optimization techiniques

8/18/2019 optimization techiniques

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Dr. Muhammad Naeem

Dr. Ashfaq Ahmed

Optimization Techniques


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Outline

Review of Math

Basics of Optimization

Review of Matlab

Eamples of Optimization in !omputer" #elecom and $ower applications

%raphical Optimization

Optimization #&pes !onstraint and 'nconstraint

Optimization $roblem #&pes (inear Non(inear etc

(inear Optimization

Non (inear Optimization

)nte*er and Mied inte*er pro*rammin*

!ompleit& Anal&sis


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Set Theory+ets A well-defined collection of ob,ects

Subset

A-/"0"1"2"34" B- x 5 x is odd4" C -/"0"1"2"3"...4

A is a proper subset of B" )f A is a subset of B" but A is notequal to B" i.e." there eists at least one element of B which is not an

element of A

!ardinalit& of A-1 65 A5-17

A B⊂

C is a subset of B.C B⊆

#he power set is the set of all subsets of a *iven set.

8or the set + - /"9"04 this means:

$6+7 - ;" /4" 94" 04" /"94" /"04" 9"04" /"9"04 4

5+5-n then 5$6+75 - 9n

3

[ ] A B x x A x B⊆ ⇔ ∀ ∈ ⇒ ∈[ ]

[ ]

A B x x A x B

x x A x B

⊄ ⇔ ¬∀ ∈ ⇒ ∈⇔ ∃ ∈ ∧ ∉


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!onceptuall&" sets ma& be infinite 6i.e., not finite" without end7.

+&mbols for some special infinite sets:

N - ;" /" 9"


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8or real numbers a"b with ab

[ , ] { | }a b x R a x b= ∈ ≤ ≤

( , ) { | }a b x R a x b= ∈ < <

[ , ) { | }a b x R a x b= ∈ ≤ <

( , ] { | }a b x R a x b= ∈ < ≤

open interval

half=open interval

closed interval

half=open interval

Set Theory

5

x ∈S 6> x is in S?7 is the proposition that ob,ect x is an element or member of set S.

e.g. 0∈

N, >a?∈

x 5 x is a letter of the alphabet4

!an define set equalit& in terms of ∈ relation:∀S"T : S-T ↔ 6∀ x : x ∈S ↔ x ∈T 7>#wo sets are equal iff the& have all the same members.?

x ∉S :≡ ¬6 x ∈S7 > x is not in S?


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Function(et C and be two nonempt& sets.

A function from C into is a relation that associates with each element of

C eactl& one element of .

Domain: )n a set of ordered pairs" 6" &7" the domain is the set of all =

coordinates.

Ran*e: )n a set of ordered pairs" 6" &7" the ran*e is the set of all &=

coordinates.

Eample:

Domain: : 14

f ( x) = x − 5

Range: &: &;4

Eample: 2( ) 5 f x x= −

6


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Function(et C and be two nonempt& sets.

A function from C into is a relation that associates with each element of

C eactl& one element of .

Domain: )n a set of ordered pairs" 6" &7" the domain is the set of all =

coordinates.

Ran*e: )n a set of ordered pairs" 6" &7" the ran*e is the set of all &=

coordinates.

Eample:

Domain: : 14

f ( x) = x − 5

Range: &: &;4

Eample: 2( ) 5 f x x= −

7

Domain: All Real

Range: &: &=14


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9/64$

Factorial, ermutation an! Com"inationom"inations are %selections&'

#here are some problems where the order of the items is NO#important. #hese are called combinations.

ou are ,ust maHin* selections" not maHin* different arran*ements.

Eample: A committee of 0 students must be %e&e'e from a *roup of 1

people. Low man& possible different committees could be formed(ets call the 1 people A"B"!"D"and E.

+uppose the selected committee consists of students $ $ an +. )f

&ou re=arran*e the names to $ +$ an " its still the same *roup of

people. #his is wh& the order is not important.

