Metnum-Teori Integrasi Trapesium Simpson

8/18/2019 Metnum-Teori Integrasi Trapesium Simpson

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1

POKOK BAHASANINTEGRASI NUMERIK

-Metode Trapesium

- Metode Simpson


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Pengantar•

Pengintegralan numerik merupakan alat ataucara yang digunakan oleh ilmuwan untukmemperoleh awa!an hampiran "aprok#ima#i$dari pengintegralan yang tidak dapat

di#ele#aikan #ecara analitik%• Metode integra#i numerik di!edakan&

'% Metode Newton Cotes & dida#arkan padapenggantian (ung#i yang komplek# atau ta!el

data dengan (ung#i polinomial #ederhana yangmudah diintegralkan%

)% Metode Gauss & untuk mengintegralkan(ung#i "tidak ta!el data$


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Dasar Pengintegralan Numerik

Penjumlahan berbobot dari nilai ungsi

x 0 x 1 x n x n-1 x

f ( x ) )(...)()(

)()(

1100

0

nn

i

n

i

i

b

a

x f c x f c x f c

x f cdx x f

+++=

≈∑∫ =


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Melakukan pengintegralan pada !agian*!agian kecil+

#eperti #aat awal !elaar integral , penumlahan !agian*!agian%

Metode Numerik hanya menco!a untuk le!ih cepat danle!ih mendekati awa!an ek#ak%

Dasar PengintegralanNumerik

0

2

4

6

8

10

12

3 5 7 9 11 13 15


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Formula Newton-Cotes

- Berdasarkan pada

Nilai hampiran f ( x ) dengan polinomial

Dasar PengintegralanNumerik

dx x f dx x f I b

a n

b

a ∫≅ )()(

n

n

1n

1n10n x a x a x aa x f )(


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f n ( x ) !i#a (ung#i linear

f n ( x ) !i#a (ung#i kuadrat


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f n ( x ) !i#a uga (ung#i ku!ik

atau polinomial yang le!ihtinggi


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Polinomial dapat dida#arkan padadata


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!ormula Newton"Cotes Aturan Trape#ium & -inier

Aturan Simp#on.# '/0 & Kuadrat Aturan Simp#on.# 0/1 & Ku!ik


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Aturan #ra$esium

A$roksimasi garis lurus %linier&

x 0 x 1 x

f ( x )

L(x)

)()(

)()()()(

10

1100i

1

0i

i

b

a

x f x f 2

h

x f c x f c x f cdx x f

∑

=


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Aturan Kom$osisi#ra$esium

x 0 x 1 x

f ( x )

x 2h h x 3h h x 4

)()()()()(

)()()()()()(

)()()()(

n1ni 10

n1n2110

x

x

x

x

x

x

b

a

x f x f 2 x 2f x f 2 x f

2

h

x f x f 2

h x f x f

2

h x f x f

2

h

dx x f dx x f dx x f dx x f

n

1n

2

1

1

0

∫

n

abh

=


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function f = example1(x)

% a = 0, b = pi

f=x.^2.*sin(2*x);


dx x 2sin x 02

)(


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» a=0; b=pi; dx=(b-a)100;

» x=a!dx!b; "=example1(x);

» #=t$ap(example1,a,b,1)

# =

-&.''0e-01


# =

-1.2&e-01» #=t$ap(example1,a,b,)

# =

-&.+'+

» #=t$ap(example1,a,b,+)

# =

-.'+

» #=t$ap(example1,a,b,1)# =

-.+'12

» #=t$ap(example1,a,b,&2)

# =

-.1+

» #=t$ap(example1,a,b,)

# =

-.&0+

» #=t$ap(example1,a,b,12+)

# =

-.&&+


# = -.&


# =

-.&'


# =

-.&+» =uad+(example1,a,b)

