Tutorials junk

Plotting

2014-07-15T16:35:00.000-07:00

>> This is a little bit advanced topic, In this lecture we will learn about plotting in MATLAB. It is very easy. For plotting we just use simple plot function 'plot(x,y)'. Where x the range of values for which the function is to be plotted and y is the function. Example:

x = [0:0.1:52];
y = 0.4*sqrt(1.8*x);

plot(x,y)

When you hit enter a new window is pop up which shows the graph as shown in pic below. Click the image.

See very simple.

Nomenclature for a typical xy plot:

MATLAB allows you to add tittle, labels along the x-axis and y-axis etc to make your graph more descriptive. Some commands is shown in figure.

Grid Command:

The grid command displays grid lines (like mesh) at the tick marks corresponding to the tick labels. Type grid on to add grid lines. Type grid off to stop plotting grid lines.

Axis Command:

You can use the axis command to override the MATLAB selections for the axis limits. The basic syntax is axis( [ xmin xmax ymin ymax]). you can set x-axis and y-axis range from this command. This command sets the scaling for the x- and y-axes to the minimum and maximum values indicated. Note that, unlike an array, this command does not use commas to separate the values.

Title and label commands:

The title command allows you to put a title on the graph. And xlabel and ylabel commands generate labels along x-axis and y-axis.

Example:

click image to see example

Subplot command:

Subplot command is used to obtain several smaller “subplots” in the same figure. The syntax is subplot(m,n,p). Here m is number of rows, n is number of columns and p is the number which tell where to put graph. This command divides the Figure window into an array of rectangular panes with m rows and n columns. For example, subplot(3,2,5) creates an array of six panes, three panes deep and two panes across, and directs the next plot to appear in the fifth pane (in the bottom-left corner).

Example:

Here subplot(1,2,1) means it generate an array of first row and second column and placed x and y graph to first row and first column. Now for subplot(1,2,2) means it generate an array of first row and second column and placed x and z graph to first row and second column. Click the image to view.

Note: In this example the command abs() is used. Its purpose is to take the absolute value

Plotting Polynomials with the polyval Function:

polyval is the built in command for graphing the polynomial function.

Example:

Data Markers and Line Types:

To plot y versus x with a solid line and u versus v with a dashed line, type plot(x,y,u,v,’--’), where the symbols ’--’ represent a dashed line.

To plot y versus x with asterisks (*) connected with a dotted line, you must plot the data twice by typing plot(x,y,’*’,x,y,’:’).

To plot y versus x with green asterisks (*) connected with a red dashed line, you must plot the data twice by typing plot(x,y,’g*’,x,y,’r--’).

Specifiers for data markers, line types, and colors:

Some of data marker, line types and colors shown in figure below. Click to see.

Labeling Curves and Data:
When more than one curve or data set is plotted on a graph, we must distinguish between them. If we use different data symbols or line types, then we must either provide a legend. To create a legend, use legend command. The basic form of this legend is
>> legend(‘string1’, ‘string2’)
% string1 and string2 are text strings of your choice.

The legend command automatically obtains from the plot the line type used for each data set and displays a sample of this line type in the legend box next to the string you selected.

Example:

A example is shown in figure below click to see.

Plot the Complex Numbers:

With only one argument, say plot(y) the plot function will plot the values in the vector y versus their indices 1,2,3...... and so on. If y is complex plot(y) plots the imaginary parts versus the real parts. Thus plot yin this case is equivalent to plot(real(y), imag(y)).
This situation is the only time when the plot function handles the imaginary parts in all other variants of the plot function, it ignore the imaginary parts.

Example:

Click to see image.

plot and fplot command:

MATLAB has a smart command for plotting functions. fplot analyzes the function to be plotted and decides how many plotting points to use so that the plot will show all the features of the function.

Syntax:

>> fplot (‘string’, [xmin xmax])
>> fplot (‘string’, [xmin xmax ymin ymax])

Example:

In this example we use this function cos(tan(x))-tan(sin(x)) to plot by using fplot and plot command.

With Plot:

x=[0:0.1:10];
f = cos(tan(x))-tan(sin(x))
plot(x,f)

With fplot:

x=[0:0.1:10];
f = 'cos(tan(x))-tan(sin(x))'
fplot(f, [1 2])

Hold and figure command:

Hold command is used to create plot that needs two or more plots. Means we want to plot 2 or more graphs in the same window

Example:

x= [0:0.1:10];
y1=sin(x);
y2=cos(x);
plot(x,y1,'go')
hold
plot(x,y2,'r*', x,y2, ':')

legend('Sin(x)', 'Cos(x)')

Figure command is used to draw 2 or more graphs on different window as shown in example below.

Example:

x= [0:0.1:10];
y1=sin(x);
y2=cos(x);
figure
plot(x,y1,'go')
legend('Sin(x)')
hold on
figure
plot(x,y2,'r*', x,y2, ':')

legend('Cos(x)')

text and gtext command:

Text command is used to place the string text in the figure window at a point specified by coordinates x,y.

Syntax:

>> text(x,y,’text’)

Gtext command is used to place the string text in the Figure window at a point specified by the mouse. When you click the mouse on the window the text will appear there.

Syntax:

>> gtext(‘text’)

Example:

>> x= [0:0.1:10];
y1=cos(x);
y2=sin(x);
plot(x,y1,'go')
hold
plot(x,y2,'r*', x,y2, ':')
legend('cos(x)', 'sin(x)')
text(5,0,'plot of cos and sine')
gtext('This is the text to be
pasted by mouse')

Logarithmic Plots:

Why use log scales?
Rectilinear scales cannot properly display variations over wide ranges.

A log-log plot can display wide variations in data values.

It is important to remember the following points when using log scales:

1. You cannot plot negative numbers on a log scale, because the logarithm of a negative number is not defined as a real number.

2. You cannot plot the number 0 on a log scale, because log10 0 = ln 0 = -infinity. You must choose an appropriately small number as the lower limit on the plot.

3. The tick-mark labels on a log scale are the actual values being plotted, they are not the logarithms of the numbers. For example, the range of x values in the plot in previous figure is from 10-1 = 0.1 to 102 = 100.

Syntax:
MATLAB has three commands for generating plots having log scales. The appropriate command depends on which axis must have a log scale.

1.Use the loglog(x,y) command to have both scales logarithmic.

2. Use the semilogx(x,y) command to have the x scale logarithmic and the y scale rectilinear.

3. Use the semilogy(x,y) command to have the y scale logarithmic and the x scale rectilinear.

Example:

x=0:0.5:50;
y=5*x.^2;
subplot(2,2,1), plot(x,y)
title('Polynomial - linear/linear')
ylabel('y'),grid
subplot(2,2,2), semilogx(x,y)
title('Polynomial - log/linear')
ylabel('y'),grid
subplot(2,2,3), semilogy(x,y)
title('Polynomial - linear/log')
ylabel('y'),grid
subplot(2,2,4), loglog(x,y)
title('Polynomial - log/log')
ylabel('y'),grid

Saving Figures in Matlab:

To save a figure that can be opened in subsequent MATLAB sessions, save it in a figure file with the .fig file name extension. To do this, select Save from the Figure window File menu or click the Save button (the disk icon) on the toolbar. If this is the first time you are saving the file, the Save As dialog box appears. Make sure that the type is MATLAB Figure (*.fig). Specify the name you want assigned to the figure file. Click OK.

Requirements for a Correct Plot:

The following list describes the essential features of any plot:

1. Each axis must be labeled with the name of the quantity being plotted and its units! If two or more quantities having different units are plotted (such as when plotting both speed and distance versus time), indicate the units in the axis label if there is room, or in the legend or labels for each curve.

2. Each axis should have regularly spaced tick marks at convenient intervals—not too sparse, but not too dense—with a spacing that is easy to interpret and interpolate. For example, use 0.1, 0.2, and so on, rather than 0.13, 0.26, and so on.

3. If you are plotting more than one curve or data set, label each on its plot or use a legend to distinguish them.

4. If you are preparing multiple plots of a similar type or if the axes’ labels cannot convey enough information, use a title.

5. If you are plotting measured data, plot each data point with a symbol such as a circle, square, or cross (use the same symbol for every point in the same data set). If there are many data points, plot them using the dot symbol.

6. Sometimes data symbols are connected by lines to help the viewer visualize the data.

Strings

2014-04-21T09:57:00.000-07:00

>> In this tutorial we will discuss about the strings in MATLAB. Creating a character string is quite simple in MATLAB. In fact, we have used it many times. For example, you type the following in the command prompt as shown. Click on the picture to see.

MATLAB considers all variables as arrays, and strings are considered as character arrays. Let us use the whos command to check the variable created above:

Interestingly, you can use numeric conversion functions like uint8 or uint16 to convert the characters in the string to their numeric codes. The char function converts the integer vector back to characters:

Example:

Create a script file and type the following code into it:

my_string = 'Tutorial' 's Point';

str_ascii = uint8(my_string) % 8-bit ascii values

str_back_to_char= char(str_ascii)

str_16bit = uint16(my_string) % 16-bit ascii values

str_back_to_char = char(str_16bit)

When you run the file, it displays the following result. Click on the picture to see.

