Matlab tutorial 2

MATLAB TUTORIAL 2
By Norhan Abdalla

Outline:
■ Built In Functions
■ Complex Numbers
■ Matrices Algebra
■ Plotting

Math Functions
■ abs(x) Finds the absolute value of x
■ sqrt(x)Finds the square root of x
■ nthroot(x,n)Finds the real nth root of x.
■ sign(x)Returns 0 if (x=0) ,returns 1 if (x>0) ,or returns -1 if (x<0)
■ rem(x,y)Computes the remainder of x/y
■ exp(x)Computes the value of 𝑒𝑥
■ log(x)Computes ln(x) –the natural logarithm –
■ log10(x)Computes 𝑙𝑜𝑔10(𝑥) –the common logarithm –
This
function will
not return
complex
results
The e
notation and
exp(x) are
not the
same.

Practice:
■ Define this vector x1 = [ -2 ,-1 ,0 , 1 , 2]
A. Find the absolute value of each element of the vector.
B. Find the square root of each element of the vector.
■ Use the ‘nthroot’ function to find the square root of -3 and 3.
■ Create a vector x2 from -9 to 12 with an increment of 3.
A. Find ln(x) –the natural logarithm –
B. Find Log10(x) –the common logarithm –
■ Use the sign function to determine which elements in x2 are positive.

Rounding Functions
■ round(x) Rounds x to the nearest integer
■ fix(x)Rounds /truncates x to the nearest integer toward zero
■ floor(x) Rounds x to the nearest integer toward negative infinity
■ ceil(x)Rounds x to the nearest integer toward positive infinity

Discrete Mathematics
■ factor(x)Finds the prime factors of x
■ gcd(x,y)Finds the greatest common divisor of x and y
■ lcm(x,y)Finds the least common multiple of x and y
■ rats(x)Represents x as a fraction
■ factorial(x)Finds the value of x!

Discrete Mathematics
■ nchoosek(n,k)Finds the number of possible combinations
of k items from a group of n items.
❖ Hint:
• This instruction is used for combinations ( 𝐶(𝑛,𝑟) =
𝑛!
𝑟! 𝑛−𝑟 !
)
• For permutations (𝑃(𝑛,𝑟) =
𝑛!
𝑛−𝑟 !
)
you can use factorial(n) /factorial(n-r)
■ primes(x) Finds all the prime numbers less than x
■ isprime(x)Checks to see if x is a prime number.

Practice:
■ Factor the number 322.
■ Find the greatest common divisor of 322 and 6 .
■ Is 322 a prime number?
■ Find the number of possible groups containing 3 people from a group of
29,when order does not matter.

Trigonometric Functions
■ sin(x)Finds the sin of x when x is in Radians
■ asin(x)Finds the arcsin/ inverse of sine of x when x =[-1,1]
• The function returns an angle in radians Ɵ=[−
ᴨ
2
,
ᴨ
2
]
■ sinh(x)Finds the hyperbolic sine of x when x is in radians
■ asinh(x)Finds the inverse of the hyperbolic sin of x
■ sind(x)Finds the sin of when x is expressed in degrees
■ asind(x)Finds the inverse sin of x when x=[-1,1]
• The function returns the angle in degrees
You can use
similar
functions in
cos(x), tan(x)

Practice:
■ sin(2Ɵ) for Ɵ = 3ᴨ
■ Cos(Ɵ) for 0≤ 2ᴨ ; let Ɵ change in steps of 0.2ᴨ
■ 𝑠𝑖𝑛−1
(1)
■ 𝑐𝑜𝑠−1
𝑥 for -1≤ x ≤ 1; let Ɵ change in steps of 0.2
■ Find the cosine of 45°
■ Find the angle whose sine is 0.5. Is your answer in degrees or radians?
■ Find the cosecant of 60.
Calculate the following :

Data Analysis Functions
Maxima & Minima
■ max(x) Finds the largest value in each
column.
❖ Hint:
• Using the transpose extracts the maximum of
each row, max(x’)
■ [a,b] = max(x)Assigns the maximum
elements in each column to a
and assigns their locations to b
■ max(x,y)Creates a x*y matrix, then
compares between the two vectors
You can use
similar
functions for
minima.
Use “min”

Practice:
• Create the following matrix:
x= [4,90,85,75;2,55,65,75;3,78,82,79;1,84,92,93]
■ What is the maximum value in each column?
■ In which row does that maximum occur?
■ What is the maximum value in each row?
■ In which column does the maximum occur?
■ What is the maximum value in the entire table?

Averages
■ mean(x) Finds the mean value of each
column of the matrix.
❖ Hint:
• mean(mean(x))Finds the mean value of
matrix x
■ median(x)Finds the median value in each
column
■ mode(x)Finds the value that occurs most
often in the array

Practice:
• Define the following matrix:
x= [4,90,85,75;2,55,65,75;3,78,82,79;1,84,92,93]
■ What is the mean value in each column?
■ What is the median value in each column?
■ What is the mean value in each row?
■ What is the median value in each row?
■ What is returned when you call “ mode” function?
■ What is the mean for the entire matrix?