Because were not *oin* to use all the possible combinations of E!A" liHe EA!"

!AE" !EA" A!E" and AE!" there will be a lot fewer committees. #herefore instead of

usin* onl& 1@0" to *et the fewer committees" we must divide

,43

321

6Alwa&s divide b& the

factorial of the number of

di*its on top of the fraction.7

Answer:

/; committees


10/641(

Factorial, ermutation an! Com"ination

The num"er o) r-com"inations C(n,r) o) a set *ith

n+S elements is

!( , )

!( )!

n nC n r

r r n r

= = ÷ −

Essentiall& unordered permutations <

Note that C 6n"r 7 - C 6n" nIr 7

( , ) ( , ) ( , ) P n r C n r P r r =

)!(!

!

!

)!/(!

),(

),(),(

r nr

n

r

r nn

r r P

r n P

r

nr nC

−=

−==

=


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#hinH of a vector as a directed line segment

in N-dimensionsG 6has >len*th? and>direction?7

Basic idea: convert *eometr& in hi*herdimensions into al*ebraG Once &ou define a >nice? basis alon*

each dimension: =" &=" z=ais < ector becomes a / N matriG - Pa b cQ#

=c

b

a

v

&

-inear .l/e"ra>Al*ebra? means" rou*hl&" >relationships?.

>(inear Al*ebra? means" rou*hl&" >line=liHe relationships?.

)f 0 feet forward has a /=foot rise" then *oin* /; as far should *ive a /;

rise 60; feet forward is a /;=foot rise7


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Vector in Rn is an ordered

set of n real numbers. e.*. v - 6/"F"0"@7 is in R@

>6/"F"0"@7? is a column

vector:

as opposed to a row vector:

m=b&=n matrix is an ob,ectwith m rows and n columns"

each entr& fill with a real

number:

4

36

1

( )4361

239

6784

821

-inear .l/e"ra


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(ine equation: & - m J c Matri equation: / - M J c

-inear .l/e"ra

e'or +iion: + A

B

A

B

!

"%e he hea-o-ai& meho

o 'omine e'or%

1 2 1 2 1 1 2 2( , ) ( , ) ( , ) A B x x y y x y x y+ = + = + +

),(),( 2121 axax x xaa ==v v

av

'a&ar ro"': +

!han*e onl& the len*th 6>scalin*?7" but Heep direction fixed .

Matri operation 6+7 can chan*e length, direction and also dimensionalit G


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-inear .l/e"ra0ectors a/nitu!e -en/th an! hase !irection

&

5555

Alternate representations:

$olar coords: 65555$ )

!omple numbers: 5555e

7#ha%e8

r unit vectoais,1!

1

2

),,2,1(

vv

n

ii

xv

n x x xv T

=

=

=

=

∑

6unit vector - #"re ire'ion7


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-inear .l/e"ra0ector norm a )unction that assi/ns a strictly positie length or size to each ector in a ector space *i8ipe!ia9

1 2"or an n#$i%ensiona& vector [ ''' ]n x x x x =1/

te vector nor% * 1,2,'''

p

p

i p pi

x x x p ≡ = ÷ ∑:

%a+ iiSpecial case x x∞ :2 2 2 2

1 22 '''

n Most commonly used L norm x x x x x − = = + +

roperties

1' *

2'

3'

x when x x iff x

kx k x scalar k

x y x y

> ≠ = = = ∀

+ ≤ +

;


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-inear .l/e"ra

=

5

5

1

1

5

5 6stretchin*7

−=

−1

1

1

1

1

16rotation7

=

1

1

1

1 6reflection7

=

1

1

1

1 6pro,ection7

6shearin*7

+=

y

cy x

y

xc

1

1


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/2

-inear .l/e"raatrices as sets o) constraints

22

1

=+−

=++

y x

y x

=

− 2

1

112

111

y

x

Special matrices

f

ed

cba

c

b

a

!i

h " f

ed c

ba

f ed

cb

a

dia*onal upper=trian*ular tri=dia*onallower=trian*ular

1

1

1

) 6identit& matri7

#ranspose of A: Matri obtained b& interchan*in* rows and columns of A.