Rectangular Character Array:

The strings we have discussed so far are one-dimensional character arrays; however, we need to store more than that. We need to store more dimensional textual data in our program. This is achieved by creating rectangular character arrays. Simplest way of creating a rectangular character array is by concatenating two or more one-dimensional character arrays, either vertically or horizontally as required. You can combine strings vertically in either of the following ways:

>> Using the MATLAB concatenation operator [ ] and separating each row with a semicolon (;). Please note that in this method each row must contain the same number of characters. For strings with different lengths, you should pad with space characters as needed.

>> Using the char function. If the strings are different length, char pads the shorter strings with trailing blanks so that each row has the same number of characters.

Example:

>> Create a script file and type the following code into it:

doc_profile = ['Zara Ali '; ...

'Sr. Surgeon '; ...

'R N Tagore Cardiology Research Center']

doc_profile = char('Zara Ali', 'Sr. Surgeon', ...

'RN Tagore Cardiology Research Center')

>> When you run the file, it displays the following result:

doc_profile =

Zara Ali

Sr. Surgeon

R N Tagore Cardiology Research Center

doc_profile =

Zara Ali

Sr. Surgeon

RN Tagore Cardiology Research Center

>> You can combine strings horizontally in either of the following ways:

1. Using the MATLAB concatenation operator, [ ] and separating the input strings with a comma or a space. This method preserves any trailing spaces in the input arrays.

2. Using the string concatenation function, strcat. This method removes trailing spaces in the inputs

Example:

>> Create a script file and type the following code into it:

name = 'Zara Ali ';

position = 'Sr. Surgeon ';

worksAt = 'R N Tagore Cardiology Research Center';

profile = [name ', ' position ', ' worksAt]

profile = strcat(name, ', ', position, ', ', worksAt)

>> When you run the file, it displays the following result:

profile =

Zara Ali , Sr. Surgeon , R N Tagore Cardiology Research Center

profile =

Zara Ali,Sr. Surgeon,R N Tagore Cardiology Research Center

Combining Strings into a Cell Array:

From our previous discussion, it is clear that combining strings with different lengths could be a pain as all strings in the array has to be of the same length. We have used blank spaces at the end of strings to equalize their length. However, a more efficient way to combine the strings is to convert the resulting array into a cell array. MATLAB cell array can hold different sizes and types of data in an array. Cell arrays provide a more flexible way to store strings of varying length.

The cellstr function converts a character array into a cell array of strings.

Example

>> Create a script file and type the following code into it:

name = ' Zara Ali ';
position = ' Sr. Surgeon ';
worksAt = ' R N Tagore Cardiology Research Center ';
profile = char(name, position, worksAt) ;
profile = cellstr(profile) ;
disp(profile)

>> When you run the file, it displays the following result:

' Zara Ali '
' Sr. Surgeon '

' R N Tagore Cardiology Research Center '

String Functions in MATLAB

MATLAB provides numerous string functions creating, combining, parsing, comparing and manipulating strings. Following table provides brief description of the string functions in MATLAB. Click image to see.

Examples

The following examples illustrate some of the above-mentioned string functions:

FORMATTING STRINGS
>> Create a script file and type the following code into it:

A = pi*1000*ones(1,5);
sprintf(' %f \n %.2f \n %+.2f \n %12.2f \n %012.2f \n', A)

>> When you run the file, it displays the following result:

ans =
3141.592654
3141.59
+3141.59
3141.59
000003141.59

JOINING STRINGS
>> Create a script file and type the following code into it:

%cell array of strings
str_array = {'red','blue','green', 'yellow', 'orange'};

% Join strings in cell array into single string
str1 = strjoin("-", str_array)
str2 = strjoin(",", str_array)

>> When you run the file, it displays the following result:

str1 =
red blue green yellow orange
str2 =
red , blue , green , yellow , orange

FINDING AND REPLACING STRINGS
>> Create a script file and type the following code into it:

students = {'Zara Ali', 'Neha Bhatnagar', ...
'Monica Malik', 'Madhu Gautam', ...
'Madhu Sharma', 'Bhawna Sharma',...
'Nuha Ali', 'Reva Dutta', ...
'Sunaina Ali', 'Sofia Kabir'};

% The strrep function searches and replaces sub-string.
new_student = strrep(students(8), 'Reva', 'Poulomi')
% Display first names
first_names = strtok(students)

>> When you run the file, it displays the following result:

new_student =
'Poulomi Dutta'
first_names =
Columns 1 through 6
'Zara' 'Neha' 'Monica' 'Madhu' 'Madhu' 'Bhawna'
Columns 7 through 10
'Nuha' 'Reva' 'Sunaina' 'Sofia'

COMPARING STRINGS
>> Create a script file and type the following code into it:

str1 = 'This is test'
str2 = 'This is text'
if (strcmp(str1, str2))
sprintf('%s and %s are equal', str1, str2)
else
sprintf('%s and %s are not equal', str1, str2)
end

>> When you run the file, it displays the following result:

str1 =
This is test
str2 =
This is text
ans =

This is test and This is text are not equal

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Colon notation

2014-03-29T15:24:00.000-07:00

>> The colon( : ) is one of the most useful operator in MATLAB. It is used to create vectors, subscript arrays, and specify for iterations. If you want to create a row vector, containing integers from 1 to 10, you write:

If you want to specify an increment value other than one, for example:

You can use the colon operator to create a vector of indices to select rows, columns or elements of arrays.

Try this example yourself:

0:pi/8:pi

ans = .........?

The following picture describes its use for this purpose (let us we have a matrix A):

Example:

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Arrays operation

2014-03-29T15:05:00.001-07:00

1. Array Addition and Subtraction:

For example

Array subtraction is performed in a similar way. The addition shown in above equation is performed in MATLAB as follows:

2. Array multiplication:

Multiplying a matrix A by a scalar w produces a matrix whose elements are the elements of A multiplied by w. For example:

This multiplication is performed in MATLAB as follows:

Multiplication of an array by a scalar is easily defined and easily carried out. However, multiplication of two arrays is not so straightforward.

MATLAB uses two definitions of multiplication:

a. array multiplication (also called element-by-element multiplication)

b. matrix multiplication.

MATLAB has two forms of arithmetic operations on arrays. One form, called array operations, which are also called element-by-element operations. And matrix operations. Each form has its own applications.

Division and exponentiation must also be carefully defined when you are dealing with operations between two arrays.

Array or Element-by-element multiplication is defined only for arrays having the same size. The definition of the product x.*y, where x and y each have n elements, is
x.*y = [x(1)y(1), x(2)y(2), ... , x(n)y(n)]
if x and y are row vectors.

For example,

If x and y are column vectors, the result of x.*y is a column vector. For example z = (x').*(y') gives

Note that x' is a column vector with size 3 × 1 and thus does not have the same size as y, whose size is 1 × 3. Thus for the vectors x and y the operations x'.*y and y.*x' are not defined in MATLAB and will generate an error message.

The array operations are performed between the elements in corresponding locations in the arrays. For example, the array multiplication operation A.*B results in a matrix C that has the same size as A and B and has the elements ci j = ai j bi j . For example,

The built-in MATLAB functions such as sqrt(x) and exp(x) automatically operate on array arguments to produce an array result the same size as the array argument x.
Thus these functions are said to be vectorized functions.

For example, in the following session the result y has the same size as the argument x.

However, when multiplying or dividing these functions, or when raising them to a power, you must use element-by-element operations if the arguments are arrays.
For example, to compute
z = (ey sin x) cos2x,
you must type
z = exp(y).*sin(x).*(cos(x)).^2.

You will get an error message if the size of x is not the same as the size of y. The result z will have the same size as x and y.

3. Array division:

The definition of array division is similar to the definition of array multiplication except that the elements of one array are divided by the elements of the other array. Both arrays must have the same size. The symbol for array right division is ./
For example, if

x = [8, 12, 15] y = [–2, 6, 5]

then z = x./y gives

z = [8/(–2), 12/6, 15/5] = [–4, 2, 3]

4. Array exponentiation:

MATLAB enables us not only to raise arrays to powers but also to raise scalars and arrays to array powers. To perform exponentiation on an element-by-element basis, we must use the ( .^ ) symbol. For example,
if x = [3, 5, 8], then typing x.^3 produces the array [33, 53, 83] = [27, 125, 512].

We can raise a scalar to an array power. For example,
if p = [2, 4, 5], then typing 3.^p produces the array [32, 34, 35] = [9, 81, 243].

Remember that ( .^ ) is a single symbol. The dot in 3.^p is not a decimal point associated with the number 3. The following operations, with the value of p given here, are equivalent and give the correct answer:

>> 3.^p

>> 3.0.^p

>> 3..^p

>> (3).^p

>> 3.^[2,4,5]

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Arrays

2014-03-29T14:10:00.000-07:00

>> So far we almost cover all the major things to understan MATLAB. In this section we will learn about arrays. We already discussed vectors and matrices. In MATLAB, all variables of all data types are multidimensional arrays. A vector is a one-dimensional array and a matrix is a two-dimensional array. we will discuss multidimensional arrays. However, before that, let us discuss some special types of arrays.

