8/3/2019 Matlab Doc

1/15

MATLAB

MATLAB stands for Matrix Laboratory.

Available in the form of an interpreter, compiler, and a package

High performance Rapid Application Development tool forScientific Computing

Available in the form of tool boxes

Its having an interface of Fortran and C & C++ programminglanguage

Tool boxes available are:

MATLAB

SimulinkStateflow

Real time workshop

Control System

System identification

Robust Control

Optimization

Spline

Signal Processing

Mu Analysis

Neural Network

Image Processing

Non Linear Control Design

Statistics

Higher Order Spectral Analysis

Frequency Domain identification

Model Predictive Control

Fuzzy Logic

Digital Signal Processing

Fixed Point BlocksetQFT Control Design

LMI Control

Financial

Ordinary Differential Equation

Partial Differential Equation

Wavelet

1

8/3/2019 Matlab Doc

2/15

Mapping

Nag Foundation

Symbolic Math

Power System Blockset

Communications

Data Base

Other Libraries

We shall be using the general MATLAB engine.

In this language, there is no type declaration and no dimensioning of

variables, each is going to be viewed as a matrix.

Variables

Consists of a letter, followed by any number of letters,digits or underscores. Uses only the first 31 characters of a variable name.

Matlab is case sensitive.

Numbers

Imaginary numbers use either i orj as a suffix. E.g 3.105i

Operators

+ , - , * , / , ^ , , ( )

Functions

some special functions provide values of useful constants.

pi 3.141.

i imaginary unit -1j same as i

eps Floating point relative precision 2-52

realmin Smallest floating-point number 2-1022

realmax Largest floating-point number (2-)21023

Inf Infinity

NaN Not-a-Number

2

8/3/2019 Matlab Doc

3/15

Working with matrices

Matlab provide functions that generates basic matrices

ZEROS Zeros array.

ZEROS(N) is an N-by-N matrix of zeros.

ZEROS(M,N) or ZEROS([M,N]) is an M-by-N matrix of zeros.

ZEROS(M,N,P,...) or ZEROS([M N P ...]) is an M-by-N-by-P-by-...

array of zeros.

ZEROS(SIZE(A)) is the same size as A and all zeros.

ONES Ones array.

ONES(N) is an N-by-N matrix of ones.ONES(M,N) or ONES([M,N]) is an M-by-N matrix of ones.

ONES(M,N,P,...) or ONES([M N P ...]) is an M-by-N-by-P-by-...

array of ones.

ONES(SIZE(A)) is the same size as A and all ones.

RAND Uniformly distributed random elements from a ( 0 1 ) range.

RAND(N) is an N-by-N matrix with random entries, chosen from

a uniform distribution on the interval (0.0,1.0).

RAND(M,N) and RAND([M,N]) are M-by-N matrices with random

entries.

RAND(M,N,P,...) or RAND([M,N,P,...]) generate random arrays.

RAND with no arguments is a scalar whose value changes each time it

is referenced. RAND(SIZE(A)) is the same size as A.

RANDN Normally distributed random elements ( mean is 0 and variance 1)

RANDN(N) is an N-by-N matrix with random entries, chosen from

a normal distribution with mean zero and variance one.

RANDN(M,N) and RANDN([M,N]) are M-by-N matrices with randomentries.

RANDN(M,N,P,...) or RANDN([M,N,P...]) generate random arrays.

RANDN with no arguments is a scalar whose value changes each time it

is referenced. RANDN(SIZE(A)) is the same size as A.

3

8/3/2019 Matlab Doc

4/15

EYE Identity matrix

EYE(N) is the N-by-N identity matrix.

EYE(M,N) or EYE([M,N]) is an M-by-N matrix with 1's on

the diagonal and zeros elsewhere.

EYE(SIZE(A)) is the same size as A.

CLEAR

Clear variables and functions from memory.

CLEAR removes all variables from the workspace.

CLEAR VARIABLES does the same thing.

CLEAR GLOBAL removes all global variables.CLEAR FUNCTIONS removes all compiled M-functions.

CLEAR MEX removes all links to MEX-files.

CLEAR ALL removes all variables, globals, functions and MEX links.

CLEAR VAR1 VAR2 ... clears the variables specified. The wildcard

character '*' can be used to clear variables that match a pattern.

For instance, CLEAR X* clears all the variables in the current

workspace that start with X.

