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Transcript of 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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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
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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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1$
-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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2$
-inear .l/e"ra
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3(
-inear .l/e"ra
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32
=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= +
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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.
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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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3$
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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4(
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.
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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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44
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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46
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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47
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
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
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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athTerminolo/ies)n a nutshell 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
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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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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.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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.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
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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
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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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.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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.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
.T-.D
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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