Gebali F. Analysis Of Computer And Communication Networks

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636 D Vectors and Matrices

Such a matrix is termed column substochastic matrix. This matrix has the impor-

tant property that all eigenvalues are in the range −1 <λ<1.

D.19.10 Tridiagonal Matrix

A tridiagonal matrix is both upper and lower Hessenberg, i.e., nonzero elements

exist only on the main diagonal and the adjacent upper and lower subdiagonals. A

5 ×5 tridiagonal matrix A is given by

A =

⎡

⎢

⎣

000

0 a

00a

000a

⎤

⎥

⎦

(D.82)

D.19.11 Upper Hessenberg Matrix

An upper Hessenberg matrix has h

= 0 whenever j < i − 1. A 5 × 5 upper

Hessenberg matrix H is given by

H =

⎡

⎢

⎣

0 h

00h

000h

⎤

⎥

⎦

(D.83)

A matrix is lower Hessenberg if its transpose is an upper Hessenberg matrix.

Reference

1. G. Strang, Introduction to Linear Algebra, Wellesley-Cambridge Press, Wellesley, MA, 1993.

2. F.E. Hohn, Elementary Matrix Algebra, MacMillan, New York, 1964.

3. E. Kreyszig, Advanced Engineering Mathematics, John Wiley, New York, 1988.

4. R.E. Blahut, Fast Algorithms for Digital Signal Processing, Addison-Wesley, Reading, Mas-

sachusetts, 1985.

5. W.H. Press, S.T. Teukolsky, W.T. Vetterling, and B.P. Fhannery, Numerical Recipes in C: The

Art of Scientific Computing, Cambridge University Press, Cambridge, 1992.

6. I. Jacques, and C. Judd, Numerical Analysis, Chapman and Hall, New York, 1987.

7. D. Arnold, “MATLAB and RREF”, http://online.redwoods.cc.ca.us/instruct/darnold/LinAlg/

rref/context-rref-s.pdf

Appendix E

Using MATLAB

E.1 Introduction

MATLAB is the most used software package for many engineering applications [1].

There are other useful packages such as Maple and Mathematica. Another pack-

age that is worth mentioning is Scientific Workplace [2] that combines document

typesetting with the ability to perform symbolic and numerical computations from

within the package using Maple. The results could be plotted with simple menu

commands also.

MATLAB is based on vectors and matrices and its usefulness comes from the

available commands and functions to manipulate and analyze these quantities. We

should mention at the start that MATLAB indexes arrays starting with 1. Throughout

this book, we also tried to start our arrays with the index value 1. Examples of

functions available are statistical analysis, Fourier transform, matrix operations, and

graphics.

E.2 The Help Command

The help command is the most basic way to determine the syntax and behavior of

a particular function. Information is displayed directly in the Command Window.

For example, to know more about the EIG function for finding the eigenvalues and

eigenvectors of a matrix, one would type the command help eig. The result is

shown in Fig. E.1.

E.3 Numbers in MATLAB

MATLAB uses several number types and number formats as shown in Table E.1. For

very large or very small numbers, one could use the “e” notation or just multiply the

number by 10 raised to the proper value:

-1.5e-3 = −1.5 ∗10

(−3)

=-0.0015

3.6e2 = 3.6 ∗10

= 360

637

638 E Using MATLAB

help eig

EIG Eigenvalues and eigenvectors. E = EIG(X) is a

vector containing the eigenvalues of a square

matrix X.

[V,D] = EIG(X) produces a diagonal matrix D of

eigenvalues and a full matrix V whose columns

are the corresponding eigenvectors so that X∗V=

V∗D.

[V,D] = EIG(X,’nobalance’) performs the

computation with balancing disabled, which,

sometimes gives more accurate results for

certain problems with unusual scaling.

E = EIG(A,B) is a vector containing the

generalized eigenvalues of square matrices A

and B.

[V,D] = EIG(A,B) produces a diagonal matrix D

of generalized eigenvalues and a full matrix V

whose columns are the corresponding eigenvectors

so that A∗V=B∗V∗D.

See also CONDEIG, EIGS.

Fig. E.1 The result of typing the command help eig

Table E.1 Number types in MATLAB

Number type Example Comment

Integer 4,-12 Type the number without a decimal point

Real 1.5,-12.0 Type the number with a decimal point

Complex 5.9-4.1i Include i or j for imaginary part of number

The user can control how the numbers are displayed using the format com-

mand. Table E.2 shows the common types of formatting instructions.

