Matlab plotting

Matlab Plotting
Presented by:
Shahid Sultan
Research Scholar
N.I.T Srinagar
1

MATLAB
Topics Covered:
1. Plotting basic 2-D plots.
The plot command.
The fplot command.
Plotting multiple graphs in the same plot.
Formatting plots.
Two Dimensional Plots
2

MAKING X-Y PLOTS
MATLAB has many functions and commands that can be used to
create various types of plots.
In our class we will only create two dimensional x – y plots.
3

Plotting
𝑠 = 2 sin 3𝑡 + 2 + 15𝑡 − 7
4
t 0 1 2 3 4 5
s -5.18 6.08 24.97 36 54.98 66.07

8 10 12 14 16 18 20 22 24
0
200
400
600
800
1000
1200
DISTANCE (cm)
INTENSITY
(lux)
Light Intensity as a Function of Distance
Comparison between theory and experiment.
Theory
Experiment
Plot title
y axis
label
x axis
label
Text
Tick-mark label
EXAMPLE OF A 2-D PLOT
Data symbol
Legend
Tick-mark
5

TWO-DIMENSIONAL plot() COMMAND
where x is a vector (one dimensional array), and y is a vector.
Both vectors must have the same number of elements.
 The plot command creates a single curve with the x values on
the abscissa (horizontal axis) and the y values on the ordinate
(vertical axis).
 The curve is made from segments of lines that connect the
points that are defined by the x and y coordinates of the
elements in the two vectors.
The basic 2-D plot command is:
plot(x,y)
6

 If data is given, the information is entered as the elements of the
vectors x and y.
 If the values of y are determined by a function from the values
of x, than a vector x is created first, and then the values of y
are calculated for each value of x. The spacing (difference)
between the elements of x must be such that the plotted curve
will show the details of the function.
CREATING THE X AND Y VECTORS
7

PLOT OF GIVEN DATA
Given data:
>> x=[1 2 3 5 7 7.5 8 10];
>> y=[2 6.5 7 7 5.5 4 6 8];
>> plot(x,y)
A plot can be created by the commands shown below. This can be
done in the Command Window, or by writing and then running a
script file.
Once the plot command is executed, the Figure Window opens with
the following plot.
x
y
1 2 3 5 7 7.5 8
6.5 7 7 5.5 4 6 8
10
2
8

LINE SPECIFIERS IN THE plot() COMMAND
Line specifiers can be added in the plot command to:
 Specify the style of the line.
 Specify the color of the line.
 Specify the type of the markers (if markers are desired).
plot(x,y,’line specifiers’)
10

Line Specifier Line Specifier Marker Specifier
Style Color Type
Solid - red r plus sign +
dotted : green g circle o
dashed -- blue b asterisk *
dash-dot -. Cyan c point .
magenta m square s
yellow y diamond d
black k
plot(x,y,‘line specifiers’)
11

 The specifiers are typed inside the plot() command as strings.
 Within the string the specifiers can be typed in any order.
 The specifiers are optional. This means that none, one, two, or
all the three can be included in a command.
EXAMPLES:
plot(x,y) A solid blue line connects the points with no markers.
plot(x,y,’r’) A solid red line connects the points with no markers.
plot(x,y,’--y’) A yellow dashed line connects the points.
plot(x,y,’*’) The points are marked with * (no line between the
points.)
plot(x,y,’g:d’) A green dotted line connects the points which are
marked with diamond markers.
12

Year
Sales (M)
1988 1989 1990 1991 1992 1993 1994
127 130 136 145 158 178 211
PLOT OF GIVEN DATA USING LINE
SPECIFIERS IN THE plot() COMMAND
>> year = [1988:1:1994];
>> sales = [127, 130, 136, 145, 158, 178, 211];
>> plot(year,sales,'--r*')
Line Specifiers:
dashed red line and
asterisk markers.
13

PLOT OF GIVEN DATA USING LINE
SPECIFIERS IN THE plot() COMMAND
Dashed red line and
asterisk markers.
14

% A script file that creates a plot of
% the function: 3.5^(-0.5x)*cos(6x)
x = [-2:0.01:4];
y = 3.5.^(-0.5*x).*cos(6*x);
plot(x,y)
CREATING A PLOT OF A FUNCTION
Consider: 4
2
for
)
6
cos(
5
.
3 5
.
0



 
x
x
y x
A script file for plotting the function is:
Creating a vector with spacing of 0.01.
Calculating a value of y
for each x.
Once the plot command is executed, the Figure Window opens with
the following plot.
15