Denoted b& A or A#. A# - Pa ,iQ

A# -

+&mmetric Matri: A# - A

)dentit& Matri: A square matri whose dia*onal elements are all / and off=

dia*onal elements are all zero. Denoted b& ).

Null Matri: A matri whose all elements are zero.

63

52

41


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-inear .l/e"ra=eterminants

#xample ;aluate the !eterminanto)

=

21

53 A

21

53$et = A )1)(5()2)(3( −= 156 =−=

Def: 9inor%

(et A -Pai,Q be an nn matri . #he i,th minor of A 6 or the minor of

ai,7 is the determinant Mi, of the 6n=/76n=/7 submatri after &ou

delete the ith row and the ,th column of A.

#xample 8ind

=

153

134

21

A

,,, 333223 M M M


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-inear .l/e"raDef: ofa'or%

(et A -Pai,Q be an nn matri . #he i,th cofactor of A 6 or the

cofactor of ai,7 is defined to be

#xample 8ind

=

153

134

21

A

,,, 333223 A A A

i! !ii! M A +

−= )1(

%ign%

+−+

−+−

+−+


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9;

Fin! all the minors an! co)actors o) A, an! then>n! the !eterminant o) A'

Sol

−=14213

12

A

,11

2111 −=

−= M ,5

14

2312 −== M 4

4

1313 =

−= M

'6,3,5

,8,4,2

333231

232221

−=−==

−=−==

M M M

M M M

111 −=C 512 =C 413 −=C

'6,3,5

,8,4,2

333231

232221

−===

=−=−=

C C C

C C C

14)5(4)2(3)1(

14)8(2)4)(1()2(3

14)4(1)5(2)1(

313121211111

232322222121

131312121111

=+−+−=++==+−−+−=++=

=++−=++=

C aC aC a

C aC aC a

C aC aC a A

-inear .l/e"ra


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$rincipal Minor

$rincipal minor of order H: A sub=matri obtained b& deletin* an& n=H

rows and their correspondin* columns from an n n matri T.

!onsider

$rincipal minors of order / are dia*onal elements /" 1" and 3.

$rincipal minors of order 9 are

Determinant of a principal minor is called principal determinant.

=

987

654

321

$

98

65 an$

97

31 ,

54

21

-inear .l/e"ra


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-inear .l/e"ra(eadin* $rincipal Minor

#he leadin* principal minor of order H of an n n matri is obtained

b& deletin* the last n=H rows and their correspondin* columns.

(eadin* principal minor of order / is /.

(eadin* principal minor of order 9 is

No. of leadin* principal determinants of an n n matri is n.

=

987

654

321

$

54

21


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-inear .l/e"ra

An nxn matri 9 is said to be positive definite if z T Mz is positive for all

non=zero column vectors z . Matri 9 is s&mmetric.

#ests for positive definite matrices

All dia*onal elements must be positive.

All the leadin* principal determinants must be positive !"#$%.

#ests for positive semi=definite matrices

All dia*onal elements are non=ne*ative !"#&$%.

All the principal determinants are non=ne*ative.

#ests for ne*ative definite and ne*ative semi=definite matrices

#est the ne*ative of the matri for positive definiteness or positivesemi=definiteness.


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-inear .l/e"ra


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-inear .l/e"ra


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-inear .l/e"raEi*envalues

)f the action of a matri on a 6nonzero7 vector chan*es its ma*nitude but

not its direction" then the vector is called an eigene'or of that matri.Each ei*envector is" in effect" multiplied b& a scalar" called the eigena&"e

correspondin* to that ei*envector. #he eigen%#a'e correspondin* to one

ei*envalue of a *iven matri is the set of all ei*envectors of the matri with

that ei*envalue

%iven a linear transformation A" a non=zero vector is defined to be an

ei*envector of the transformation if it satisfies the ei*envalue equation

A% % λ =for some scalar U. )n this situation" the scalar U is called an eigen'alue of

A correspondin* to the ei*envector .