Special Arrays in MATLAB:

In this section, we will discuss some functions that create some special arrays. For all these functions, a single argument creates a square array, double arguments create rectangular array.

>> zeros() = The zeros() function creates an array of all zeros.

Sometimes we want to initialize a matrix to have all zero elements. The zeros command creates a matrix of all zeros. Typing zeros(n) creates an n × n matrix of zeros, whereas typing zeros(m,n) creates an m × n matrix of zeros. For example:

>> ones() = The ones() function creates an array of all ones.

The syntax of the ones command is the same, except that it creates arrays filled with ones. For example:

>> eye() = The eye() function creates an identity matrix.

The functions eye(n) and eye(size(A)) create an n × n identity matrix and an identity matrix the same size as the matrix A. For example:

>> rand() = The rand() function creates an array of uniformly distributed random numbers on (0,1). For example:

>> A magic square is a square that produces the same sum, when its elements are added row-wise, column-wise or diagonally.

magic() = The magic() function creates a magic square array. It takes a singular argument that gives the size of the square. The argument must be a scalar greater than or equal to 3. For example:

Multidimensional Arrays:

An array having more than two dimensions is called a multidimensional array in MATLAB. Multidimensional arrays in MATLAB are an extension of the normal two-dimensional matrix. Generally to generate a multidimensional array, we first create a two-dimensional array and extend it. For example, let's create a two-dimensional array a.

The array a is a 3-by-3 array; we can add a third dimension to a, by providing the values like:

We can also create multidimensional arrays using the ones(), zeros() or the rand() functions. For example,

We can also use the cat() function to build multidimensional arrays. It concatenates a list of arrays along a specified dimension:

Syntax for the cat() function is: B = cat(dim, A1, A2...)
Where,

a. B is the new array created.

b. A1, A2, ... are the arrays to be concatenated.

c. dim is the dimension along which to concatenate the arrays.

for example:

Array Functions:

MATLAB provides the following functions to sort, rotate, permute, reshape, or shift array contents.

Examples:

The following examples illustrate some of the functions mentioned above.

>> Length, Dimension and Number of elements:

>> Circular Shifting the Array Elements:

Sorting Arrays:

This is used to sort the arrays for example,

Cell Array:

Cell arrays are arrays of indexed cells where each cell can store an array of a different dimension and data type.

The cell function is used for creating a cell array. Syntax for the cell function is:

C = cell(dim)
C = cell(dim1,...,dimN)
D = cell(obj)
Where,

a. C is the cell array;

b. dim is a scalar integer or vector of integers that specifies the dimensions of cell array C;

c. dim1, ... , dimN are scalar integers that specify the dimensions of C;

d. obj is One of the following: >> Java array or object >> .NET array of type System.String or System.Object

Example:

Accessing Data in Cell Arrays:

There are two ways to refer to the elements of a cell array:

a. Enclosing the indices in first bracket (), to refer to sets of cells

b. Enclosing the indices in braces {}, to refer to the data within individual cells

When you enclose the indices in first bracket, it refers to the set of cells. Cell array indices in smooth parentheses refer to sets of cells. For example:

You can also access the contents of cells by indexing with curly braces. For example:

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Matrix Operations

2014-03-29T13:06:00.000-07:00

>> In this section we will discuss the following basic and commonly used matrix operations:

1. Addition and Subtraction of Matrices

2. Division of Matrices

3. Scalar Operations of Matrices

4. Transpose of a Matrix

5. Concatenating Matrices

6. Matrix Multiplication

7. Determinant of a Matrix

8. Inverse of a Matrix

1. Addition and Subtraction of Matrices:

You can add or subtract matrices. Both the operand matrices must have the same number of rows and columns.

Example:

2. Division of Matrices

You can divide two matrices using left (\) or right (/) division operators. Both the operand matrices must have the same number of rows and columns.

Example:

3. Scalar Operations of Matrices:

When you add, subtract, multiply or divide a matrix by a number, this is called the scalar operation. Scalar operations produce a new matrix with same number of rows and columns with each element of the original matrix added to, subtracted from, multiplied by or divided by the number.

Example:

4. Transpose of a Matrix:

The transpose operation switches the rows and columns in a matrix. It is represented by a single quote( ' ).

Example:

6. Matrix Multiplication:

Consider two matrices A and B. If A is an m x n matrix and B is a n x p matrix, they could be multiplied together to produce an m x n matrix C. Matrix multiplication is possible only if the number of columns n in A is equal to the number of rows n in B.

In matrix multiplication, the elements of the rows in the first matrix are multiplied with corresponding columns in the second matrix.

Each element in the (i, j)th position, in the resulting matrix C, is the summation of the products of elements in ith row of first matrix with the corresponding element in the jth column of the second matrix.

In MATLAB, matrix multiplication is performed by using the * operator.

Example:

Matrix-Matrix Multiplication:

In the product of two matrices AB, the number of columns in A must equal the number of rows in B. The row-column multiplications form column vectors, and these column vectors form the matrix result. The product AB has the same number of rows as A and the same number of columns as B. For example,

Use the operator * to perform matrix multiplication in MATLAB.
Do it yourself

A = [6,-2;10,3;4,7];
B = [9,8;-5,12];

A*B = ?

Matrix multiplication does not have the commutative property; that is, in general, AB is not equal to BA. A simple example will demonstrate this fact:

Reversing the order of matrix multiplication is a common and easily made mistake.

7. Determinant of a Matrix:

Determinant of a matrix is calculated using the det function of MATLAB. Determinant of a matrix A is given by det(A).

Example:

8. Inverse of a Matrix:

The inverse of a matrix A is denoted by A^−1 such that the following relationship holds:

(AA)^−1 = A^−1A = 1

The inverse of a matrix does not always exist. If the determinant of the matrix is zero, then the inverse does not exist and the matrix is singular.

In MATLAB, inverse of a matrix is calculated using the inv function. Inverse of a matrix A is given by inv(A).

Example:

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Matrices

2014-03-29T12:24:00.000-07:00

>> A matrix is a two-dimensional array of numbers. Vectors are special cases of matrices having one row or one column.

>> In MATLAB, you create a matrix by entering elements in each row as comma or space delimited numbers and using semicolons to mark the end of each row. For example, let us create a 4-by-5 matrix a:

>> If the matrix is small you can type it row by row, separating the elements in a given row with spaces or commas and separating the rows with semicolons. For example, typing
A = [2,4,10;16,3,7];

Remember, spaces or commas separate elements in different columns, whereas semicolons separate elements in different rows.

>> To create matrices from Vectors suppose a = [1,3,5] and b = [7,9,11] (row vectors). Note the difference between the results given by [a b] and [a;b] in the following session:

You need not use symbols to create a new array. For example, you can type

D = [[1,3,5];[7,9,11]];

Referencing the Elements of a Matrix:

To reference an element in the mth row and nth column, of a matrix mx, we write:

mx(m, n);

For example, to refer to the element in the 2nd row and 5th column, of the matrix a, as created in the last section, we type:

To reference all the elements in the mth column we type A(:,m).

Let us create a column vector v, from the elements of the 4th row of the matrix a:

You can also select the elements in the mth through nth columns, for this we write:

a(:,m:n)

In the same way, you can create a sub-matrix taking a sub-part of a matrix.

n the same way, you can create a sub-matrix taking a sub-part of a matrix.

For example, let us create a sub-matrix sa taking the inner subpart of a:

3 4 5
4 5 6

To do this, write:

Array addressing:

The colon operator selects individual elements, rows, columns, or ''subarrays'' of arrays. Here are some examples:
v(:) = represents all the row or column elements of the vector v.

v(2:5) = represents the second through fifth elements; that is v(2), v(3), v(4), v(5).

A(:,3) = denotes all the elements in the third column of the matrix A.

A(:,2:5) = denotes all the elements in the second through fifth columns of A.

A(2:3,1:3) = denotes all the elements in the second and third rows that are also in the first through third columns.

Examples:

Try these operation yourself.
>> a= [4 5 6; 1 2 3; 7 8 9]
>> a(:)
>> a= [4 5 6; 1 2 3; 7 8 9]
>> a(2:5)
>> a(: , 3)
>> a= [4 5 6; 1 2 3; 7 8 9]
>> a(: , 2:3)
>> a(: , 2:5)

Output…….?

Deleting a Row or a Column in a Matrix:

You can delete an entire row or column of a matrix by assigning an empty set of square braces [ ] to that row or column. Basically, [ ] denotes an empty array.

For example, let us delete the fourth row of a:

Next, let us delete the fifth column of a:

Example:

In this example, let us create a 3-by-3 matrix m, then we will copy the second and third rows of this matrix twice to create a 4-by-3 matrix.