A PLOT OF A FUNCTION
4
2
for
)
6
cos(
5
.
3 5
.
0



 
x
x
y x
16

CREATING A PLOT OF A FUNCTION
If the vector x is created with large spacing, the graph is not accurate.
Below is the previous plot with spacing of 0.3.
x = [-2:0.3:4];
y = 3.5.^(-0.5*x).*cos(6*x);
plot(x,y)
17

THE fplot COMMAND
fplot(‘function’,limits)
The fplot command can be used to plot a function
with the form: y = f(x)
 The function is typed in as a string.
 The limits is a vector with the domain of x, and optionally with limits
of the y axis:
[xmin,xmax] or [xmin,xmax,ymin,ymax]
 Line specifiers can be added.
18

PLOT OF A FUNCTION WITH THE fplot() COMMAND
>> fplot('x^2 + 4 * sin(2*x) - 1')
3
3
for
1
)
2
sin(
4
2





 x
x
x
y
A plot of:
19

PLOTTING MULTIPLE GRAPHS IN THE SAME PLOT
Two methods:
1. Using the plot command.
2. Using the hold on, hold off commands.
20

USING THE plot() COMMAND TO PLOT
MULTIPLE GRAPHS IN THE SAME PLOT
Plots three graphs in the same plot:
y versus x, v versus u, and h versus t.
 By default, MATLAB makes the curves in different colors.
 Additional curves can be added.
 The curves can have a specific style by adding specifiers after
each pair, for example:
plot(x,y,u,v,t,h)
plot(x,y,’-b’,u,v,’—r’,t,h,’g:’)
21

4
2 

 x
Plot of the function, and its first and second
derivatives, for , all in the same plot.
10
26
3 3


 x
x
y
4
2 

 x
x = [-2:0.01:4];
y = 3*x.^3-26*x+6;
yd = 9*x.^2-26;
ydd = 18*x;
plot(x,y,'-b',x,yd,'--r',x,ydd,':k')
vector x with the domain of the function.
Vector y with the function value at each x.
4
2 

 x
Vector yd with values of the first derivative.
Vector ydd with values of the second derivative.
Create three graphs, y vs. x (solid blue
line), yd vs. x (dashed red line), and ydd
vs. x (dotted black line) in the same figure.
22

-2 -1 0 1 2 3 4
-40
-20
0
20
40
60
80
100
120
23

hold on Holds the current plot and all axis properties so that
subsequent plot commands add to the existing plot.
hold off Returns to the default mode whereby plot commands
erase the previous plots and reset all axis properties
before drawing new plots.
USING THE hold on, hold off, COMMANDS
TO PLOT MULTIPLE GRAPHS IN THE SAME PLOT
This method is useful when all the information (vectors) used for
the plotting is not available a the same time.
24

Plot of the function, and its first and second
derivatives, for all in the same plot.
10
26
3 3


 x
x
y
4
2 

 x
x = [-2:0.01:4];
y = 3*x.^3-26*x+6;
yd = 9*x.^2-26;
ydd = 18*x;
plot(x,y,'-b')
hold on
plot(x,yd,'--r')
plot(x,ydd,':k')
hold off
Two more graphs are created.
First graph is created.
USING THE hold on, hold off, COMMANDS
TO PLOT MULTIPLE GRAPHS IN THE SAME PLOT
25

8 10 12 14 16 18 20 22 24
0
200
400
600
800
1000
1200
DISTANCE (cm)
INTENSITY
(lux)
Light Intensity as a Function of Distance
Comparison between theory and experiment.
Theory
Experiment
Plot title
y axis
label
x axis
label
Text
EXAMPLE OF A FORMATTED 2-D PLOT
Data symbol
Legend
Tick-mark
Tick-mark label
26

FORMATTING PLOTS
A plot can be formatted to have a required appearance.
With formatting you can:
 Add title to the plot.
 Add labels to axes.
 Change range of the axes.
 Add legend.
 Add text blocks.
 Add grid.
27

FORMATTING PLOTS
There are two methods to format a plot:
1. Formatting commands.
In this method commands, that make changes or additions to
the plot, are entered after the plot() command. This can be
done in the Command Window, or as part of a program in a
script file.
2. Formatting the plot interactively in the Figure Window.
In this method the plot is formatted by clicking on the plot and
using the menu to make changes or add details.
28