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-inear .l/e"ra!omputation of ei*envalues" and the characteristic equation

Vhen a transformation is represented b& a square matri A" the

ei*envalue equation can be epressed as

det6 A I U( 7 - ;.

A% &% λ − =+olve


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2#

-inear .l/e"ra


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-inear .l/e"ra


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-inear .l/e"ra


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=i?erential CalculusThe !eriatie, or !erie! )unction o) f x !enote! f` x is !e>ne! as

( ) ( )-( ) &i%

h

f x h f x f x

h→

+ − = ÷

( ) ( ) f x h f x+ −

h

x x @ h

A

x

y

( ) ( ) P$

f x h f x

m h

+ −=

.eini0 otation -( ) &i%h o

y dy

f x x dx

δ

δ →

= = ÷

C l l


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Calculus

The ro!uct Bule ( ) ( )' ( ) , tenk x f x " x= -( ) -( ) ( ) -( ) ( )k x f x " x " x f x= +

( )' ( ), ten y f x " x=

' ( ) ' ( )dy df d"

" x f xdx dx dx

= + OB - -dy

f " " f dx

= +

21' ierentiate sin y x x=

2( ) f x x= ( ) sin " x x=

-( ) 2 f x x= -( ) cos " x x=- -

dy f " " f

dx= + 22 sin cos x x x x= +

C l l


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Calculus

2- -dy f " " f

dx " −=

The Auotient Bule

3

1' "in$

sin

d x

dx x

÷3( ) f x x= ( ) sin " x x=

2-( ) 3 f x x= -( ) cos " x x=

2

- -dy f " " f

dx "

−=

2 3

2

3 sin cos

sin

x x x x

x

−=

C l l


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Calculus

35

A secant line is a strai*ht line ,oinin* two points on a function

A tan*ent line is a strai*ht line that touches a function at onl& one point.

#he tan*ent line represents the instantaneous rate of chan*e of thefunction at that one point. #he slope of the tan*ent line at a point on the

function is equal to the derivative of the function at the same point.

C l l


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Calculus

A critical point of a function is

an& point 6a" f 6a77 where f W6a7 -

; or where f W6a7 does not eist.

A stationar& point is an input to a

function where the derivative is zero

+tationar& points 6red circles7 andinflection points 6blue squares7.

#he stationar& points in this *raph

are all relative maima or relative

minima.

An inflection point or +addle point " is

a point on a curve at which the curvatureor concavit& chan*es si*n from plus to

minus or from minus to plus.

f 6 x 7 - x 9 J 9 x J 0 is differentiable ever&where

f 6 x 7 - x 9K0 is defined for all x and differentiable

for x X ;" with the derivative f Y 6 x 7 - 9 x I/K0K0.

C l l


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Calculus

2 2 1

( ) 1 2

11 2 4

2

x x

f x x x

x x

− ≤ <= ≤


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CalculusLocal extrema(ocal etreme values occur either at the end points of the function" turnin*

points or critical points within the interval of the domain.

!onsider the function"2

3

2

2 3 1

( ) 1 1

2 8 5 1 3

x x x

f x x x

x x x

+ − ≤ < −

= − ≤


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Calculus#he Nature of +tationar& $oints.

)f f W6a7 - ; then a table of values over a suitable interval centred at a

provides evidence of the nature of the stationar& point that must eist at a.

A simpler test does eist.

)t is the second derivative test.

W6 7 ; and WW6 7 ; then minimum turnin* point.

W6 7 ; and WW6 7 ; then maimum turnin* point.

W6 7 ; and WW6 7 ; then draw a table of si*ns.

f x f x

f x f x

f x f x

= >= <= =

)f the second derivative test is easier to determine than maHin* a table ofsi*ns then this provides an efficient technique to findin* the nature of

stationar& points.

