = 0;>> b = sin(pi);>> abs(a-b) < 1.0e-14ans = 1

Note: This problemdoes not occur whencomparing integers!

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Logic Operators

• These permit combinations of relational tests toyield a composite “True” or “False” condition.

• Set of logical operators:

& AND

OR

xor Exclusive OR

∼∼∼∼ Not

111

Truth Table for Logic Operators

Inputs AND OR XOR NOT

i 1 i 2 i 1 & i 2 i 1 i 2 xor(i 1 , i 2) ∼∼∼∼i 1

0 0 0 0 0 1

0 1 0 1 1 1

1 0 0 1 1 0

1 1 1 1 0 0

112

Sample Results of Logic Operators

Operation Result

(4 < 6) & (3 > 2) 1

(4 < 6) (3 > 9) 1

∼∼∼∼(4 < 6) 0

∼∼∼∼(4 < 6) (3 > 9) 0

[(3 < 1) 2 = = 2] [(2 <= 4) 7 ∼∼∼∼= 9] [1 1]

[1 0] & 1 [1 0]