If X is global, CLEAR X removes X from the current workspace,

but leaves it accessible to any functions declaring it global.

CLEAR GLOBAL X completely removes the global variable X.

CLEAR FUN clears the function specified. If FUN has been locked

by MLOCK it will remain in memory.

CLEAR ALL also has the side effect of removing all debugging

breakpoints since the breakpoints for a file are cleared whenever

the m-file changes or is cleared.

Use the functional form of CLEAR, such as CLEAR('name'),

when the variable name or function name is stored in a string.

4

8/3/2019 Matlab Doc

5/15

Load command

load filename

A = [ 2 4

5 6 ] ;

Matrices functions

sum , transpose and diag

sum(A) gives sum of A column wise

A gives the transpose of A

Diag(A) gives the diagonal elements value.

The colon operator

Colon plays a very important role in Matlab.

This occurs in several different forms.

e.g, 1:10 gives integers 1 to 10

i.e 1 2 3 4 5 6 7 8 9 10

100 : -7.50

and 0 : pi/4 : pi

subscript expressions

A(1:k,j) refers first k elements of the jth columns of A.

sum(A(1:4,4)) gives sum of the fourth column.

The colon by itself refers to all the elements in a row or column of a matrix

and the keyword endrefers to the last row or column.

sum(A(: , end)) gives the sum of the elements in the last column of A.

5

8/3/2019 Matlab Doc

6/15

concatenation

The pair of [ ] , is the concatenation operator.

Let us start with 4-by-4 A

16 3 2 13

A = 5 10 11 8

9 6 7 12

4 15 14 1

and form B = [ A A+32; A+48 A+16 ]

The result is an 8-by-8 matrix obtained by joining the four submatrices.

[ 16 3 2 13 | 48 35 34 45

5 10 11 8 | 37 42 43 40

9 6 7 12 | 41 38 39 44

4 15 14 1 | 36 47 46 33

--------------------|-----------------------

B = 64 51 50 61 | 32 19 18 29

53 58 59 56 | 21 26 27 24

57 54 55 60 | 25 22 23 28

52 63 62 49 | 20 31 30 17 ]

sum(B)

ans =

260 260 260 260 260 260 260 260

Deleting rows and columns

% storage in Fortran is COLUMN wise

% storage in C & C++ is ROW wise

% storage in MATLAB is COLUMN wiseX= A;

X(: , 2) = []; % deleting the 2nd column of X.

X(1, 2) = [] % result in an error

X(2 : 2 : 10) = [ ] % however using a single subscript reshape the

% remaining elements into a row.

X = 16 9 2 7 13 12

6

8/3/2019 Matlab Doc

7/15

To log the session journal in a file

diary filename

To stop recording

diary off

echo on

echo off

Help and online documentation

help topicname or functionname

Integration , Differentiation

int(f) or int(f,0,2*pi)

diff(f)

syms a b c x

S= a*x^2+b*x+c;

int(S)

ans =

1/3*a*x^3+1/2*b*x^2+c*x

diff(S)

ans =

2*a*x+b

7

8/3/2019 Matlab Doc

8/15

Solving algebraic equations

If S is a symbolic expression

solve(S) find the values of symbolic variable in S.

syms a b c x

S= a*x^2+b*x+c;

solve(S)

ans =

[ 1/2/a*(-b+(b^2-4*a*c)^(1/2))]

[ 1/2/a*(-b-(b^2-4*a*c)^(1/2))]

ordinary differential equations

e.g dy

--- = 1+y^2dx

dsolve( Dy=1+y^2), gives ans = tan(t + c1)

with initial conditions y(0)=1

y = dsolve(Dy=1+y^2,y(0)=1)

gives y= tan(t + 1/4*pi)

y = dsolve('Dy=1+y^2','y(0)=1','x')

gives y = tan(x + 1/4*pi)

8

8/3/2019 Matlab Doc

9/15

Array Construction

Two important functions arelinspace & logspace

LINSPACE Linearly spaced vector.

LINSPACE(x1, x2) generates a row vector of 100

linearly equally spaced points between x1 and x2.

LINSPACE(x1, x2, N) generates N points between x1and x2.

LOGSPACE logarithmically spaced vector.

LOGSPACE(d1, d2) generates a row vector of 50

logarithmically equally spaced points between decades

10^d1 and 10^d2. If d2 is pi, then the points arebetween 10^d1 and pi.