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Vectors

2014-03-29T11:37:00.000-07:00

>> In this lecture we will discuss about the vectors in MATLAB and their operation. A vector is a one-dimensional array of numbers. MATLAB allows creating two types of vectors:

1. Row vectors

2. Column vectors

1. Row vectors:

To create a row vector, separate the elements by commas or by space. For example,

2. Column vectors:

To create a column vector by using the transpose notation ( ' ). Or you can also create a column vector by separating the elements by semicolons. For example,

Referencing the Elements of a Vector:

You can reference one or more of the elements of a vector in several ways. The ith component of a vector v is referred as v(i). For example:

When you reference a vector with a colon, such as v(:), all the components of the vector are listed.

MATLAB allows you to select a range of elements from a vector.

For example, let us create a row vector rv of 9 elements, then we will reference the elements 3 to 7 by writing rv(3:7) and create a new vector named sub_rv.

>> You can create vectors by ''appending'' one vector to another. For example, to create the row vector u whose first three columns contain the values of r = [2, 4, 20] and whose fourth, fifth, and sixth columns contain the values of w = [9, -6, 3], you type u = [r, w]. The result is the vector u = [2, 4, 20, 9, -6, 3].

>> The colon operator (:) easily generates a large vector of regularly spaced elements.
Typing x = [m:q:n] creates a vector x of values with a spacing q. The first value is m. For example, typing x = [0: 2: 8] creates the vector x = [0, 2, 4, 6, 8], whereas typing x = [0: 2: 7] creates the vector x = [0, 2, 4, 6].

>> To create a row vector z consisting of the values from 5 to 8 in steps of 0.1, type z = [5: 0.1: 8]. If the increment q is omitted, it is presumed to be 1. Thus typing y = [-3: 2] produces the vector y = [-3, -2, -1, 0, 1, 2].

>> The linspace command also creates a linearly spaced row vector, but instead you specify the number of values rather than the increment. The syntax is linspace(x1, x2, n), where x1 and x2 are the lower and upper limits and n is the number of points. For example, linspace(5,8,31) is equivalent to [5:0.1:8]. If n is omitted, the spacing is 1.

>> The logspace command creates an array of logarithmically spaced elements. Its syntax is logspace(a, b, n), where n is the number of points between 10a and 10b. For example, x = logspace(-1,1,4) produces the vector x = [0.1000, 0.4642, 2.1544, 10.000]. If n is omitted, the number of points defaults to 50.

Vector Operations:

let us discuss the following vector operations:

a. Addition and Subtraction of Vectors

b. Scalar Multiplication of Vectors

c. Transpose of a Vector

d. Appending Vectors

e. Magnitude of a Vector

f. Vector Dot Product

g. Vectors with Uniformly Spaced Elements

a. Addition and Subtraction of Vectors:

You can add or subtract two vectors. Both the operand vectors must be of same type and have same number of elements.

Example:

b. Scalar Multiplication of Vectors:

When you multiply a vector by a number, this is called the scalar multiplication. Scalar multiplication produces a new vector of same type with each element of the original vector multiplied by the number.

Example:

c. Transpose of a Vector:

The transpose operation changes a column vector into a row vector and vice versa. The transpose operation is represented by a single quote(').

Example:

d. Appending Vectors:

MATLAB allows you to append vectors together to create new vectors. If you have two row vectors r1 and r2 with n and m number of elements, to create a row vector r of n plus m elements, by appending these vectors, you write:

r = [r1, r2]

You can also create a matrix r by appending these two vectors, the vector r2, will be the second row of the matrix:

r = [r1; r2]

However, to do this, both the vectors should have same number of elements.

Similarly, you can append two column vectors c1 and c2 with n and m number of elements. To create a column vector c of n plus m elements, by appending these vectors, you write:

c = [c1; c2]

You can also create a matrix c by appending these two vectors; the vector c2 will be the second column of the matrix:

c = [c1, c2]

However, to do this, both the vectors should have same number of elements.

Example:

Create a script file with the following code:

r1 = [ 1 2 3 4 ];
r2 = [5 6 7 8 ];
r = [r1,r2]
rMat = [r1;r2]

c1 = [ 1; 2; 3; 4 ];
c2 = [5; 6; 7; 8 ];
c = [c1; c2]

cMat = [c1,c2]

you will get

e. Magnitude of a Vector:

Magnitude of a vector v with elements v1, v2, v3, …, vn, is given by the equation:

|v| = √ (v1^2 + v2^2 + v3^2 + … + vn^2)

You need to take the following steps to calculate the magnitude of a vector:

Take the product of the vector with itself, using array multiplication (.*). This produces a vector sv, whose elements are squares of the elements of vector v.

sv = v.*v;

Use the sum function to get the sum of squares of elements of vector v. This is also called the dot product of vector v.

dp= sum(sv);

Use the sqrt function to get the square root of the sum which is also the magnitude of the vector v.

mag = sqrt(s);

Example:

e. Magnitude of a Vector:

Dot product of two vectors a = (a1, a2, …, an) and b = (b1, b2, …, bn) is given by:

a.b = ∑(ai.bi)

Dot product of two vectors a and b is calculated using the dot function.

dot(a, b);

Example:

g. Vectors with Uniformly Spaced Elements:

MATLAB allows you to create a vector with uniformly spaced elements. To create a vector v with the first element f, last element l, and the difference between elements is any real number n, we write:

v = [f : n : l]

Example:

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Loops

2014-03-29T10:15:00.000-07:00

>> In this lecture we will study about the loops. There may be a situation when you need to execute a block of code several number of times. In general, statements are executed sequentially. The first statement in a function is executed first, followed by the second, and so on. Programming languages provide various control structures that allow for more complicated execution paths.

A loop statement allows us to execute a statement or group of statements multiple times and following is the general form of a loop statement in most of the programming languages.

MATLAB provides following types of loops to handle looping requirements.

1. While loop.

2. For loop.

3. Nested loop.

1. While loop:

An expression is true when the result is nonempty and contains all nonzero elements (logical or real numeric). Otherwise, the expression is false. The while loop repeatedly executes statements until the condition is true.

Syntax:

The syntax of a while loop in MATLAB is:

while <expression>

<statements>

end

Example:

2. For loop:

A for loop is a repetition control structure that allows you to efficiently write a loop that needs to execute a specific number of times.

Syntax:

The syntax of a for loop in MATLAB is:

for index = values

<program statements>
...
end

Examples:

3. Nested loop:

MATLAB allows to use one loop inside another loop. Following section shows few examples to illustrate the concept.

Syntax:

The syntax for a nested for loop statement in MATLAB is as follows:

for m = 1:j

<statements>;

end

end

The syntax for a nested while loop statement in MATLAB is as follows:

while <expression1>

while <expression2>

<statements>

end

end

Example:

for i=2:100

for j=2:100

if(~mod(i,j))

break; % if factor found, not prime

end

end

if(j > (i/j))

fprintf('%d is prime\n', i);

end

end

the solution will be

Loop Control Statements:

Loop control statements change execution from its normal sequence. When execution leaves a scope, all automatic objects that were created in that scope are destroyed.

MATLAB supports the following control statements.

a. Break statement.

b. Continue statement.

a. Break statement:

The break statement terminates execution of for or while loop. Statements in the loop that appear after the break statement are not executed.

In nested loops, break exits only from the loop in which it occurs. Control passes to the statement following the end of that loop.

Example:

b. Continue statement:

The continue statement is used for passing control to next iteration of for or while loop. The continue statement in MATLAB works somewhat like the break statement. Instead of forcing termination, however, 'continue' forces the next iteration of the loop to take place, skipping any code in between.

Example:

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Decisions

2014-03-29T05:19:00.000-07:00

>> Decision making structures require that the programmer specify one or more conditions to be evaluated by the program, along with a statement or statements to be executed if the condition is determined to be true, and optionally, if the condition is determined to be false other statements to be executed.

MATLAB provides following types of decision making statements.

1. if ... end statement.

2. if...else...end statement.

3. If... elseif...elseif...else...end statements.

4. nested if statements.

5. switch statement.

6. nested switch statements

1. if ... end statement.

An if ... end statement consists of an if statement and a boolean expression followed by one or more statements. It is delimited by the end statement.

Syntax:

The syntax of an if statement in MATLAB is given below. If the expression evaluates to true, then the block of code inside the if statement will be executed. If the expression evaluates to false, then the first set of code after the end statement will be executed.

if <expression>

% statement(s) will execute if the boolean expression is true

<statements>

end

Example:

2. if...else...end statement.

An if statement can be followed by an optional else statement, which executes when the expression is false. If the boolean expression evaluates to true, then the if block of code will be executed, otherwise else block of code will be executed.

Syntax:

The syntax of an if...else statement in MATLAB is:

if <expression>

% statement(s) will execute if the boolean expression is true

<statement(s)>

else

<statement(s)>

% statement(s) will execute if the boolean expression is false

end

Example:

3. If... elseif...elseif...else...end statements.

An if statement can be followed by an (or more) optional elseif... and an else statement, which is very useful to test various condition.

When using if... elseif...else statements, there are few points to keep in mind:

a. An if can have zero or one else's and it must come after any elseif's.

b. An if can have zero to many elseif's and they must come before the else.

c. Once an else if succeeds, none of the remaining elseif's or else's will be tested.