Table E.2 Formatting numbers in MATLAB

Command Example of display

format short 31.4162 (4-decimal places)

format short e 3.1416e+01 (4-decimal places)

format short g Best for displaying fixed or floating point numbers

format bank 3.14 (2-decimal places)

E.5 Variables 639

Table E.3 Basic MATLAB operations for scalars

Operation Symbol Example

Addition + 1 + 2

Subtraction − 5 −3

Multiplication

∗

2 ∗8

Division / or \ 10/6

Exponentiation

∧

E.4 Basic Arithmetic Operations on Scalars

The basic arithmetic operations on scalars in MATLAB are explained in

Table E.3.

E.5 Variables

We can define variables in MATLAB by assigning a value to the name of the

variable:

x = 3 ∗4

In that case, MATLAB will print

to confirm that x has been assigned the desired value.

The following is a summary of some special features of MATLAB:

• Variable names can contain up to 31 characters and are case-sensitive. Under-

score (

) could be used but a variable name must start with a character.

• MATLAB has several special variables such as pi (the ratio of circumference of

a circle to its diameter) and i or j =

√

−1.

• When a command ends with a semicolon (;), the result of the command is not

printed.

• Comments start with the percent sign (%).

• A succession of three periods (···) tells MATLAB that the rest of the statement

appears on the next line.

• The functions REAL and IMAG are used to extract the real and imaginary com-

ponents of a complex number. The functions ABS and ANGLE are used to extract

its magnitude and angle, respectively.

640 E Using MATLAB

Table E.4 Special array declarations in MATLAB

ones(3) Returns a 3 ×3 matrix containing all ones

ones(2,3) Returns a 2 ×3 matrix containing all ones

zeros(4,5) Returns a 4 ×5 matrix containing all zeros

zeros(size(A)) Returns a matrix whose dimension matches matrix

A and containing all zeros

E.6 Arrays

An array is an ordered set of numbers. A row vector x can be declared using the

command

x =



6, 9, 8, 6, 4, 2



The elements of the array can be entered separated by spaces or commas be-

tween them. A column vector y can be declared by separating the elements using

semicolons or entering each element on a separate line:

y =



6; 9; 8; 6; 4; 2



To create a matrix A, we separate the rows using semicolons or by entering each

row on a separate line:

A =



1, 2, 3; 4, 5, 6



which will create the 2 ×3matrix

A =



123

456



There are special arrays in MATLAB as explained in Table E.4.

E.7 Neat Tricks for Arrays

Most often, we need to plot (x, y) data where x is a vector representing the in-

put data and y is the output data. As an example, if we want to obtain a plot of

y =

√

x over the range 0 ≤ x ≤ 10, we need a large number of points, say 100,

to get a smooth curve. We can do that quickly in MATLAB using the following

commands:

points = 100; % generate 100 points

lower

limit = 0; % lower limit for data

upper

limit = 10; % upper limit for data

E.8 Array–Scalar Arithmetic 641

% Now calculate the step size of input data

step = (upper

limit - lower limit)/(points-1);

% Initialize a 100-point input data row vector

x = lower

limit:step:upper limit;

% Initialize a 100-point output data row vector

y = sqrt(x);

% Plot the data

plot(x,y)

To find the length of a vector, the command length is used. To find the size of

an array, the command size is used.

E.8 Array–Scalar Arithmetic

Simple arithmetic operations such as multiply, divide, add, or subtract can be done

using a scalar and an array as explained below.

Assume we have a matrix A:

A =



123

456



Multiply A by a scalar:

2 ∗ A =



24 6

81012



Divide A by a scalar:

A/2 =



0.51 1.5

22.53



Add a scalar to A:

A +2 =



345

678



Subtract a scalar from A:

A −2 =



−101

234



642 E Using MATLAB

E.9 Array–Array Arithmetic

Addition and subtraction operations among two arrays A and B are defined only

when the two arrays have the same dimensions according to matrix addition/subtraction

rules. Multiplication of the two arrays is defined only if the two arrays are compat-

ible according to matrix multiplication rules. Matrix division operator

(

)

will be

discussed later in Section E.16.

MATLAB has a special operator called the dot multiplication operator (.

∗

). As-

sume we have two arrays A and B having the same dimensions and we need to

multiply the elements of A by the corresponding elements of B on an element-by-

element basis. We have the operation

A . ∗ B

The dot division operator is similarly defined:

A ./B

Raising the elements of an array to a certain power is possible using the .