FORMATTING COMMANDS
title(‘fuctions’)
Adds the string as a title at the top of the plot.
xlabel(‘string’)
Adds the string as a label to the x-axis.
ylabel(‘string’)
Adds the string as a label to the y-axis.
axis([xmin xmax ymin ymax])
Sets the minimum and maximum limits of the x- and y-axes.
29

FORMATTING COMMANDS
legend(‘string1’,’string2’,’string3’)
Creates a legend using the strings to label various curves (when
several curves are in one plot). The location of the legend is
specified by the mouse.
text(x,y,’string’)
Places the string (text) on the plot at coordinate x,y relative to
the plot axes.
gtext(‘string’)
Places the string (text) on the plot. When the command
executes the figure window pops and the text location is clicked
with the mouse.
30

FORMATTING Example
123
x = [10: 0.1: 22];
y = 95000 ./ x .^ 2;
x_data = [10: 2: 22];
y_data = [950 640 460 340 250 180 140];
plot(x, y, '-', x_data, y_data, 'ro--')
xlabel('DISTANCE (cm)')
ylabel('INTENSITY (lux)')
title('Light Intensity as a Function of Distance')
axis( [8 24 0 1200] )
legend('Theory', 'Experiment')
The plot that is obtained is shown again in the next slide.
Labels for the axes
Title of the plot
Set limits of the axes
Create legend
31

EXAMPLE OF A FORMATTED PLOT
123
32

EXAMPLE OF A FORMATTED PLOT
33

FORMATTING A PLOT IN THE FIGURE WINDOW
Once a figure window is open, the figure can be formatted interactively.
Use Figure,
Axes, and
Current Object-
Properties in
the Edit menu
Click here to start the
plot edit mode.
Use the insert menu to
34

PLOTTING MULTIPLE PLOTS ON ONE PAGE
Several plots on one page can be created with the subplot command.
subplot(m,n,p) This command creates mxn plots in the Figure
Window. The plots are arranged in m rows and n
columns. The variable p defines which plot is active.
The plots are numbered from 1 to mxn. The upper left
plot is 1 and the lower right plot is mxn. The numbers
increase from left to right within a row, from the first
row to the last.
35

PLOTTING MULTIPLE PLOTS ON ONE PAGE
subplot(3,2,1) subplot(3,2,2)
subplot(3,2,3)
subplot(3,2,5)
subplot(3,2,4)
subplot(3,2,6)
For example, the
command:
subplot(3,2,p)
Creates 6 plots
arranged in 3 rows
and 2 columns.
36

EXAMPLE OF MULTIPLE PLOTS ON ONE PAGE
The script file of the figure above is shown in the next slide.
37

% Example of using the subplot command.
x1=20:100; % Creating a vector x1
y1=sin(x1); % Calculating y1
subplot(2,3,1) % Creating the first plot
plot(x1,y1) % Plotting the first plot
axis([0 20 -2 2]) % Formatting the first plot
text(2,1.5,'A plot of sin(x)')
y2=sin(x1).^2; % Calculating y2
subplot(2,3,2) % Creating the second plot
plot(x1,y2) % Plotting the second plot
axis([0 20 -2 2]) % Formatting the second plot
text(2,1.5,'A plot of sin^2(x)')
y3=sin(x1).^3; % Calculating y2
SCRIPT FILE OF MULTIPLE PLOTS ON ONE PAGE
(The file continues on the next slide)
38

subplot(2,3,3) % Creating the third plot
plot(x1,y3) % Plotting the third plot
axis([0 20 -2 2]) % Formatting the third plot
text(2,1.5,'A plot of sin^3(x)')
subplot(2,3,4) % Creating the fourth plot
fplot('abs(sin(x))',[0 20 -2 2]) % Plotting the fourth plot
text(1,1.5,'A plot of abs(sin(x))') % Formatting the fourth plot
subplot(2,3,5) % Creating the fifth plot
fplot('sin(x/2)',[0 20 -2 2]) % Plotting the fifth plot
text(1,1.5,'A plot of sin(x/2)') % Formatting the fifth plot
subplot(2,3,6) % Creating the sixth plot
fplot('sin(x.^1.4)',[0 20 -2 2]) % Plotting the fifth plot
text(1,1.5,'A plot of sin(x^1^.^2)') % Formatting the sixth plot
SCRIPT FILE OF MULTIPLE PLOTS ON ONE PAGE
(CONT,)
39