Syntax:

if <expression 1>

% Executes when the expression 1 is true

<statement(s)>

elseif <expression 2>

% Executes when the boolean expression 2 is true

<statement(s)>

Elseif <expression 3>

% Executes when the boolean expression 3 is true

<statement(s)>

else

% executes when the none of the above condition is true

<statement(s)>

end

Example:

4. nested if statements:

It is always legal in MATLAB to nest if-else statements which means you can use one if or elseif statement inside another if or elseif statement(s).

Syntax:

The syntax for a nested if statement is as follows:

if <expression 1>

% Executes when the boolean expression 1 is true

if <expression 2>

% Executes when the boolean expression 2 is true

end

end

Example:

5. switch statement:

A switch block conditionally executes one set of statements from several choices. Each choice is covered by a case statement.

>> An evaluated switch_expression is a scalar or string.

>> An evaluated case_expression is a scalar, a string or a cell array of scalars or strings.

The switch block tests each case until one of the cases is true. A case is true when:

>> For numbers, eq(case_expression,switch_expression).

>> For strings, strcmp(case_expression,switch_expression).

>> For objects that support the eq function, eq(case_expression,switch_expression).

>> For a cell array case_expression, at least one of the elements of the cell array matches switch_expression, as defined above for numbers, strings and objects.

When a case is true, MATLAB executes the corresponding statements and then exits the switch block.

The otherwise block is optional and executes only when no case is true.

Syntax:

The syntax of switch statement in MATLAB is:

switch <switch_expression>

case <case_expression>

<statements>

case <case_expression>

<statements>
...
...

otherwise

<statements>

end

Example:

6. nested switch statements:

It is possible to have a switch as part of the statement sequence of an outer switch. Even if the case constants of the inner and outer switch contain common values, no conflicts will arise.

Syntax:

The syntax for a nested switch statement is as follows:

switch(ch1)

case 'A'

fprintf('This A is part of outer switch');

switch(ch2)

case 'A'

fprintf('This A is part of inner switch' );

case 'B'

fprintf('This B is part of inner switch' );

end

case 'B'

fprintf('This B is part of outer switch' );

end

Example:

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Data types

2014-03-29T03:56:00.000-07:00

>> MATLAB does not require any type declaration or dimension statements. Whenever MATLAB encounters a new variable name, it creates the variable and allocates appropriate memory space. If the variable already exists, then MATLAB replaces the original content with new content and allocates new storage space, where necessary. For example,
Total = 42
The above statement creates a 1-by-1 matrix named 'Total' and stores the value 42 in it.

Data Types Available in MATLAB:

MATLAB provides 15 fundamental data types. Every data type stores data that is in the form of a matrix or array. The size of this matrix or array is a minimum of 0-by-0 and this can grow up to a matrix or array of any size.

The following table shows the most commonly used data types in MATLAB:

int8	8-bit signed integer
uint8	8-bit unsigned integer
int16	16-bit signed integer
uint16	16-bit unsigned integer
int32	32-bit signed integer
uint32	32-bit unsigned integer
int64	64-bit signed integer
uint64	64-bit unsigned integer
single	single precision numerical data
double	double precision numerical data
logical	logical values of 1 or 0, represent true and false respectively
char	character data (strings are stored as vector of characters)
cell array	array of indexed cells, each capable of storing an array of a different dimension and data type
structure	C-like structures, each structure having named fields capable of storing an array of a different dimension and data type
user classes	objects constructed from a user-defined class
java classes	objects constructed from a Java class
function handle	pointer to a function

Example:

Write the following code in script file

str = 'Hello World!'
n = 2345
d = double(n)
un = uint32(789.50)
rn = 5678.92347

c = int32(rn)

you will get

Data Type Conversion:

MATLAB provides various functions for converting from one data type to another. The following table shows the data type conversion functions:

Determination of Data Types:

MATLAB provides various functions for identifying data type of a variable. Following table provides the functions for determining the data type of a variable:

Example:

Create a script file with the following code and check the result.

x = 3
isinteger(x)
isfloat(x)
isvector(x)
isscalar(x)
isnumeric(x)

x = 23.54
isinteger(x)
isfloat(x)
isvector(x)
isscalar(x)
isnumeric(x)

x = [1 2 3]
isinteger(x)
isfloat(x)
isvector(x)
isscalar(x)

x = 'Hello'
isinteger(x)
isfloat(x)
isvector(x)
isscalar(x)

isnumeric(x)

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Operators

2014-03-28T16:02:00.001-07:00

>> In this section we will tell you about the MATLAB operators. An operator is a symbol that tells the compiler to perform specific mathematical or logical manipulations. MATLAB is designed to operate primarily on whole matrices and arrays. Therefore, operators in MATLAB work both on scalar and non-scalar data.

MATLAB allows the following types of elementary operations:

1. Arithmetic Operators

2. Relational Operators

3. Logical Operators

4. Bitwise Operations

5. Set Operations

1. Arithmetic Operators

MATLAB allows two different types of arithmetic operations:

a. Matrix arithmetic operations

b. Array arithmetic operations

Matrix arithmetic operations are same as defined in linear algebra. Array operations are executed element by element, both on one dimensional and multi-dimensional array.

The matrix operators and arrays operators are differentiated by the period (.) symbol. However, as the addition and subtraction operation is same for matrices and arrays, the operator is same for both cases.

The following table gives brief description of the operators:

+	Addition or unary plus. A+B adds A and B. A and B must have the same size, unless one is a scalar. A scalar can be added to a matrix of any size.
-	Subtraction or unary minus. A-B subtracts B from A. A and B must have the same size, unless one is a scalar. A scalar can be subtracted from a matrix of any size.
*	Matrix multiplication. C = A*B is the linear algebraic product of the matrices A and B. For nonscalar A and B, the number of columns of A must equal the number of rows of B. A scalar can multiply a matrix of any size.
.*	Array multiplication. A.*B is the element-by-element product of the arrays A and B. A and B must have the same size, unless one of them is a scalar.
^	Matrix power. X^p is X to the power p, if p is a scalar. If p is an integer, the power is computed by repeated squaring. If the integer is negative, X is inverted first. For other values of p, the calculation involves eigenvalues and eigenvectors, such that if [V,D] = eig(X), then X^p = V*D.^p/V.
.^	Array power. A.^B is the matrix with elements A(i,j) to the B(i,j) power. A and B must have the same size, unless one of them is a scalar.
\	Backslash or matrix left division. If A is a square matrix, A\B is roughly the same as inv(A)*B, except it is computed in a different way. If A is an n-by-n matrix and B is a column vector with n components, or a matrix with several such columns, then X = A\B is the solution to the equation AX = B. A warning message is displayed if A is badly scaled or nearly singular.
/	Slash or matrix right division. B/A is roughly the same as B*inv(A). More precisely, B/A = (A'\B')'.
.\	Array left division. A.\B is the matrix with elements B(i,j)/A(i,j). A and B must have the same size, unless one of them is a scalar.
./	Array right division. A./B is the matrix with elements A(i,j)/B(i,j). A and B must have the same size, unless one of them is a scalar.
:	Generates regularly spaced elements and represents an entire row or column.
( )	Encloses function arguments and array indices; overrides precedence.

'	Matrix transpose. A' is the linear algebraic transpose of A. For complex matrices, this is the complex conjugate transpose.

.'	Array transpose. A.' is the array transpose of A. For complex matrices, this does not involve conjugation.

Example:

If you enter the following operation in MATLAB

a = 10;
b = 20;
c = a + b
d = a - b
e = a * b
f = a / b
g = a \ b
x = 7;
y = 3;

z = x ^ y

you will get

Apart from the above-mentioned arithmetic operators, MATLAB provides the following commands/functions used for similar purpose. There are a variety of other functions available in MATLAB that are not so commonly used, but which will definitely be useful:

round(x)	Rounds a number to the nearest integer
ceil(x)	Rounds a number up to the nearest integer
floor(x)	Rounds a number down to the nearest integer
fix(x)	Rounds a number to the nearest integer towards zero
rem(x,y)	The remainder left after division
mod(x,y)	The signed remainder left after division
abs(x)	The absolute value of x
sign(x)	The sign of x
factor(x)	The prime factors of x
uplus(a)	Unary plus; increments by the amount a
plus (a,b)	Plus; returns a + b
uminus(a)	Unary minus; decrements by the amount a
minus(a, b)	Minus; returns a - b
times(a, b)	Array multiply; returns a.*b
mtimes(a, b)	Matrix multiplication; returns a* b
rdivide(a, b)	Right array division; returns a ./ b
ldivide(a, b)	Left array division; returns a.\ b
mrdivide(A,B)	Solve systems of linear equations xA = B for x
mldivide(A,B)	Solve systems of linear equations Ax = B for x
power(a, b)	Array power; returns a.^b
mpower(a, b)	Matrix power; returns a ^ b
diff(X,n)	Applies diff recursively n times, resulting in the nth difference.
idivide(a, b, 'round')	Fractional quotients are rounded to the nearest integers.
sum(A,dim)	Sums along the dimension of A specified by scalar dim.
divide(a, b) idivide(a, b,'fix')	Integer division with rounding option; is the same as a./b except that fractional quotients are rounded toward zero to the nearest integers.

