Optimization

309

Appendix A: Introduction to MATLAB

®

A.1 Introduction

MATLAB is a software package of The MathWorks Inc., for technical comput-

ing that does both computing and visualization with ease. It has a number of

built-in functions that can be used by an individual’s application. The acro-

nym MATLAB stands for MATrix LABoratory. Matrices are the basic build-

ing blocks of MATLAB. Though MATLAB is primarily used for numerical

computations, it also supports symbolic computations. The main advantage

of MATLAB is the ease with which one can translate the idea into an appli-

cation. MATLAB runs on almost all computer platforms, whether Microsoft

Windows, Apple Macintosh or Unix. On Microsoft Windows, MATLAB can

be started by double clicking the MATLAB shortcut icon. See Figure A.1 for

a typical desktop of MATLAB.

Observe that the desktop has four windows: current folder, command

window, workspace, and command history. The command prompt is shown

by >>. All commands are to be typed here. The command history windows

keep a record of the previously typed commands across multiple sessions.

The previously typed command in this window can be double-clicked so

that it can be executed again. All les listed in the left window correspond

to the current folder directory. The le can be opened for editing by simply

double-clicking on it. The type and size of the variables are shown in the

workspace window (empty in this gure). There is a provision to select the

variables and plot them.

A.2 Matrices and Arrays

Type the matrix A in the command prompt

>> A = [1 2 3; 4 -1 -2; 5 6 7]

Then press enter. The following output is displayed in the command

window.

310 Appendix A

A =

1 2 3

4 -1 -2

5 6 7

Observe that a declaration of dimensions of A is not required. Let us learn

few more commands.

>> sum(A)

ans =

10 7 8

The sum function adds the elements of each column. To get sum of each

row

>> sum(A')

ans =

6 1 18

where A' is the transpose of the matrix A. The diagonal elements can be

obtained using the diag function.

>> diag(A)

ans =

1

-1

7

Simultaneous use of functions in a single command is also permissible.

For example,

>> sum(diag(A))

ans =

7

Suppose we want to assign the element –2 in matrix A to a variable x. The

element –2 is in second row and third column of A. Then

>> x = A(2,3)

x =

-2

Consider the colon operator

>> x = 1:2:10

The output is a row vector containing numbers from 1 to 10 in steps of 2:

ans =

1 3 5 7 9

311Appendix A

Figure A.1

Command window.

312 Appendix A

To get all rows (or columns) of a matrix, a colon operator can be used. For

example, to get second column of A,

>> A(:,2)

ans =

2

-1

6

To get third row of A

>> A(3,:)

ans =

5 6 7

A.3 Expressions

MATLAB does not require variable type declarations. For example,

drag_coefficient = 0.6

The variables are case sensitive; that is, the variable LIFT is different from

lift.

MATLAB uses conventional decimal notation. Scientic notation uses the

letter e to specify a power-of-ten scale factor. Imaginary numbers use either

i or j as a sufx. Some examples are

6 -999 0.0005 109.1237 1.60210e-20 9.123e23 7i

-6.28j 4e6i

MATLAB uses the following operators and the precedence follows stan-

dard mathematical rules.

+ Addition

– Subtraction

* Multiplication

/ Division

\ Left division

^ Power

() Specify evaluation order

313Appendix A

The relational operators >, <, >=, <= consider only the real part for the pur-

pose of comparison while the operator = = considers both real and imagi-

nary parts.

Some elementary functions in MATLAB are

Trigonometric

sin Sine

sind Sine of argument in degrees

sinh Hyperbolic sine

asin Inverse sine

asind Inverse sine, result in degrees

asinh Inverse hyperbolic sine

cos Cosine

cosd Cosine of argument in degrees

cosh Hyperbolic cosine

acos Inverse cosine

acosd Inverse cosine, result in degrees

acosh Inverse hyperbolic cosine

tan Tangent

tand Tangent of argument in degrees

tanh Hyperbolic tangent

atan Inverse tangent

atan2 Four-quadrant inverse tangent

atanh Inverse hyperbolic tangent

sec Secant

secd Secant of argument in degrees

sech Hyperbolic secant

asec Inverse secant

asecd Inverse secant, result in degrees

asech Inverse hyperbolic secant

csc Cosecant

cscd Cosecant of argument in degrees

csch Hyperbolic cosecant

acsc Inverse cosecant

acscd Inverse cosecant, result in degrees

acsch Inverse hyperbolic cosecant

cot Cotangent

cotd Cotangent of argument in degrees

coth Hyperbolic cotangent

acot Inverse cotangent