Sums & Products
■ sum(x)Sums the elements in each
column
■ prod(x) Computes the product of each
column
■ cumsum(x)Computes the cumulative
sums of the elements
■ cumprod(x) Computes the cumulative
products of the elements
❖ Hint:
• sum(x(:,1)) Finds the sum of all the
rows in the first column of matrix
• sum(x(1,:)) Finds the sum of all the
columns in the first row of matrix x

Sorting
■ sort(x)Sorts the matrix column wise in
ascending order
■ sort(x,’descend’)Sorts the elements in
each column in descending order
■ sortrows(x) Sorts the matrix row wise
in ascending order on the bases of
column 1( keeps the row intact)
■ sortrows(x,n)Sorts the rows based on
the nth column, but maintains the rows
unchanged.
• If n is negative , the values are sorted in
descending order.
• If n is not specified , the default column
is 1

Size
■ size(x)Determines the number of rows
and columns in matrix x
■ [a,b]= size(x) Determines the number
of rows and columns in matrix x then
assigns rows to a and columns to b
■ length(x)Determines the number of the
largest dimension in a matrix
■ numel(x)Determines the total number
of elements in a matrix x

Practice:
• define the following matrix:
x=
[4,90,85,75;2,55,65,75;3,78,82,79;1,84,92,93]
■ Use the size function to determine the
number of rows and columns in this matrix.
■ Use the sort function to sort each column in
ascending order.
■ Use the sort function to sort each column in
descending order.
■ Use the sortrows function to sort the matrix
so that the first column is in ascending order.
■ Use the sortrows function to sort the matrix
from the 4th exercise in descending order;
based on the third column.

Statistical Functions
■ std(x) Computes the standard deviation of the
values in a vector x.
■ var(x) Calculates the variance of x.
• For 2d matrix use the operator (:)
■ std(x(:)) Computes the standard deviation of all
the values in a matrix x
■ var(x(:)) Calculates the variance of all the
values in a matrix x

Practice:
• Define the following matrix:
x= [4,90,85,75;2,55,65,75;3,78,82,79;1,84,92,93]
■ Find the standard deviation for each column.
■ Find the variance for each column.
■ Calculate the square root of the variance you
found for each column.

Special Functions
■ i/j Means imaginary number
■ Inf Means infinity , which occurs during a
calculational overflow or when a number is
divided by zero
■ NaN Means not a number, occurs when a
calculation is undefined
■ clock Returns current time
■ date Returns current date

Complex Numbers
■ abs(x)Computes the absolute value of a
complex
■ angle(x)Computes the angle from the
horizontal in radius
■ complex(x,y) Generates a complex number
with a real component x and an imaginary
component y
■ conj(x)Generates the complex conjugate of a
complex number
■ imag(A) Extracts the imaginary component
of A
■ real(A) Extracts the real component of A
imaginary
real

Practice:
■ Create the following complex numbers:
A= 1+j
B=2-3j
C= 8+2j
■ Create a vector D of complex numbers whose real components are 2,4
and 6 and whose imaginary components are -3 , 8 and -16.
■ Find the magnitude (absolute value) of each of the vectors you created
above.
■ Use the transpose operator to find the complex conjugate of vector D.

Dot & Cross Product:
■ Dot Product : vector * vector = scalar
■ Cross Product: vector * vector= vector
• Remember:
A* B ≠ B* A
Practice:
• Define the following vectors:
• Ԧ
𝐴 = [ 1, 2, 3, 4 ] , 𝐵 = [ 12, 20 ,15, 7 ]
• Find the dot product of Ԧ
𝐴 .𝐵
A & B must
have the
same
number of
elements

Matrix Inverse & Determinant
■ What is an inverse of a matrix?
• It is the matrix when you multiply it by the original
matrix you get an identity matrix
• Identity matrix : is a matrix consists of ones down the
main diagonal and zeros in all the other locations
■ In order for a matrix to have an inverse , it has to be a
square matrix (m*m)
■ When a matrix has an inverse ,it is called an invertible
matrix
■ When a matrix has no inverse ,it is called a singular
matrix
■ To get the inverse of a matrix ,the determinant should
not be 0
■ If the determinant of a matrix is 0, the matrix has no
inverse

Special Matrices
■ diag(A) Extracts the diagonal of
a two dimensional matrix A
■ fliplr(A) Flips a matrix into its
mirror image
■ flipud(A) Flips a matrix
vertically
■ magic matrix : is a matrix with
unusual properties.
• The sum of columns is the same as
the sum of rows and also the sum
of diagonals.
• The sum of the diagonal line is
equal to the sum of the diagonal
line after flipping it.
• Here is the
proof
• Try : diag(A, 1)