2. Relation operation:

Relational operators can also work on both scalar and non-scalar data. Relational operators for arrays perform element-by-element comparisons between two arrays and return a logical array of the same size, with elements set to logical 1 (true) where the relation is true and elements set to logical 0 (false) where it is not.

The following are the relational operators:

< Less than
<= Less than or equal to
> Greater than
>= Greater than or equal to
== Equal to
~= Not equal to

Examples:

There are a variety of other functions available in MATLAB which is given below:

eq(a, b)	Tests whether a is equal to b
ge(a, b)	Tests whether a is greater than or equal to b
gt(a, b)	Tests whether a is greater than b
le(a, b)	Tests whether a is less than or equal to b
lt(a, b)	Tests whether a is less than b
ne(a, b)	Tests whether a is not equal to b
isequal	Tests arrays for equality
isequaln	Tests arrays for equality, treating NaN values as equal

Example:

Create a script file and type the following and check it:

% comparing two values
a = 100;
b = 200;
if (ge(a,b))
max = a
else
max = b
end

% comparing two different values
a = 340;
b = 520;
if (le(a, b))
disp(' a is either less than or equal to b')
else
disp(' a is greater than b')
end

3. Logical operations:

MATLAB offers two types of logical operators and functions:

Element-wise - these operators operate on corresponding elements of logical arrays.

Short-circuit - these operators operate on scalar, logical expressions.

Element-wise logical operators operate element-by-element on logical arrays. The symbols &, |, and ~ are the logical array operators AND, OR, and NOT respectively.

Short-circuit logical operators allow short-circuiting on logical operations. The symbols && and || are the logical short-circuit operators AND and OR respectively.

Example:

Functions for Logical Operations:

Below are some function for logical operations.

and(A, B)	Finds logical AND of array or scalar inputs; performs a logical AND of all input arrays A, B, etc. and returns an array containing elements set to either logical 1 (true) or logical 0 (false). An element of the output array is set to 1 if all input arrays contain a nonzero element at that same array location. Otherwise, that element is set to 0.
not(A)	Finds logical NOT of array or scalar input; performs a logical NOT of input array A and returns an array containing elements set to either logical 1 (true) or logical 0 (false). An element of the output array is set to 1 if the input array contains a zero value element at that same array location. Otherwise, that element is set to 0.
or(A, B)	Finds logical OR of array or scalar inputs; performs a logical OR of all input arrays A, B, etc. and returns an array containing elements set to either logical 1 (true) or logical 0 (false). An element of the output array is set to 1 if any input arrays contain a nonzero element at that same array location. Otherwise, that element is set to 0.
xor(A, B)	Logical exclusive-OR; performs an exclusive OR operation on the corresponding elements of arrays A and B. The resulting element C(i,j,...) is logical true (1) if A(i,j,...) or B(i,j,...), but not both, is nonzero.
all(A)	Determine if all array elements of array A are nonzero or true. If A is a vector, all(A) returns logical 1 (true) if all the elements are nonzero and returns logical 0 (false) if one or more elements are zero. If A is a nonempty matrix, all(A) treats the columns of A as vectors, returning a row vector of logical 1's and 0's. If A is an empty 0-by-0 matrix, all(A) returns logical 1 (true). If A is a multidimensional array, all(A) acts along the first nonsingleton dimension and returns an array of logical values. The size of this dimension reduces to 1 while the sizes of all other dimensions remain the same.
all(A, dim)	Tests along the dimension of A specified by scalar dim.
any(A)	Determine if any array elements are nonzero; tests whether any of the elements along various dimensions of an array is a nonzero number or is logical 1 (true). The any function ignores entries that are NaN (Not a Number). If A is a vector, any(A) returns logical 1 (true) if any of the elements of A is a nonzero number or is logical 1 (true), and returns logical 0 (false) if all the elements are zero. If A is a nonempty matrix, any(A) treats the columns of A as vectors, returning a row vector of logical 1's and 0's. If A is an empty 0-by-0 matrix, any(A) returns logical 0 (false). If A is a multidimensional array, any(A) acts along the first nonsingleton dimension and returns an array of logical values. The size of this dimension reduces to 1 while the sizes of all other dimensions remain the same.
any(A,dim)	Tests along the dimension of A specified by scalar dim.
false	Logical 0 (false)
false(n)	is an n-by-n matrix of logical zeros
false(m, n)	is an m-by-n matrix of logical zeros.
false(m, n, p, ...)	is an m-by-n-by-p-by-... array of logical zeros.
false(size(A))	is an array of logical zeros that is the same size as array A.
false(...,'like',p)	is an array of logical zeros of the same data type and sparsity as the logical array p.
ind = find(X)	Find indices and values of nonzero elements; locates all nonzero elements of array X, and returns the linear indices of those elements in a vector. If X is a row vector, then the returned vector is a row vector; otherwise, it returns a column vector. If X contains no nonzero elements or is an empty array, then an empty array is returned.
nd = find(X, k) ind = find(X, k, 'first')	Returns at most the first k indices corresponding to the nonzero entries of X. k must be a positive integer, but it can be of any numeric data type.
ind = find(X, k, 'last')	returns at most the last k indices corresponding to the nonzero entries of X.
[row,col] = find(X, ...)	Returns the row and column indices of the nonzero entries in the matrix X. This syntax is especially useful when working with sparse matrices. If X is an N-dimensional array with N > 2, col contains linear indices for the columns.
[row,col,v] = find(X, ...)	Returns a column or row vector v of the nonzero entries in X, as well as row and column indices. If X is a logical expression, then v is a logical array. Output v contains the non-zero elements of the logical array obtained by evaluating the expression X.
islogical(A)	Determine if input is logical array; returns true if A is a logical array and false otherwise. It also returns true if A is an instance of a class that is derived from the logical class.
logical(A)	Convert numeric values to logical; returns an array that can be used for logical indexing or logical tests.
true	Logical 1 (true)
true(n)	is an n-by-n matrix of logical ones.
true(m, n)	is an m-by-n matrix of logical ones.
true(m, n, p, ...)	is an m-by-n-by-p-by-... array of logical ones.
true(size(A))	is an array of logical ones that is the same size as array A.
true(...,'like', p)	is an array of logical ones of the same data type and sparsity as the logical array p.

4. Bitwise operations:

Bitwise operator works on bits and performs bit-by-bit operation. The truth tables for &, |, and ^ are as follows:

p	q	p & q	p \| q	p ^ q
0	0	0	0	0
0	1	0	1	1
1	1	1	1	0
1	0	0	1	1

Assume if A = 60; and B = 13; Now in binary format they will be as follows:

A = 0011 1100

B = 0000 1101

Then the function operate as follow:

A&B = 0000 1100

A|B = 0011 1101

A^B = 0011 0001

~A = 1100 0011

MATLAB provides various functions for bit-wise operations like 'bitwise and', 'bitwise or' and 'bitwise not' operations, shift operation, etc.

The following table shows the commonly used bitwise operations:

bitand(a, b)	Bit-wise AND of integers a and b
bitcmp(a)	Bit-wise complement of a
bitget(a,pos)	Get bit at specified position pos, in the integer array a
bitor(a, b)	Bit-wise OR of integers a and b
bitset(a, pos)	Set bit at specific location pos of a
bitshift(a, k)	Returns a shifted to the left by k bits, equivalent to multiplying by 2^k. Negative values of k correspond to shifting bits right or dividing by 2\|k\| and rounding to the nearest integer towards negative infinite. Any overflow bits are truncated.
bitxor(a, b)	Bit-wise XOR of integers a and b
swapbytes	Swap byte ordering

Example:

4. Set operations:

MATLAB provides various functions for set operations, like union, intersection and testing for set membership, etc.

The following table shows some commonly used set operations:

intersect(A,B)	Set intersection of two arrays; returns the values common to both A and B. The values returned are in sorted order.
intersect(A,B,'rows')	Treats each row of A and each row of B as single entities and returns the rows common to both A and B. The rows of the returned matrix are in sorted order.
ismember(A,B)	Returns an array the same size as A, containing 1 (true) where the elements of A are found in B. Elsewhere, it returns 0 (false).
ismember(A,B,'rows')	Treats each row of A and each row of B as single entities and returns a vector containing 1 (true) where the rows of matrix A are also rows of B. Elsewhere, it returns 0 (false).
issorted(A)	Returns logical 1 (true) if the elements of A are in sorted order, and logical 0 (false) otherwise. Input A can be a vector or an N-by-1 or 1-by-N cell array of strings. A is considered to be sorted if A and the output of sort(A) are equal.
issorted(A, 'rows')	Returns logical 1 (true) if the rows of two-dimensional matrix A are in sorted order, and logical 0 (false) otherwise. Matrix A is considered to be sorted if A and the output of sortrows(A) are equal.
setdiff(A,B)	Set difference of two arrays; returns the values in A that are not in B. The values in the returned array are in sorted order.
setdiff(A,B,'rows')	Treats each row of A and each row of B as single entities and returns the rows from A that are not in B. The rows of the returned matrix are in sorted order. The 'rows' option does not support cell arrays.
setxor	Set exclusive OR of two arrays
union	Set union of two arrays
unique	Unique values in array

Example:

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Script files or M - files

2014-03-27T12:39:00.000-07:00

>> Till now we are working in MATLAB command window. In previous lecture you learn how to enter command. Now this lecture show you how to save your entire work in a script or M - file.