=

-.&+ MATLABfunction



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n = 2

I = -1.4239 e-15

Exact = -4. 9348

dx x 2sin x 0

2 )(∫


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n = 4

I = -3.8758

E!a = -4. 9348

dx x 2sin x 0

2 )(∫


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n = 8

I = -4."785

E!a = -4. 9348

dx x 2sin x 0

2 )(∫


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n = 1"

I = -4.8712

E!a = -4. 9348

dx x 2sin x 0

2 )(∫


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•

Hitung integral dari


dx xe I 4

0

x 2

∫

#..)().().(

).().()(.$

#..)().(

)().()().(

)().()(.$

#..)()(

)()()($

#..)()()($

#..)()($

66 2 9553554 f 753 f 253 f 2

50 f 2250 f 20 f 2

h I 250h16 n

5010 76 57644 f 53 f 2

3 f 252 f 22 f 251 f 2

1 f 250 f 20 f 2

h I 50h8n

7139 7972884 f 3 f 2

2 f 21 f 20 f 2

h I 1h4n

75132 23121424 f 2 f 20 f 2

h I 2h2n

12357 66 23847 4 f 0 f 2

h I 4h1n

At K i i


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» x=0!0.0!; "=example2(x);

» x1=0!!; "1=example2(x1);

» x2=0!2!; "2=example2(x2);» x&=0!1!; "&=example2(x&);

» x=0!0.!; "=example2(x);

» /=plot(x,",x1,"1,-*,x2,"2,$-s,x&,"&,c-o,x,",m-d);

» set(/,ineidt3,&,4a$5e$6i7e,12);

» xlabel(x); "label("); title(f(x) = x exp(2x));

» #=t$ap(example2,0,,1)# = 2.&++e800» #=t$ap(example2,0,,2)# = 1.212e800» #=t$ap(example2,0,,)

# = '.2+++e800&» #=t$ap(example2,0,,+)# = .'+e800&» #=t$ap(example2,0,,1)# = .&e800&


At K i i


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dx xe I 4

0

x 2

∫


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21

POKOK BAHASANINTEGRASI NUMERIK

-

Metode Trapesium- Metode Simpson


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Aturan Sim$son

Menggunakan $olinomial order

lebih tinggi untuk menghubungkantitik"titik'

Sim$son ()* digunakan $olinomial

order + %$arabola& ,ang melalui *titik'

x 0 x 1 x

f ( x )

x 2h h

L( x )


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Aturan Sim$son

Aturan Sim$son *)- . Bila terda$at +

titik tambahan dengan jarak ,angsama antara %a& sd %b&/ makakeem$at titik tsb dihubungkandengan $olinomial order tiga'

x 0 x 1 x

f(x)

x 2h h

L(x)

x 3h


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Aturan Sim$son ()*

A$roksimasi dengan ungsi

$arabola

x 0 x 1 x

f ( x )

x 2h h

L( x )

)()()(

)()()()()(

210

221100i

2

0i

i

b

a

x f x f 4 x f 3

h

x f c x f c x f c x f cdx x f

∑

=


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Ingatkah Interpola#i Polinomial-agrange2

• Secara umum order n&

3rder )&


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Aturan Sim$son ()*.$olinomial lagrange order + ,ang melalui * titik

x 0

x 1

x

f ( x )

x 2

h h

L( x )