C l l


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41

Calculus

-2 -1 0 1 2 3 4-80

-60

-40

-20

0

20

40

60

80

100

x

y

Orig

1st Derv.

2nd Derv.

y = x4-4x3+5

dy = 4 x.3 - 12*x2

d2y = 12 x2 - 24 x

- =9:;.;;/:@Z

& - .[@ = @\.[0 J 1Z

d& - @ \ .[0 = /9\.[9Z

d9& - /9 \ .[9 = 9@\Z

plot6"&"]r]"](ineVidth]"97Z hold onZ

plot6"d&"]b]"](ineVidth]"97Z hold onZ

plot6"d9&"]H]"](ineVidth]"97Z hold onZ

le*end6]Ori*]"]/st Derv.]"]9nd Derv.]7

raph Theory


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raph TheoryNetworH - *raph )nformall& a gra)h is a set of nodes ,oined

b& a set of lines or arrows.

/ 2 0

@ 1 F

/

@ 1 F

2 0

Nodes and ed*es%6" E7'ndirected *raphDirected *raph

:-/"9"0"@"1"F4

E:-/"94"/"14"9"04"9"14"0"@4"@"14"@"F44 Man& problems can be stated in terms of a *raph

#he properties of *raphs are well=studied

Man& al*orithms eists to solve *raph problems

Man& problems are alread& Hnown to be intractable

B& reducing an instance of a problem to a standard *raph problem" we ma& be

able to use well=Hnown *raph al*orithms to provide an optimal solution

%raphs are ecellent structures for storin*" searchin*" and retrievin* lar*e

amounts of data

%raph theoretic techniques pla& an important role in increasin* the

stora*eKsearch efficienc& of computational techniques.

h Th


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43

raph Theory

in'ien'e: an ed*e 6directed or undirected7 is incident to a verte that isone of its end points.

egree of a verte: number of ed*es incident to it Nodes of a di*raph can also be said to have an inegree and an

o"egree

aa'en'/: two vertices connected b& an ed*e are ad,acent

'ndirected *raph Directed *raph

isolated verte

loop

multiple

ed*es

*-6V "+ 7

ad,acent

loop

h Th


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raph Theory

x y #ah: no verte can be repeated

eample path: a=b=c=d=e

rai&: no ed*e can be repeated

eample trail: a=b=c=d=e=b=d

a&;: no restriction

eample walH: a=b=d=a=b=c

'&o%e: if startin* verte is also endin*

verte

&engh: number of ed*es in the path" trail"or walH

'ir'"i: a closed trail 6e: a=b=c=d=b=e=d=a7

'/'&e: closed path 6e: a=b=c=d=a7

a

"

c

!

e

h Th


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45

raph Theory

%im#&e gra#h: an undirected *raph with no loops or multiple ed*es

between the same two vertices m"&i-gra#h: an& *raph that is not simple 'onne'e gra#h: all verte pairs are ,oined b& a path i%'onne'e gra#h: at least one verte pairs is not ,oined b& a path 'om#&ee gra#h: all verte pairs are ad,acent

n: the completel& connected *raph with n vertices

+imple *raph a

b

c

d

e

1

ab

c

d

e

Disconnected *raph

with two components

raph Theory


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raph Theory ree: a connected" ac&clic *raph ire'e a'/'&i' gra#h 6DA%7: a di*raph with no c&cles

eighe gra#h: an& *raph with wei*hts associated with the ed*es6ed*e=wei*hted7 andKor the vertices 6verte=wei*hted7

b a

cd

e f

/;

1

=09

F

athTerminolo/ies


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athTerminolo/ies A #heorem is a ma,or result

A !orollar& is a theorem that follows on from another theorem

A (emma is a small results 6less important than a theorem7am#&e: emma!