>> You can perform operations in MATLAB in two ways:

1. In the interactive mode, in which all commands are entered directly in the Command window.

2. By running a MATLAB program stored in script file. This type of file contains MATLAB commands, so running it is equivalent to typing all the commands—one at a time—at the Command window prompt. You can run the file by typing its name at the Command window prompt.

>> MATLAB allows writing two kinds of program files:

1. Scripts or M-file - script files or M-file are program files with .m extension. In these files, you write series of commands, which you want to execute together. Scripts do not accept inputs and do not return any outputs. They operate on data in the workspace.

2. Functions - functions files are also program files with .m extension. Functions can accept inputs and return outputs. Internal variables are local to the function.

You can use the MATLAB Editor or any other text editor to create your .m files.

Creating and Running Script File or M-file:

To open script file or M-file you have to follow the following steps:

1. Open MATLAB.

2. Click on file at left most top side.

3. Click on new and then on script or m-file

4. A new file is open this is script or m-file where you can work and save your programms. In this file type the code to be execute, save file then compile it. The answer will show on command window.

Keep in mind when using script files:

1. The name of a script file must begin with a letter, and may include digits and the underscore character, up to 31 characters.

2. Do not give a script file the same name as a variable.

3. Do not give a script file the same name as a MATLAB command or function. You can check to see if a command, function or file name already exists by using the exist command.

Debugging Script Files:

Program errors usually fall into one of the following categories.

1. Syntax errors such as omitting a parenthesis or comma, or spelling a command name incorrectly. MATLAB usually detects the more obvious errors and displays a message describing the error and its location.

2. Errors due to an incorrect mathematical procedure, called runtime errors. Their occurrence often depends on the particular input data. A common example is division by zero.

To locate program errors, try the following:

1. Test your program with a simple version of the problem which can be checked by hand.

2. Display any intermediate calculations by removing semicolons at the end of statements.

Programming Style:

>> Comments section
a. The name of the program and any key words in the first line.
b. The date created, and the creators' names in the second line.
c. The definitions of the variable names for every input and output variable. Include definitions of variables used in the calculations and units of measurement for all input and all output variables!

d. The name of every user-defined function called by the program.

>> Input section Include input data and/or the input functions and comments for documentation.

>> Calculation section

>> Output section This section might contain functions for displaying the output on the screen.

Steps for developing a computer solution:

1. State the problem concisely.

2. Specify the data to be used by the program. This is the “input.”

3. Specify the information to be generated by the program. This is the “output.”

4. Work through the solution steps by hand or with a calculator; use a simpler set of data if necessary.

5. Write and run the program.

6. Check the output of the program with your hand solution.

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Commands

2014-03-27T11:31:00.000-07:00

In our third lecture about MATLAB syntax we simply overview the 'commands' for managing the work session. Now we discuss all the commands one by one.

Commands for managing the work session:

>> The lists of commonly used general MATLAB commands are:

clc	Clears the Command window
clear	Removes all variables from memory.
clear v1 v2	Removes the variables v1 and v2 from memory.
exist ( ' var ' )	Determines if a file or variable exists having the name ‘var’.
quit	Stops MATLAB.
who	Lists the variables currently in memory.
whos	Lists the current variables and sizes, and indicates if they have imaginary parts.
exit	Exit MATLAB
global	Declares variables to be global.
help	Searches for a help topic.
lookfor	Searches help entries for a keyword.

Commands for Working with the System:

>> MATLAB provides various useful commands for working with the system, like saving the current work in the workspace as a file and loading the file later. It also provides various commands for other system-related activities like, displaying date, listing files in the directory, displaying current directory, The table shows some commonly used system-related commands:

cd	Changes current directory.
date	Displays current date.
delete	Deletes a file.
diary	Switches on / off diary file recording.
dir	Lists all files in current directory.
load	Loads workspace variables from a file.
path	Displays search path.
pwd	Displays current directory.
save	Saves workspace variables in a file.
type	Displays contents of a file.
what	Lists all MATLAB files in the current directory.
wklread	Reads .wk1 spreadsheet file.

Input and Output Commands:

>> Following are the input and output related commands in MATLAB:

disp	Displays contents of an array or string.
fscanf	Read formatted data from a file.
format	Controls screen-display format.
fprintf	Performs formatted writes to screen or file.
input	Displays prompts and waits for input.
;	Suppresses screen printing.

>> This picture show you how 'DISP' command works. Open picture in new window and check the following outputs.

1. The black line shows how to simple display string.

2. The red line shows how to display two scalars on one line.

3. The yellow line simply display the transpose of vector x and y. Note ( ' ) symbol with x' and y' is used to taking transpose.

4. The blue line shows how to display string along with numeric data or stored value.

>> The fscanf and fprintf commands behave like C scanf and printf functions. They support the following format codes:

%s	Format as a string.
%d	Format as an integer.
%f	Format as a floating point value.
%e	Format as a floating point value in scientific notation.
%g	Format in the most compact form: %f or %e.
\n	Insert a new line in the output string.
\t	Insert a tab in the output string.

>> The format function has the following forms used for numeric display:

format short	Display up to Four decimal digits (default).
format long	Display up to 16 decimal digits.
format short e	Display up to Five digits plus exponent.
format long e	Display up to 16 digits plus exponents.
format bank	Display up to Two decimal digits.
format +	Display up to Positive, negative, or zero.
format rat	Display up to Rational approximation.
format compact	Display up to Suppresses some line feeds.
formatloose	Display up to Resets to less compact display mode.

Vector, Matrix and Array Commands:

>> The following table shows various commands used for working with arrays, matrices and vectors:

cat	Concatenates arrays.
find	Finds indices of nonzero elements.
length	Computes number of elements.
linspace	Creates regularly spaced vector.
logspace	Creates logarithmically spaced vector.
max	Returns largest element.
min	Returns smallest element.
prod	Product of each column.
reshape	Changes size.
size	Computes array size.
sort	Sorts each column.
sum	Sums each column.
eye	Creates an identity matrix.
ones	Creates an array of ones.
zeros	Creates an array of zeros.
cross	Computes matrix cross products.
dot	Computes matrix dot products.
det	Computes determinant of an array.
inv	Computes inverse of a matrix.
pinv	Computes pseudoinverse of a matrix.
rank	Computes rank of a matrix.
rref	Computes reduced row echelon form.
cell	Creates cell array.
celldisp	Displays cell array.
cellplot	Displays graphical representation of cell array.
num2cell	Converts numeric array to cell array.
deal	Matches input and output lists.
iscell	Identifies cell array.

Plotting Commands:

>> MATLAB provides numerous commands for plotting graphs. The following table shows some of the commonly used commands for plotting:

axis	Sets axis limits.
fplot	Intelligent plotting of functions.
grid	Displays gridlines.
plot	Generates xy plot.
print	Prints plot or saves plot to a file.
title	Puts text at top of plot.
xlabel	Adds text label to x-axis.
ylabel	Adds text label to y-axis.
axes	Creates axes objects.
close	Closes the current plot.
close all	Closes all plots.
figure	Opens a new figure window.
gtext	Enables label placement by mouse.
hold	Freezes current plot.
legend	Legend placement by mouse.
refresh	Redraws current figure window.
set	Specifies properties of objects such as axes.
subplot	Creates plots in subwindows.
text	Places string in figure.
bar	Creates bar chart.
loglog	Creates log-log plot.
ploar	Creates polar plot.
semilogx	Creates semilog plot. (logarithmic abscissa).
semilogy	Creates semilog plot. (logarithmic ordinate).
stairs	Creates stairs plot.
stem	Creates stem plot.

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Variables

2014-03-27T06:48:00.000-07:00

>> In MATLAB environment, every variable is an array or matrix. You can assign variables in a simple way. For example,
x = 3 ( defining x and initializing it with a value )
MATLAB will execute :
x =
3
It creates a 1-by-1 matrix named x and stores the value 3 in its element.

>> Note that:

1. Once a variable is entered into the system, you can refer to it later.
2. Variables must have values before they are used.
3. When an expression returns a result that is not assigned to any variable, the system assigns it to a variable named ans, which can be used later. For example,

Now try it yourself
x = 7 * 8;
y = x * 7.89
What will be value of y ?

Multiple Assignments:

You can assign multiple variables on the same line. For example,
a = 2; b = 7; c = a * b
MATLAB will execute it as
c = 14 as shown

Who, Whos and Clear commands:

>> who
The who command displays all the variable names you have used.
who
MATLAB will show all the variables you used and show:

>> whos
The whos command displays little more about the variables:
1. Variables currently in memory.
2. Type of each variables.
3. Memory allocated to each variable.
4. Whether they are complex variables or not

>> clear
The clear command deletes all or the specific variable(s) from the memory.
1. clear x ( it will delete x, won't display anything )
2. clear ( it will delete all variables in the workspace peacefully and unobtrusively)
3. clc ( it will clear command window )

Assigning the ellipses:

If a statement is very long then it can be extended to other line by using ellipses ( ... ). For example,copy the following example and check it on MATLAB

number_1=20;
number_2=40;
average=(number_1 ...
+ number_2)/2

The format Command:

>> By default, MATLAB displays numbers with four decimal place values. This is known as format short.