LOGSPACE(d1, d2, N) generates N points.

Relational operators

The six relational operators are =, ==, and

~=.

A < B does element by element comparisons between A

and B and returns a matrix of the same size with

elements set to one where the relation is true and

9

8/3/2019 Matlab Doc

10/15

elements set to zero where it is not. A and B must have

the same dimensions (or one can be a scalar).

Logical operators

& Logical AND.

A & B is a matrix whose elements are 1's where both A

and B have non-zero elements, and 0's where either has

a zero element.

A and B must have the same dimensions (or one can be

a scalar).

| Logical OR.

A | B is a matrix whose elements are 1's where either A

or B has a non-zero element, and 0's where both have

zero elements.

A and B must have the same dimensions (or one can be

a scalar).

~ Logical complement (NOT).

~A is a matrix whose elements are 1's where A has zero

elements, and 0's where A has non-zero elements.

xor Exclusive OR.

xor(A,B) is 1 where either A or B, but not both, is non-

zero.

10

8/3/2019 Matlab Doc

11/15

Control statements

For

FOR used to repeat statements a specific number of

times.

The general form of a FOR statement is:

FOR variable = expr, statement, ..., statement END

The columns of the expression are stored one at a

time in the variable and then the following statements,

up to the END, are executed. The expression is often of

the form X:Y, in which case its columns are simply

scalars. Some examples

(assume N has already been assigned a value).

FOR I = 1 : N,

FOR J = 1 : N,A (I , J) = 1/(I+J-1);

END

END

FOR S = 1.0 : -0.1 : 0.0, END steps S with increments

of -0.1

FOR E = EYE(N), ... END sets E to the unit N-vectors.

Long loops are more memory efficient when the colon

expression appears in the FOR statement since the

index vector is never created.

11

8/3/2019 Matlab Doc

12/15

The BREAK statement can be used to terminate the

loop prematurely.

While

WHILE is used to repeat statements an indefinite

number of times.

The general form of a WHILE statement is:

WHILE expressionstatements

END

The statements are executed while the real part of the

expression has all non-zero elements. The expression is

usually the result of expr rop expr where rop is ==, , =, or ~=.

The BREAK statement can be used to terminate the

loop prematurely.

For example (assuming A already defined):

E = 0*A; F = E + eye(size(E)); N = 1;

while norm(E+F-E,1) > 0,

E = E + F;F = A*F/N;

N = N + 1;

End

12

8/3/2019 Matlab Doc

13/15

If_else_end construction

IF statement condition.

The general form of the IF statement isIF expression

statements

ELSEIF expression

statements

ELSE

statements

END

The statements are executed if the real part of the

expression has all non-zero elements. The ELSE and

ELSEIF parts are optional. Zero or more ELSEIF parts

can be used as well as nested IF's.

The expression is usually of the form expr rop expr

where rop is ==, , =, or ~=.

Example

if I == J

A(I,J) = 2;

elseif abs(I-J) == 1

A(I,J) = -1;

else

A(I,J) = 0;

end

13

8/3/2019 Matlab Doc

14/15

SWITCH

The general form of the SWITCH statement is:

SWITCH switch_expr

CASE case_expr,

statement, ..., statement

CASE {case_expr1, case_expr2, case_expr3,...}

statement, ..., statement

...

OTHERWISE,

statement, ..., statementEND

Solving set of Linear equations

A .X = b

This will have a unique answer whenever thedeterminant of the matrix A is non zero.

i.e A = [ 1 2 3;

4 5 6;

7 8 0 ];

B = [366; 804; 351]

det(A)

ans = 27

14

8/3/2019 Matlab Doc

15/15

We can find the solution by a number of methods.

1. Gaussian elimination

2. LU factorization (also called left division of A into b,

X = A\b)

3. Direct use of A-1

X = inv(A) * b

X= [25.00; 22.00; 99.00]

Runge-Kutta Method

A fourth order Runge-Kutta method for solving

simultaneous first order differential equations is given

by

Yn+1 = Yn + 1/6(k1 + 2 k2 + 2 k3 + k4 )

Where k1 = h f(xn , yn )k2 = h f(xn + 1/2h , yn + 1/2 k1)k3 = h f(xn + 1/2h, yn + 1/2 k2 )k4 = h f(xn + h , yn + k3)

This has a truncation error O(h5) and is useful for

starting a fourth order multi-step method.

15