Area Plot
• In the Area plotting graph, you can use basic functions. It is a
very easy draw.
• In the MATLAB plotting, there is a function
area() to plot Area.
% To create the area plot for the given equation Sin(t)Cos(2t). %
Enter the value of range of variable 't'.
t=[0:0.2:20];
a=[sin(t).*cos(2.*t)];
area(a)
title('Area Plot')
40

Stem Plot
• In Stem plot, the discrete sequence data and variables are
used. This plot is created by using the stem() function.
% Consider the variable range of 'x' and 'y',
x=[3 1 6 7 10 9 11 13 15 17];
y=[14 7 23 11 8 16 9 3 23 17];
stem(x,y,'r')
title('Stem Plot')
xlabel('X axis')
ylabel('Y axis')
42

Bar Plot
• A bar plot or bar chart is a graph that represents the category
of data with rectangular bars with lengths and heights that is
proportional to the values which they represent.
x=[1 3 5 7 10 13 15];
y=[0 0.5 1 1.5 3 2 2];
bar(x,y)
title('Bar Plot')
xlabel('X axis')
ylabel('y axis')
44

Barh Plot
• Barh plot is short abbreviations of Horizontal bar. Here I am
using the Barh function for the horizontal plane.
x=[1 3 5 7 10 13 15];
y=[0 0.5 1 1.5 3 2 2];
barh(x,y)
title('Barh Plot')
xlabel('X axis')
ylabel('y axis')
46

Stairs Plot
• This is again one of the MATLAB 2D plots that look more like
stairs.
x=[0 1 2 4 5 7 8];
y=[1 3 4 6 8 12 13];
stairs(x,y)
title('Stairs Plot')
xlabel('X axis')
ylabel('Y axis')
48

Pie Plot
• In mathematics, the pie chart is used to indicate data in
percentage (%) form.
x=[10 20 25 40 75 80 90];
pie(x)
title('Pie Plot')
50

Scatter Plot
• A scatter plot (aka scatter chart, scatter graph) uses dots to
represent values for two different numeric variables. The
position of each dot on the horizontal and vertical axis
indicates values for an individual data point. Scatter plots are
used to observe relationships between variables.
x=[1 2 3 5 7 9 11 13 15];
y=[1.2 3 4 2.5 3 5.5 4 6 7];
scatter(x,y,'g')
title('Scatter Plot')
xlabel('X axis')
ylabel('Y axis')
52

Solving Basic Algebraic Equations in
MATLAB
• The solve function is used for solving algebraic
equations. In its simplest form, the solve
function takes the equation enclosed in
quotes as an argument.
• For example, let us solve for x in the equation
x-5 = 0
solve('x-5=0') or
y = solve('x-5 = 0') or
solve('x-5')
54

• If the equation involves multiple symbols, then
MATLAB by default assumes that you are solving
for x
• The solve function has another form −
solve(equation, variable)
where, you can also mention the variable.
• For example, let us solve the equation
v – u – 3t2 = 0, for v.
In this case, we should write −
solve('v-u-3*t^2=0', ‘v')
55

Solving Quadratic Equations in
MATLAB
• The solve function can also solve higher order equations. It
is often used to solve quadratic equations. The function
returns the roots of the equation in an array.
• The following example solves the quadratic equation x2 -7x
+12 = 0.
eq = 'x^2 -7*x + 12 = 0';
s = solve(eq);
disp('The first root is: ');
disp(s(1));
disp('The second root is: '),
disp(s(2));
56

• When you run the file, it displays the following
result −
The first root is:
3
The second root is:
4
57

Solving System of Equations in
MATLAB
• The solve function can also be used to generate solutions of
systems of equations involving more than one variables.
• Let us solve the equations −
5x + 9y+3z = 5
3x – 6y+7y = 4
• Create a script file and type the following code −
s = solve('5*x + 9*y = 5','3*x - 6*y = 4‘);
s.x
s.y
When you run the file, it displays the following result −
ans = 22/19
ans = -5/57
58

Polynomials
• MATLAB represents polynomials as row
vectors containing coefficients ordered by
descending powers. For example, the
equation P(x) = x4 + 7x3 - 5x + 9 could be
represented as −
p = [1 7 0 -5 9];
59

Evaluating Polynomials
• The polyval function is used for evaluating a
polynomial at a specified value. For example,
to evaluate our previous polynomial p, at x =
4, type −
p = [1 7 0 -5 9];
polyval(p,25)
60