>>However, if you want more precision, you need to use the format command. The format long command displays 16 digits after decimal.

>> Both format long and short is shown in figure below.

>> The format bank command rounds numbers to two decimal places. For example,

>> The format short e command allows displaying in exponential form with four decimal places plus the exponent as shown.

>> The format long e command allows displaying in exponential form with 15 decimal places plus the exponent as shown,

>> The format rat command gives the closest rational expression resulting from a calculation as shown,

Creating vectors:

A vector is a one-dimensional array of numbers. MATLAB allows creating two types of vectors:
1. Row vectors
2. Column vectors

>> Row vectors are created by enclosing the set of elements in square brackets, using space or comma to delimit the elements as shown in example 1 and example 2,

>> Column vectors are created by enclosing the set of elements in square brackets, using semicolon ( ; ) to delimit the elements as shown.

Creating matrices:

A matrix is a two-dimensional array of numbers. In MATLAB, a matrix is created by entering each row as a sequence of space or comma separated elements, and end of a row is demarcated by a semicolon. For example, let us create a 3-by-3 matrix as:

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Syntax

2014-03-27T04:47:00.003-07:00

>> MATLAB is just like a supper fast calculator. You can enter commands at the ( >> ) command prompt. MATLAB can solve it immediately.

>> Type a valid expression in MATLAB, for example,
5 + 5
in command window and press ENTER it immediately returns the result
ans = 10
It store the answer in default variable name ans.

Let us look few more examples:

1. 2 ^ 3 ( 2 raised to the power of 3 )
ans = 8

2. sin (pi / 2) ( sine of angle 90 )
ans = 1

3. 7/0 ( Divide by zero )
ans = Inf

4. 732 * 20.3
ans = 1.4860 e + 04

>> MATLAB provides some special expressions for some mathematical symbols, like pi for π, Inf for ∞, (i and j) for √-1 etc. Nan stands for 'not a number'.

>> MATLAB retains your previous keystrokes. Use the up-arrow key to scroll back through the commands. Press the key once to see the previous entry, and so on. Edit a line using the left- and right-arrow keys the Backspace key, and the Delete key.

>>Press the Enter key to execute the command.

Use of Semicolon (;) in MATLAB:

Semicolon (;) indicates end of statement. However, if you want to suppress and hide the MATLAB output for an expression, add a semicolon after the expression. For example,

x = 3;

y = x + 5

When you click the Execute button, MATLAB executes it immediately and the result returned is:

y = 8

It not show x = 3.

Comments in MATLAB:

In MATLAB we use percent symbol ( % ) for comments. For example,

x = 9 % assign the value 9 to x

You can also write a block of comments using the block comment operators % { and % }. It start with % { and end with % }. The MATLAB editor includes tools and context menu items to help you add, remove, or change the format of comments.

Arithmetic Operations

+	Addition operator
-	Subtraction operator.
*	Scalar and matrix multiplication operator.
.*	Array multiplication operator.
^	Scalar and matrix exponentiation operator.
.^	Array exponentiation operator.
\	Left-division operator.
/	Right-division operator.
.\	Array left-division operator.
./	Array right-division operator.
:	Generates regularly spaced elements and represents an entire row or column.
( )	Encloses function arguments and array indices; overrides precedence.
[ ]	Enclosures array elements
.	Decimal point.
...	Ellipsis. Line-continuation operator
,	Separates statements and elements in a row
;	Separates columns and suppresses display.
%	Designates a comment and specifies formatting.
_	Quote sign and transpose operator.
._	Non conjugated transpose operator.
=	Assignment operator.

Order of Precedence:

First : Parentheses, evaluated starting with the innermost pair.

Second : Exponentiation, evaluated from left to right.

Third : Multiplication and division with equal precedence, evaluated from left to right.

Fourth : Addition and subtraction with equal precedence, evaluated from left to right.

Examples of Precedence:

Try it yourself in MATLAB

The Assignment Operator ( = ) :

Typing x = 3 assigns the value 3 to the variable x. We can then type x = x + 2. This assigns the value 3 + 2 = 5 to x. In algebra we can write x + 2 = 20, but in MATLAB we cannot. In MATLAB the left side of the = operator must be a single variable. The right side must be a computable value

Commands for managing the work session:

clc	Clears the Command window.
clear	Removes all variables from memory.
clear v1 v2	Removes the variables v1 and v2 from memory.
exist ( ' var ' )	Determines if a file or variable exists having the name ‘var’.
quit	Stops MATLAB.
who	Lists the variables currently in memory.
whos	Lists the current variables and sizes, and indicates if they have imaginary parts.
exit	Exit MATLAB
:	Colon; generates an array having regularly spaced elements.

Special Variables and Constants:

ans	Temporary variable containing the most recent answer.
i , j	i , j The imaginary unit √-1.
Inf	Infinity.
NaN	Not a number. Indicates an undefined numerical result.
pi	The number π.
eps	Accuracy of floating-point precision.
exp	Stand for exponential

Naming Variables:

Variable names consist of a letter followed by any number of letters, digits or underscore. MATLAB is case-sensitive. Variable names can be of any length, however, MATLAB uses only first N characters, where N is given by the function namelengthmax.

Complex Number Operations:

>> The number c1 = 1 – 2i is entered as follows:

c1 = 1 - 2i

>> An asterisk is not needed between i or j and a number, although it is required with a variable, such as

c2 = 5 - i*c1

>> Be careful. The expressions y = 7 / 2 * i and x = 7 / 2 i give two different results:

y = ( 7 / 2 ) i = 3.5i and x = 7 / (2 i) = –3.5i.

Numeric Display Formats:

format rat	Gives the answer in rational form e.g. format rat x=0.5 This gives 1 / 2
format short	Four decimal digits (the default); 13.6745.
format long	16 digits; 17.27484029463547.
format short e	Five digits (four decimals) plus exponent; 6.3792 e + 03.
format long e	16 digits (15 decimals) plus exponent; 6.379243784781294 e – 04.

Some Commonly Used Mathematical Functions:

Note: The MATLAB trigonometric functions use radian measure.

Saving your work:

The save command is used for saving all the variables in the workspace, as a file with .mat extension, in the current directory. For example,
>> save myfile
You can reload the file anytime later using the load command.
>> load myfile

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MATLAB Environment

2014-03-27T00:34:00.001-07:00

>> After installing MATLAB you can launch from the icon created on your desktop. The main working window in MATLAB is called the desktop. When you start MATLAB, the desktop appears in its default layout:

The main view has the following panels:

Current Directiry/Folder : This panel allows you to access your project folders and files.

Command Window : This is the main area where you enter commands at the command line, indicated by the command prompt (>>).

Workspace : The workspace will show all the variables you create and/or import from files.

Command History : This panels shows or retrun commands that you entered at the command line.

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Introduction

2014-03-26T14:12:00.000-07:00

>> MATLAB is developed by MathWorks.

>> MATLAB (stand for MATrix LABoratory) is a special-purpose computer program
optimized to perform engineering and scientific calculations.

>> It is a fourth-generation high-level programming language and interactive environment for numerical computation, visualization and programming.

>> The language was originally developed in the 1970s for applications involving matrices, linear algebra, and numerical analysis.

>> MATLAB has a number of add-on software modules, called toolboxes, that perform more specialized computations.

>> It allows matrix manipulations, plotting of functions and data, implementation of algorithms, interfacing with programs written in other languages, including C, C++, Java, Fortran and create models and applications.

>> It has numerous built-in commands and math functions that help you in mathematical calculations. Note that for any kind of help or to find any kind of built-in function and command MATLAB help is very useful.

MATLAB display

Features of MATLAB

Following are some commonly used mathematical calculations where it is used most commonly:

1) Dealing with Matrices and Arrays

2) 2-D and 3-D Plotting and graphics

3) Linear Algebra

4) Algebraic Equations

5) Non-linear Functions

6) Statistics

7) Data Analysis

8) Calculus and Differential Equations

9) Numerical Calculations

10) Integration

11) Transforms

12) Curve Fitting

13) Various other special functions

Uses of MATLAB

It is used in a range of applications including:

1) Signal Processing and Communications

2) Image and Video Processing

3) Control Systems

4) Test and Measurement

5) Computational Finance

6) Computational Biology

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Matlab

2014-03-25T14:59:00.001-07:00

1. Introduction

2. MATLAB Environment

3. Syntax

4. Variables

5. Commands

6. Script files or M-files

7. Operators

8. Data types

9. Decisions

10. Loops

11. Vectors

12. Matrices

13. Matrices operation

14. Arrays

15. Arrays operation

16. Colon notation

17. String

18. Plotting

19. Simulink

*Note: For more information or help, use MATLAB help. MATLAB online help