Special Matrices
■ Identity matrix:
• eye (m), eye(m,n)
NOTE:
• Do not name an identity matrix ‘i’
,because ‘i’ will no longer represent
−1 in any statements that follow
■ all  Determines if all the
elements are non zero
■ any  Determines if any element
is non zero
■ ones(m,n,z) Creates a 3 D matrix
with m rows, n columns , z pages

Special Matrices
■ expm(A) Returns matrix A’s
exponential
■ sqrtm(A) Returns the square root
of matrix A
■ eig(A) Returns eigenvalues and
eigenvectors of matrix A

Basic Plotting
■ Try these commands:
• x= [ 0:2:18];
• y = exp(x)-3;
• plot(x,y)
• title(‘ lab experiment ‘)
• xlabel(‘ x axis ’)
• ylabel(‘ y axis ’)
• grid on
create a graph first ,
then add title and
axis labels.

Basic Plotting
■ plot Creates an x-y plot
■ title(‘TEXT’) Adds title to a plot
■ xlabel(‘TEXT’)  Adds a label to
the x axis
■ ylabel (‘TEXT’) Adds a label to
the y axis
■ grid Adds a grid to the graph
■ pause  Pauses the execution of the
program ,allowing the user to view
the graph
■ figure Determines which figure
will be used for the current plot
■ hold Freezes the current plot , so
that an additional plot can be
overlaid.

Multiple plots
■ To plot two functions on
the same window:
■ To use new window to plot a
new graph :
• Use figure(#)

Plotting a Matrix
■ Consider the following code:
■ If the plot contains a matrix ,
MATLAB does a separate line
for each column, and the x-axis
becomes the row vector

Subplots
■ The ‘subplot’ command allows
you to subdivide the graphing
window into a grid of m rows
and n columns.
• subplot(m,n,p)
No. of rows
No. of columns
location

Practice:
■ Subdivide a figure window into two rows and one column.
■ In the top window ,plot y =tan(x) for
-1.5 ≤ x ≤ 1.5 (use an increment of 1).
■ Add a title and axis labels to your graph.
■ In the bottom window ,plot y =sinh(x) for the same range.
■ Add a title and labels to your graph.
■ Try this exercise again ,but divide the figure window vertically instead of
horizontally.

Other Functions
■ To generate a square wave :
• Use square ( 𝜔 *t , rho)
■ To generate a triangular wave :
• Use sawtooth(𝜔 *t , rho)
■ Polar Plots:
• Use polar(theta, r)

Practice:
*Hint : Use the command: polar(theta, r)
■ Declare theta from 0 to 2pi ,in steps of 0.01*pi.
1. Define an array r= 5*cos(4*theta), then plot it in polar form.
2. Define an array r= 4*cos(6*theta), then plot it in polar form.
3. Define an array r= 5 –5*sin(theta), then plot it in polar form.
4. Define an array r= sqrt(5^2*cos(2*theta)), then plot it in polar form.
■ Declare theta = pi/2:4/5*pi:4.5pi
• Define an array r which is a six-member array of ones ,then plot it in a
polar form.

Editing the Graph
■ Changing the shape of line , mark and style.
• They come in this order: ( ‘ line, point type , color’)
■ Axis scaling:
• axis([ x,min x,max y,min y,max])
Max of y-axis
Max of x-axis
Min of x-axis
Min of y-axis
Dotted line
Circled points
Black color

Line, Mark , and Color Options :
Line Type Indicator Point Type Indicator Color Indicator
solid - point . blue b
dotted : circle o green g
dash-dot -. x-mark x red r
dashed - - plus + cyan c
star * magenta m
square s yellow y
diamond d black k
triangle down v white w
triangle up ^
triangle left <
triangle right >
pentagram p
hexagram h

Editing the Graph
■ axis(v) v is the input to axis command , it
must be 4 element vector
■ axis equal  Forces the scaling on the x and y
axis to be the same
■ legend(‘ string1 , ‘string2’) Allows you to add
legends to your graph
■ text(x_coordinate , y_coordinate , ‘TEXT’)
Adds a text at point (x,y)
■ gtext(‘ string’) Similar to the ‘text’ command ,
but the location is determined by the user by
clicking in the figure window manually.

Pie and Bar Graphs
■ Try the following:

Pie and Bar Graphs
■ bar(x) When x is a vector , ‘bar’ generates a vertical bar graph and
when x is 2d matrix ,bar groups the data by row
■ barh(x) Similar to bar , but generates horizontal bars
■ bar3(x) Generates a 3d bar chart
■ barh3(x) Generates a 3d horizontal bar chart
■ pie(x) Generates a pie chart. Each element of the matrix is
represented as a slice of a pie
■ pie3(x) Generates a 3d pie chart.

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