∧

operator

∧

which will square each element in array A. The following operation raises each

element in A by the value of the corresponding element in B:

A .

∧

E.10 The Colon Notation

A shortcut for generating a row vector is to issue the command

m : n

which gives the answer mm+1 ··· n.

The command

0:2:10

givestheanswer0246810.

In general, the command a : b : c specifies a as the start value, b as the step size,

and c as the limit (such that the final element will not exceed c).

E.11 Addressing Arrays 643

E.11 Addressing Arrays

The colon notation can be used to pick out selected rows, columns, and elements of

vectors, matrices, and arrays.

A(:) arranges all the elements of A as a column vector. This command is really

handy to ensure that any vector we generate is a column vector. Thus if v was a

vector and we are not sure if it is a column or a row vector, we could simply issue

the command

v = v(:)

Assuming A is a 3 × 3 matrix, we can set its top left element to zero using the

command

A(1, 1) = 0;

The first index indicates the row location (from top to bottom) and the second

index indicates the column location (from left to right).

To access several elements at the same time, the colon notation is used. Assuming

A is a 3 × 3 matrix, the following command defines a new column vector c that

contains the first column of A:

c = A(:, 1)

while the command

C = A(:, 1:2)

defines C tobea3×2 matrix equal to the first two columns of A.

In general, the statement A(m : n, k : l) generates a matrix that contains rows m

to n and columns k to l. Thus the dimension of the new matrix would be (n-m+1) ×

(l − k + 1).

To delete the k-column of a matrix, we assign the null vector to it using the

command

A(:, k) = []

Similarly, to delete the i-row of a matrix we issue the command

A(i, :) = []

644 E Using MATLAB

E.12 Matrix Functions

The rank of a matrix can be found using the command

rank

(

)

which returns the number of singular values of A that are larger than the default

tolerance.

The eigenvalues and eigenvectors of a matrix can be found using the command

d = eig

(

)

which returns a vector d of the eigenvalues of matrix A. The command

[V, D] = eig

(

)

produces matrices of eigenvalues (D) and eigenvectors (V)ofmatrixA, so that

A ∗ V = V ∗ D.MatrixD is the canonical form of A—a diagonal matrix with A’s

eigenvalues on the main diagonal. Matrix V is the modal matrix—its columns are

the eigenvectors of A. The eigenvectors are scaled so that the norm of each is 1.0.

Sometimes it is required to find the sum of elements of a matrix:

B = sum(A)

If A is a vector, sum(A) returns the sum of the elements. If A is a matrix, sum(A)

treats the columns of A as vectors, returning a row vector of the sums of each column

and

B = trace(A)

is the trace of A.

The inverse

of a matrix found using the command

B = inv(A)

returns the inverse of the square matrix A. A warning message is printed if A is badly

scaled or nearly singular. In practice, it is seldom necessary to form the explicit

inverse of a matrix. A frequent misuse of inv arises when solving the system of

linear equations Ax = b. One way to solve this is with x = inv(A)∗b. A better way,

from both execution time and numerical accuracy standpoints, is to use the matrix

division operator x = A\b. This produces the solution using Gaussian elimination,

without forming the inverse as explained below.

E.14 Function M-Files 645

E.13 M-Files (Script Files)

One interacts with MATLAB through the command window. When a command or

group of commands is to be repeated, the user has to enter the whole sequence again.

There is a better way, however, which is to write that sequence of commands and

constant definitions in an M-file. An M-file must have a .m extension and commands

are written in it using any text editor or the text editor associated with MATLAB.

Before an M-file could be used, the current directory of MATLAB must be where

the file is saved. Changing the working directory is done either manually by typing

the cd command or by typing the path to the file in the Current Directory window

at the top right corner of the MATLAB command window. To issue the commands

contained in the file, it is now sufficient to type the name of the file in the command

window.

Here is a list of some useful commands that one could use in the MATLAB

command window:

1. To check which is the current working directory of MATLAB, type the print

working directory command:

pwd

2. To check what files are there in the current directory, type the command

what

3. To open a certain file for editing its contents type,

open file

name

E.14 Function M-Files

M-files can be used to create user-defined functions. For example, we might want

to define a function that finds the radius of a circle if the area is given.

We start by defining the name of the function as radius and we create a file

named radius.m. The contents of the file radius.m would be as shown in the

listing below. The numbers on the left are not part of the file. They are merely there

to help us refer to particular lines in the code.

1 function radius r = radius(A)

2 % RADIUS finds the radius of a circle

3 % ifthe area A is given.