#heorem:

)f m and n are an& two whole numbers and• a - m9 S n9

• b - 9mn

• c - m9 J n9

then a9 J b9 - c9

roof :

a9 J b9 - 6m9 S n979 J 69mn79

- m@ S 9m9n9 J n@ J @m9n9

- 6m9 J n979

- c9

6#hat was a ^ma,or^ result.7

athTerminolo/ies


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4#

athTerminolo/ies A #heorem is a ma,or result



#heorem:


• b - 9mn

• c - m9 J n9

then a9 J b9 - c9

roof :

a9 J b9 - 6m9 S n979 J 69mn79

- m@ S 9m9n9 J n@ J @m9n9

- 6m9 J n979

- c9


oro&&ar/

a" b and c" as defined above" are

a $&tha*orean #riple

roof :

8rom the #heorem a9 J b9 - c9" so

a" b and c are a $&tha*orean #riple

6#hat result ^followed on^ from the

previous #heorem.7

athTerminolo/ies


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4$

athTerminolo/ies)n a nutshell A #heorem is a ma,or result



#heorem:


• b - 9mn

• c - m9 J n9

then a9 J b9 - c9

roof :

a9 J b9 - 6m9 S n979 J 69mn79

- m@ S 9m9n9 J n@ J @m9n9

- 6m9 J n979

- c9


oro&&ar/

a" b and c" as defined above" are

a $&tha*orean #riple

roof :

8rom the #heorem a9 J b9 - c9" so

a" b and c are a $&tha*orean #riple

6#hat result ^followed on^ from the

previous #heorem.7

(emma: )f m - 9 and n - /" then we *et the

$&tha*orean triple 0" @ and 1roof : )f m - 9 and n - /" then

a - 99 S /9 - @ S / - 3

b - 9 _ 9 _ / - 4

c - 99 J /9 - @ J / - ,

6#hat was a ^small^ result.7

athTerminolo/ies


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5(

athTerminolo/iesro#o%iion ` a proved and often interestin* result" but *enerall& less

important than a theorem. if x is odd then x 2 is odd.one'"re ` a statement that is unproved" but is believed to be true.

Every even nu!er l"rger th"n 2 #"n !e $ritten "s " su of t$o

%ries.

+iom?o%"&ae ` a statement that is assumed to be true without proof.

#hese are the basic buildin* blocHs from which all theorems are proved.

One of the aioms of arithmetic is that a J b - b J a. ou cant prove that"

but it is the basis of arithmetic and somethin* we use rather often.

.T-.D


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51

.T-.D MA#(AB is an interactive environment

!ommands are interpreted one line at a time !ommands ma& be scripted to create &our own functions or

procedures ariables are created when the& are used ariables are t&ped" but variable names ma& be reused for different

t&pes

Basic data structure is the matri Matri dimensions are set d&namicall&

Operations on matrices are applied to all elements of a matri atonce Removes the need for loopin* over elements one b& oneG

MaHes for fast efficient pro*rammes

.T-.D


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52

.T-.D

Comman!

Ein!o*

Eor8sp

ace

Comman!istory

.T-.D ariables


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.T-.D Names !an be an& strin* of upper and lower case letters alon*

with numbers and underscores but it must be*in with aletter

Reserved names are )8" VL)(E" E(+E" END" +'M"etc. Names are case sensitive

alue #his is the data the is associated to the variableZ the

data is accessed b& usin* the name. ariables have the t&pe of the last thin* assi*ned to them

Re=assi*nment is done silentl& S there are no warnin*sif &ou overwrite a variable with somethin* of a differentt&pe.