• MATLAB also provides the polyvalm function
for evaluating a matrix polynomial. A matrix
polynomial is a polynomial with matrices as
variables.
• For example, let us create a square matrix X
and evaluate the polynomial p, at X −
p = [1 7 0 -5 9];
X = [1 2 -3 4; 2 -5 6 3; 3 1 0 2; 5 -7 3 8];
polyvalm(p, X)
61

Finding the Roots of Polynomials
• The roots function calculates the roots of a
polynomial. For example, to calculate the
roots of our polynomial p, type −
p = [1 7 0 -5 9];
r = roots(p)
62

MATLAB - Transforms
• MATLAB provides command for working with
transforms, such as the Laplace and Fourier
transforms.
• Transforms are used in science and engineering
as a tool for simplifying analysis and look at data
from another angle.
• The Fourier transform allows us to convert a
signal represented as a function of time to a
function of frequency. Laplace transform allows
us to convert a differential equation to an
algebraic equation.
63

• MATLAB provides the laplace, fourier and fft
commands to work with Laplace, Fourier and
Fast Fourier transforms.
64

The Laplace Transform
• The Laplace transform of a function of time f(t) is given by the
following integral −
• Laplace transform is also denoted as transform of f(t) to F(s). You
can see this transform or integration process converts f(t), a
function of the symbolic variable t, into another function F(s), with
another variable s.
• Laplace transform turns differential equations into algebraic ones.
To compute a Laplace transform of a function f(t), write −
laplace(f(t))
65

• In this example, we will compute the Laplace transform of
some commonly used functions.
• Create a script file and type the following code −
syms s t a b w
laplace(a)
laplace(t^2)
laplace(t^9)
laplace(exp(-b*t))
laplace(sin(w*t))
laplace(cos(w*t))
66

The Inverse Laplace Transform
• MATLAB allows us to compute the inverse
Laplace transform using the command
ilaplace.
• For example,
ilaplace(1/s^3)
67

Fourier Transform
• Fourier transforms commonly transforms a
mathematical function of time, f(t), into a new
function, sometimes denoted by or F, whose
argument is frequency with units of cycles/s
(hertz) or radians per second. The new
function is then known as the Fourier
transform and/or the frequency spectrum of
the function f.
68

• create a script file and type the following code in it −
syms x
f = exp(-2*x^2); %our function
ezplot(f,[-2,2]) % plot of our function
frt= fourier(exp(-2*x^2)) % Fourier transform
69

• The following result is displayed −
frt= (2^(1/2)*pi^(1/2)*exp(-w^2/8))/2
ezplot(ftr)
70

Inverse Fourier Transforms
• MATLAB provides the ifourier command for
computing the inverse Fourier transform of a
function. For example,
f = ifourier(-2*exp(-abs(w)))
• MATLAB will execute the above statement and
display the result −
f = -2/(pi*(x^2 + 1))
71

MATLAB - Differential
• MATLAB provides the diff command for
computing symbolic derivatives. In its simplest
form, you pass the function you want to
differentiate to diff command as an argument.
• compute the derivative of the function f(t) =
3t2 + 2t-2
syms t
f = 3*t^2 + 2*t^(-2);
diff(f)
72

syms x
syms t
f = (x + 2)*(x^2 + 3)
der1 = diff(f)
f = (t^2 + 3)*(sqrt(t) + t^3)
der2 = diff(f)
f = (x^2 - 2*x + 1)*(3*x^3 - 5*x^2 + 2)
der3 = diff(f)
f = (2*x^2 + 3*x)/(x^3 + 1)
der4 = diff(f)
f = (x^2 + 1)^17
der5 = diff(f)
f = (t^3 + 3* t^2 + 5*t -9)^(-6)
der6 = diff(f)
73

Computing Higher Order Derivatives
• To compute higher derivatives of a function f,
we use the syntax diff(f,n).
• Let us compute the second derivative of the
function y = f(x) = x .e-3x
f = x*exp(-3*x);
diff(f, 2)
74

• let us solve a problem. Given that a function y = f(x) = 3 sin(x) + 7
cos(5x). We will have to find out whether the equation f" + f = -
5cos(2x) holds true.
syms x
y = 3*sin(x)+7*cos(5*x); % defining the function
lhs = diff(y,2)+y; %evaluting the lhs of the equation
rhs = -5*cos(2*x); %rhs of the equation
if(isequal(lhs,rhs))
disp('Yes, the equation holds true');
else disp('No, the equation does not hold true');
End
disp('Value of LHS is: ');
disp(lhs);
75

Matlab plotting

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