=

=

===

=

=

1 x x

0 x x

1 x x

h

dx d

h

x x

2

abh

2ba x b x a x let

x f x x x x

x x x x

x f x x x x

x x x x x f x x x x

x x x x x L

2

1

0

1

120

2

1202

10

1

2101

200

2010

21

ξ

ξ

ξ

ξ$$

$$

)())((

))((

)())((

))(()())((

))(()(

)()(

)()()()(

)(21

2

0

x f 2

1 x f 1 x f

2

1 L

=

ξ

ξξ


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Aturan Sim$son ()*

)(

)(

)()()(

)(

)( 212

0 x f 2

1

x f 1 x f 2

1

L

=

ξ

ξξ

1

1

23

2

1

1

3

1

1

1

23

0

1

12

1

0

2

1

1

10

1

1

)23

(2

)(

)3

()()23

(2

)(

)1(2)()1)(

)1(2

)()()(

−

−−

−

−−

++

−+−=

++−+

−=≈

∫ ∫

∫ ∫ ∫

ξ ξ h x f

ξ ξ h x f

ξ ξ h x f

dξ ξ ξ

h

x f dξ ξ ( h x f

dξ ξ ξ h

x f dξ Lhdx x f b

a

ξ

)()()()( 210b

a

x f x f 4 x f 3

hdx x f


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Aturan Kom$osisi Sim$son

x 0 x 2 x

f ( x )

x 4h h x n-2h x n

%...

h x 3 x 1 x n-1

n

abh

=


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• n = 2, h = 2

• n = 4, h = 1

Aturan Kom$osisiSim$son

dx xe I 4

0

x 2

∫

#..

)()()(

)()()()()(

708 9755670

e4e34e22e403

1

4 f 3 f 42 f 21 f 40 f 3h I

86 42

#..)(

)()()(

96 57 4118240e4e24032

4 f 2 f 40 f 3

h I

84


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Aturan Sim$son *)-

A$roksimasi dengan ungsi kubik

x 0 x 1 x

f(x)

x 2h h

L(x)

x 3h

)()()()(

)()()()()()(

3210

33221100i

3

0i

i

b

a

x f x f 3 x f 3 x f 8

h3

x f c x f c x f c x f c x f cdx x f

∑

=


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Ingatkah Interpola#i Polinomial-agrange2

• Secara umum order n&

3rder 0&


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E&&o& 'eena*an

Aturan Sim$son *)-

x 0 x 1

f(x)

x 2h h

L(x)

x 3h

)())()((

))()(()(

))()((

))()((

)())()((

))()(()(

))()((

))()(()(

3

231303

2102

321202

310

1

312101

3200

302010

321

x f x x x x x x

x x x x x x x f

x x x x x x

x x x x x x

x f x x x x x x

x x x x x x x f

x x x x x x

x x x x x x x L

=

)()()()( 3210

b

a

b

a

x f x f 3 x f 3 x f 8

h3

3abh ; L(x)dx f(x)dx

=

∫

3

abh ; f

6480

ab f h

80

3 E 4

545

t

= )()(

)( )()(

ξ


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Aturan Sim$son ()*

Aturan Sim$son *)-

Aturan Sim$son *)-

dx xe4

0

x 2

∫

#..

..

.)(

)()()(

96 57 926 5216

4118240926 5216

4118240e4e240

3

2

4 f 2 f 40 f 3

hdx xe I

84

4

0

x 2

=

=

∫

71 !30926 !5216

209 !6819926 !5216

209 !6819832 !11923 )33933 !552( 3 )18922 !19( 308

)4"3( 3

)4( f )3

8( f 3 )

3

4( f 3 )0( f

8

h3dx xe I

4

0

x 2

=

=

∫

At K i i


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function # = 6imp(f, a, b, n)

% inte$al of f usin composite 6impson $ule

% n must be e9en

3 = (b - a)n;

6 = fe9al(f,a);

fo$ i = 1 ! 2 ! n-1 x(i) = a 8 3*i;

6 = 6 8 *fe9al(f, x(i));

end

fo$ i = 2 ! 2 ! n-2 x(i) = a 8 3*i;

6 = 6 8 2*fe9al(f, x(i));

end

6 = 6 8 fe9al(f, b); # = 3*6&;


At Si


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Aturan Sim$son

Aturan Kom$osisi


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MA#0AB !un1tion+ t$ap7

» x=:0 1 1. 2.0 2. &.0 &.& &. &.+ &. .0

x =


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Metnum-Teori Integrasi Trapesium Simpson

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