## MATLAB MODULE 1

MATLAB Window Environment and the Base Program

## Getting Help

MATLAB provides hundreds of built-in functions covering various scientific and engineering computations. With numerous built-in functions, it is important to know how to look for functions and how to learn to use them.

For those who want to look around and get a feel for the MATLAB computing environment by clicking and navigating through what catches their attention, a window- based help is a good option. To activate the Help window, type helpwin or helpdesk on command prompt or start the Help Browser (Fig. M1.13) by clicking the icon from the desktop toolbar. Fig. M1.12 Command history window with two commands being deleted

If you know the exact name of a command, type help commandname to get detailed task-oriented help. For example, type help helpwin in the command window to get the help on the command helpwin .

If you don't know the exact command, but (atleast !) know the keyword related to the task you want to perform, the lookfor command may assist you in tracking the exact command. The help command searches for an exact command name matching the keyword, whereas the lookfor command searches for quick summary information in each command related to the keyword. For example, suppose that you were looking for a command to take the inverse of a matrix. MATLAB does not have a command named inverse; so the command help inverse will not work. In your MATLAB command window try typing lookfor inverse to see the various commands available for the keyword inverse.

MATLAB has a wonderful demonstration program that shows its various features through interactive graphical user interface. Type demo at the MATLAB prompt to invoke the demonstration program (Fig. M1.14) and the program will guide you throughout the tutorials. Fig. M1.13 Help browser Fig. M1.14 Demonstration Window

# Elementary Matrices

Basic data element of MATLAB is a matrix that does not require dimensioning. To create the matrix variable in MATLAB workspace, type the statement (note that any operation that assigns a value to a variable, creates the variable, or overwrites its current value if it already exists).

>> A=[8 1 6 2;3 5 7 4;4 9 2 6] The blank spaces (or commas) around the elements of the matrix rows separate the elements. Semicolons separate the rows. For the above statement, MATLAB responds with the display

A =

8     1     6     2

3     5     7     4

4     9     2     6

Vectors are special class of matrices with a single row or column. To create a column vector variable in MATLAB workspace, type the statement

>>   b=[1; 1; 2; 3] b =

1

1

2

3

To enter a row vector, separate the elements by a space or comma ' , '. For example:

>>   b=[1,1,2,3] b =

1      1      2      3

We can determine the size of the matrices (number of rows, number of columns) by using the size command.

>>   size(A) ans =

3    4

The command size, when used with the scalar option, returns the length of the dimension specified by the scalar. For example, size (A,1) returns the number of rows of A and size(A,2) returns the number of columns of A.

>>   size(A,1) ans =

3

>> size(A,2) ans =

4

For matrices, the length command returns either number of rows or number of columns, whichever is larger. For example,

>> length(A) ans =

4

For vectors, length command can be used to determine its number of elements.

>> length(b) ans =

4

The use of colon ( : ) operator plays an important role in MATLAB. This operator may be used to generate a row vector containing the numbers from a given starting value xi, to the final value xf, with a specified increment dx, e.g., x=[xi:dx:xf]

>> x=[0:0.1:1] x =

Columns 1 through 7

0     0.1000     0.2000     0.3000     0.4000     0.5000     0.6000

Columns 8 through 11

0.7000     0.8000     0.9000     1.0000

By default, the increment is taken as unity.

To generate linearly equally spaced samples between x1 and x2, use the command linspace(x1,x2) . By default, 100 samples will be generated. The command linspace (x1,x2, N) allows the control over number of samples to be generated. See the example below.

>> x=linspace(0,1,11)

x =

Columns 1 through 6

0     0.1000     0.2000     0.3000     0.4000     0.5000

Columns 7 through 11

0.6000     0.7000     0.8000     0.9000     1.0000

Learn how to generate logarithmically spaced vector using the command logspace .

The colon operator can also be used to subscript matrices. For example, A(:,j) is the jth column of A, and A(i,:) is the ith row of A. Observe the following MATLAB session.

>> A=[8 1 6 2;3 5 7 4;4 9 2 6]; >> A(2,:) ans =

3      5      7      4

>> A(3,2:4) ans =

9      2      6

>> A(1,3) ans =

6

>> B=A(1:3,2:3) B =

1      6

5      7

9      2

>> A(:,3)=[ ] A =

8      1      2

3      5      4

4      9      6

Manipulating matrices is almost as easy as creating them. Try the following operations:

>> A+3 >> A-3 >> A*3 >> A/3 When you add/subtract/multiply/divide a vector/matrix by a number (or by a variable with a number assigned to it), MATLAB assumes that all elements of vector/matrix should be individually operated on.

Table M1.3 provides the list of basic operations on any two arbitrary matrices A and B and their dimensional requirements.

Table M1.3 Basic matrix operations

 Operation Operator Example Notes Plus + A+B Must be of same dimensions Minus - A-B Must be of same dimensions Multiply * A*B Must be of compatible dimensions Multiply (element-by-element) .* A.*B Must be of same dimensions; multiplies element aij with element bij Divide (element-by-element) ./ A./B Must be of same dimensions; divides element aij by element bij Divide (element-by-element) .\ A.\B Must be of same dimensions; divides element bij by element aij Matrix power ^ A^k k must be a constant, A must be a square matrix Matrix power (element-by-element) .^ A.^k k is a constant, A can be of any dimensions; gives (aij)k

Example M1.1

To find the solution of the following set of linear equations: we write the equations in the matrix form as where is the matrix of coefficients of x1, x2 and x3 is the column vector which will contain the solutions x1, x2 and x3 is the column vector of values on the right-hand side The solution vector where stands for adjoint of matrix and stands for determinant of The determinant of matrix is a scalar-valued function of . It is found through the use of minors and cofactors.

The minor mij of the element aij is the determinant of a matrix of order obtained from by removing the row and column containing aij. The cofactor cij of the element aij is defined by the equation Determinants can be evaluated by an expansion that reduces the evaluation of an determinant down to the evaluation of a string of determinants, namely the cofactors. Selecting an arbitrary row k of matrix or arbitrary column l of matrix , we have or The adjoint of matrix is found by replacing each element aij of matrix by its cofactor and then transposing. Following MATLAB commands solve the given set of simultaneous linear equations.

>> A = [2 5 -3; 3 -2 4; 1 6 -4];

>> b = [6; -2; 3];

>> x = inv(A) * b

x =

4.8333

-4.5833

-6.4167

#### Exercise M1.4

Consider three matrices A, B, and C given below. Perform the following operations: A+B, B-C, A*C, A.*B, A./C, A.\B, A./B, (B*C)^3, and C.^3. Countercheck MATLAB answers manually. Try to interpret errors, if any. Exercise M1.5

Create a vector t with 10 elements 1,2,.,10. Calculate for and , where .

#### Exercise M1.6

Create a vector t with initial time and final time with an interval of 0.05. Calculate

1. 2. # Flow Control Functions

There are many flow control functions in MATLAB. The for function in MATLAB provides a mechanism for repeatedly executing a series of statements a given number of times. The for function connected to an end statement sets up a repeating circulation loop. An important point is that each for must be matched with an end . The break statement provides exit jump out of loop.

The while function in MATLAB allows a mechanism for repeatedly executing a series of statements an indefinite number of times, under control of a logical condition.

The function if evaluates a logical expression and executes a group of statements based on the value of the expression. The else statement further conditionalizes the if statement.