The document provides an introduction to MATLAB, focusing on matrix operations, MATLAB functions, and 2-D/3-D plotting techniques. It details matrix creation, special matrices, built-in functions, polynomial operations, and user-defined functions, along with loops and conditionals. Additionally, it covers visualization techniques including 2-D and 3-D graphs, as well as examples of mathematical operations in MATLAB.

1. Lab Session 1:
Introduction to MATLAB

2. Matrices and Matrix Computations
 MATLAB treats all variables as matrices
 Matrices are assigned to variables with an ’=’
sign
 Names are case sensitive
Example: Enter
1 2 3
4 5 6
7 8 9
A

 
 
 
 
 

 
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3. Cont‟d
>> A= [1 -2 3;-4 5 6; 7 8 -9]
A =
1 -2 3
-4 5 6
7 8 -9
 New rows are indicated by a new line or a
semicolon.
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4. Cont‟d
Example: Enter the column vector
>> B=[1;-4;7] Or >> B= [1 -4 7]'
B = B =
1 1
-4 -4
7 7
1
4
7
B
 
 
 
 
 
 
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5. Cont‟d
 Elements of a matrix can be expressions:
>> C= [1 1+1 9/3 2^3]
C =
1 2 3 8
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6. Definition of an array
where a = first value, h= step size & b= last value
Examples:
>> 0:2:10
ans =
0 2 4 6 8 10
>> 10:-2:1
ans =
10 8 6 4 2
a:h:b
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7. The “linspace” Function
where a = first value, b= last value & n = # of
elements to be considered
>> linspace(1,9,5)
ans =
1 3 5 7 9
linspace (a, b, n)
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8. Some special matrices generated by
built-in functions
>> ones (2) % a 2x2 matrix of 1s
ans =
1 1
1 1
>> zeros(2,4)%a 2x4 matrix of 0s
ans =
0 0 0 0
0 0 0 0
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9. Cont‟d
>> eye(3)%a 3x3 identity matrix
ans =
1 0 0
0 1 0
0 0 1
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10. The „diag‟ Function
1. Creates a matrix with specified values on the
(main) diagonal
>> d=[1 2 3];
>> D=diag(d)
D =
1 0 0
0 2 0
0 0 3
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11. Cont‟d
Note: the’;’ in “d=[1 2 3];” is used to suppress
the echoing of the input (intermediate results)
2. Extracts the diagonal entries
>> C=[1 -2 3;-4 5 6;7 8 -9];
>> c=diag(C)
c =
1
5
-9
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12. Some Useful Matrix Functions
 inv - inverse
 rank - rank
 eig - eigenvalues & eigenvectors
 lu - LU decomposition( in partial
pivoting sense)
 qr - QR factorization
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13. Cont‟d
For example,
 >> [U, D]=eig(A)
generates a matrix U whose columns are
eigenvctors of A & a diagonal matrix D
with the eigenvalues of on its diagonal.
 [L U P]=lu(A)
returns unit lower triangular matrix L, upper
triangular matrix U, and permutation matrix P so
that P*A = L*U.
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14. Matrix Operations
1. Transpose
2. Scalar Multiplication
3. Addition
4. Multiplication & Dot product of vectors
5. Division
6. Power
7. Component by component Multiplication,
Division and Power
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15. Polynomials
 Polynomials are specified by a vector whose
entries are the coefficients of the polynomial
Example: is specified by
[1 2 -10 1].
>> f=[1 2 -10 1] ;
1
10
2 2
3



 x
x
x
y
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16. Cont‟d
>> polyval(f,-2)%evaluates the polynomial at x=-2
ans =
21
>> roots(f)%finds the roots of a polynomial
ans =
-4.3511
2.2489
0.1022
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17. Output Formats
 All calculations in MATLAB are performed
in double precision. But the format of the
displayed output can be controlled by the
following commands:
 format short = Fixed point with 4 decimal
places (is the default format)
 format long = Fixed point with 15 decimal
places
 format short e = Scientific notation with4
decimal places
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18. Cont‟d
 format long e = Scientific notation with 15
decimal places
 format rat = approximation by ratio of
small integers
Once initiated, the chosen format remains
in effect until changed.
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19. Function Files
 Some elementary mathematical MATLAB
functions:
Trigonometric: sin, cos, tan
Exponential: exp - Natural exp
log - Natural log
log10 -Common log
sqrt -square root
abs - absolute value
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20. Lab Session 2:
Introduction to MATLAB -
Continued

21. 2-D Plots
 Two dimensional graphs of functions are
generated by the command plot.
Example: Plot over the interval [-2, 2].
>> x = -2 : 0.25 : 2;
>> y = x.^2+1;
>> plot(x,y)
 The command plot( x, y, 'r+:') results in red
color with + for points and dotted lines.
1
2

 x
y
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22. Cont‟d
 Use help plot for more options.
 Other features can be added with commands
such as grid, xlabel, ylable, title, etc.
 Plots of parametrically defined curves can
also be made. For example:
 >> t = 0 : 0.1 :4*pi;
 >> x = cos(2*t);
 >>y = sin(3*t);
 >> plot(x,y)
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23. Multiple plots on a single graph
Example:
>> x=-2:0.1:2;
>> y1=x.^2+1;
>> y2=x.^2-1;
>> plot(x,y1,x,y2)
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24. Multiple plots on a single
graph
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-1
0
1
2
3
4
5
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25. 3-D Mesh Plots
 3-D mesh surface plots are drawn with the
function mesh.
Example: Graph of over the square
[-4, 4] X [-4, 4].
>> [x y]=meshgrid(-4:0.25:4,-4:0.25:4);
>> z=exp(-x.^2-y.^2);
>> mesh(x,y,z)
 Other commands related to 3-D graphs are
plot3 and surf.
2
2
y
x
e
z 