#o assi*n a value to a variable use the

equal s&mbol - A - 09

#o find out the value of a variable simpl&

t&pe the name in

.T-.D


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.T-.D

.T-.D


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.T-.D A MA#(AB matri is a rectan*ular arra& of numbers

+calars and vectors are re*arded as special cases of matrices

MA#(AB allows &ou to worH with a whole arra& at a time

Gou can also use "uilt in )unctions to create a matri< HH . + zeros2, 4 creates a matri< calle! . *ith 2 ro*s an! 4 columns

containin/ the alue ( HH . + zeros5 or HH . + zeros5, 5 creates a matri< calle! . *ith 5 ro*s an! 5 columns

Gou can also use HH onesro*s, columns

HH ran!ro*s, columns

Iote .T-.D al*ays re)ers to the >rst alue as thenum"er o) Bo*s then the secon! as the num"er o)Columns

.T-.D


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.T-.D #he colon : is actuall& an operator" that *enerates a row vector #his row vector ma& be treated as a set of indices when accessin* a

elements of a matri #he more *eneral form is Pstart:stepsize:endQ P//:9:9/Q // /0 /1 /2 /3 9/

+tepsize does not have to be inte*er 6or positive7 P99:=9.;2://Q 99.;; /3.30 /2.F /1.23 /0.29 //.F1

.T-.D


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.T-.Dathematical Operators

.!! @

Su"tract J =ii!e 'K ultiply 'L o*er 'M e'/' 'M2 means square!

Gou can use roun! "rac8ets to speci)y the or!erin *hich operations *ill "e per)orme!

Iote that prece!in/ the sym"ol K or L or M "y aN' means that the operator is applie! "et*eenpairs o) correspon!in/ elements o) ectors o)matrices

.T-.D


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5#

.T-.DCom"inin/ this *ith metho!s )rom .ccessin/ atri< ;lements

/ies *ay to more use)ul operations

HH results + zeros3, 5

HH results, 14 + ran!3, 4

HH results, 5 + results, 1 @ results, 2 @ results, 3 @results, 4

or

HH results, 5 + results, 1 'L results, 2 'L results, 3 'L

results, 4

.T-.D


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5$

.T-.D-o/ical Operators reater Than H

-ess Than P reater Than or ;qual To H+

-ess Than or ;qual To P+

Qs ;qual ++

Iot ;qual To R+

8or eample" &ou can find data that is above a certain limit:

r - results6:"/7

ind - r ;.9

Boolean Operators:

AND:

OR: 5

NO#:

.T-.D


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.T-.D #here are a number of special functions that provide useful

constants

pi - 0./@/139F1


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.T-.D #he plot function can be used in different wa&s:

plot6data7

plot6" &7

plot6data" r.=7

)n the last eample the line st&le is defined

!olour: r" b" *" c" H" & etc.

$oint st&le: . J \ o etc. (ine st&le: = == : .=

. "asic plotHH < +

(('12Lpi9

HH y + sin


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.T-.D

>> x = [0:0.1:2*pi];>> y = sin(x);

>> plot(x, y, '*!')

>> "ol# on

>> plot(x, y*2, $r.!')

>> title('%in &lots');

>> leen#('sin(x)', '2*sin(x)');

>> axis([0 .2 !2 2])

>> xlael($x);

>> ylael($y);

>> "ol# o 0 1 2 3 4 5 6-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Sin Plots

x

y

sin(x)

2*sin(x)

.T-.D


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.T-.D 8or command

'se a for loop to repeat one or more statements

#he end He&word tells Matlab where the loop finishes

ou control the number of times a loop is repeated b& definin* thevalues taHen b& the inde variable

#his uses the colon operator a*ain" so inde values do not needto be inte*er

8or eample for i - /:@ a6i7 - i \ 9 end

.T-.D


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.T-.D #he counter can be used to inde different rows or columns

E.*.

results - rand6/;"07 for i - /:0

m6i7 - mean6results6:" i77

end

..althou*h &ou could do this in one step

m - mean6results7Z

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.T-.D #he if command is used with lo*ical operators

A*ain" the end command is used to tell Matlab where the statement

ends. 8or eample" the followin* code loops throu*h a matri performin*

calculations on each column

for i - /:size6results" 97

m - results6:" i7

if m / do somethin*

else

do somethin* different

end

end

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optimization techiniques