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26. 3-D Mesh Plots
-4
-3
-2
-1
0
1
2
3
4
-4
-3
-2
-1
0
1
2
3
4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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27. Cont‟d
 A complete list of elementary functions can
be obtained with the command:
>> help elfun
Note: All function names are in lower case.
 MATLAB functions may have single or
multiple arguments
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28. User Defined Functions
 MATLAB allows users to define new
functions by constructing an M-file in the
M-file Editor/Debugger
function y = function name(input arguments)
y = ………………………..
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29. Example:
Define
function y=f(x)
y=2*x.^2+1
Once a function is saved as an M-file (as f.m),
we can use MATLAB command window to
compute. For example,
>> f(2)
ans =
9
2
( ) 2 1
f x x
 
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30. Relations and Loops
 Relational operators in MATLAB:
= =, < =, ~ =, < , >
 Logical Operators:
& = and ! = or ~ = not
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31. Loops-Flow control statements
(if, for & while statements)
1. if
General Form :
if (expression)
statements
else if (expression)
statements
else
statements
end
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32. Cont‟d
2. for:
for(loop variable=loop exp)
statements
end
3. while:
while(while expression)
condition statement
end
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33. Example:
Write a MATLAB program which displays the sum of
the first 100 positive integers.
Solution:
>> sum=0;
>> for i=1:100
sum = sum +i;
end
>> sum
sum =
5050
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34. Last
Extra !!!!!!!!!!!!
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35. 35
Polynomial
a. derivative:
>>k = polyder(p)
b. the derivative of the product of the
polynomials a and b:
>>k = polyder(a,b)
Example: a = [3 6 9]; b = [1 2 0];
>>k = polyder(a,b)
k =
12 36 42 18
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36. 36
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37. 37
c. Integration
A polynomial representing the integral of polynomial p,
>> polyint(p) assumes a constant of integration c=0.
Example: p = [1 -6 -72 -27]
ans =
0.2500 -2.0000 -36.0000 -27.0000
D . To find a zero of fun near x0, if x0 is a scalar, we use
>> x = fzero((@myfun, x0 )
 fun is the function whose zero is to be computed.
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38. 38
Example 1: Calculate π by finding the zero of the sine
function near 3.
>> x = fzero(@sin,3)
ans. x = 3.1416
Example 2: To find the zero of cosine between 1 and 2
>> x = fzero(@cos,[1 2])
ans. x = 1.5708
Note that cos(1) and cos(2) differ in sign.
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39. Example 3
To find a zero of the function f(x) = x3 – 2x – 5, write
an anonymous function f: f = @(x)x.^3-2*x-4; Then
find the zero near :
>>z = fzero(f,1)
z = 2.0000
Example 4: f=@(x)-11-22*x+17*x^2-2.5*x^3;
>> x = fzero(f,3) x = 2.4269
or >>x = fzero(@(x)-11-22*x+17*x^2-2.5*x^3,3)
or x = fzero(@(x)-11-22*x+17*x^2-2.5*x^3,[2 3])
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40. 40
Polynomial Interpolation
Example: Here are two vectors representing the census years
from 1900 to 1990 and the corresponding United States
population in millions of people.
t = 1900:10:1990;
p = [75.995 91.972 105.711 123.203 131.669 150.697
179.323 203.212 226.505 249.633];
>>interp1(t,p,1975)
The expression interpolates within the census data to estimate the
population in 1975. The result is
ans = 214.8585
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41. .
Graphs
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42. Sphere
42
 a = 2;
 u = linspace(0, 2*pi, 41);
 v = linspace(-pi/2, pi/2, 31);
 [U,V] = meshgrid(u,v);
 X = a*cos(V).*cos(U);
 Y = a*cos(V).*sin(U);
 Z = a*sin(V);
 surf(X,Y,Z); colormap(gray); shading flat
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43. Cylindrer
43
 z = linspace(0,1,41);
 cylinder(3*(z-1/3).^2)
-2
-1
0
1
2
-2
-1
0
1
2
0
0.2
0.4
0.6
0.8
1
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44.  >> [x,y]=meshgrid(-4:0.1:4,-4:0.1:4);
 >> z=sqrt(x.^2+y.^2);
 >> mesh(x,y,z);
 >> hold on;
 >> w=-sqrt(x.^2+y.^2);
 >> mesh(x,y,w);
 >> hold off
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45. -4
-2
0
2
4
-4
-3
-2
-1
0
1
2
3
4
-6
-4
-2
0
2
4
6
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46. Cont…
46
 syms s t
 % vertical cylinder of radius 1
 x = cos(s);
 y = sin(s);
 z = t;
 ezsurf(x,y,z, [0 2*pi -2 2])
 hold on
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47. vertical cylinder of radius 1
.
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-2
-1
0
1
2
x
x = cos(s), y = sin(s), z = t
y
z
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48. Two intersecting cylinders
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 u = linspace(0,2*pi,41);
 v = linspace(-2,2,41)
 [U,V] = meshgrid(u,v);
 % vertical cylinder of radius 1
 surf(cos(U), sin(U), V);
 hold on
 % horizontal cylinder of radius .5
 surf(.5*cos(U), V, .5*sin(U))
 hold off

49. 5/15/2019
-1
-0.5
0
0.5
1
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
y
x = cos(s), y = sin(s), z = t
x
z
49

50. Torus
50
 a = .5; r = 2;
 u = linspace(0, 2*pi, 41); v = u;
 [U,V] = meshgrid(u,v);
 X = cos(U).*(r + a*cos(V));
 Y = sin(U).*(r + a*cos(V));
 Z = a*sin(V);
 surf(X,Y,Z);
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51. .
 Torus
 END!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
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