Kuliah 4 Interpolasi-NewtonDividedDifference

7/24/2019 Kuliah 4 Interpolasi-NewtonDividedDifference

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KAEDAH

TERBAHAGI

NEWTON(Newtons Divided Difference Polynomial Method)(Newtons Divided Difference Polynomial Method)


2/26

Bentuk umum:

0 1 0 2 0 1 0 1 1( ) ( ) ( )( ) ... ( )( )...( )n n nf x b b x x b x x x x b x x x x x x = + + +

0 1, ,... nb b b

Data yang diberikan digunakan untuk mendapatkan nilai-nilaipekali;

2

Data yang diberi adalah

0 0 1 1( , ( )),( , ( ))...( , ( ))n nx f x x f x x f x n + 1 data


3/26

NEWTONS DIIDED DI!!E"EN#E $ET%OD

Daripada data& diper'leh:

0 0 0

1 0

1 1 1 01 0

1 02 1

( )

( ) ( )[ , ]

( ) ( )( ) ( )

x x b f x

f x f xx x b f x x

x x

f x f xf x f x

= =

= = =

3

2 1 1 02 2 2 1 0

2 0

1 1 0

[ , , ]

[ , , , ]n n n n

x x b f x x xx x

x x b f x x x x

= = =

= =


4/26

Diberi yang mana),,( 00 yx ),,( 11 yx

)()( 0101 xxbbxf +=

)( 00 xfb =

INTERPOLASI NEWTON LINEAR

4

01

01

1)()(

xxxfxfb

=


5/26

CONTO !(")

%ala(u menaik )ebuah r'ket diberikan )ebagai *ung)i ma)a

)eperti dalam +adual 1, #ari hala(u pada t#!$ )aat menggunakan

kaedah bea terbahagi Ne.t'n /interp'la)i linear0 ,

t v(t)

s m/s

%

0 0

10 227.04

15 362.78

20 517.3522.5 602.97

30 901.67

+adual 1

el'ity 2), time data *'r the r'ket e3ample


6/26

,15=t 78.362=tv

0 1 0 2 0 1 0 1 1( ) ( ) ( )( ) ... ( )( )...( )n n nf x b b x x b x x x x b x x x x x x = + + +

$

,201 =t 35.517)( 1 =tv

)( 00 tvb = 78.362=

01

01

1

)()(

tt

tvtv

b

= 914.30=

)()( 010 ttbbtv +=


7/26

10 12 14 16 18 20 22 24350

400

450

500

550517.35

362.78

y s

f range( )

f x desired( )

x s1

10+x s0

10 x s range, x desired,

)()( 010 ttbbtv +=

),15(914.3078.362 += t 2015 t

At 16=t

)1516(914.3078.362)16( +=v

69.393= m/s

7


8/26

INTERPOLASI NEWTON KUADRATIK

Diberi ),,( 00 yx ),,( 11 yx dan ),,( 22 yx memenuhi interpolasi kuadratik.

))(()()( 1020102 xxxxbxxbbxf ++=

)( 00 xfb =

8

01

011

)()(xx

xfxfb

=

02

01

01

12

12

2

)()()()(

xx

xx

xfxf

xx

xfxf

b

=


9/26

CONTO !(&)



kaedah bea terbahagi Ne.t'n /interp'la)i kuadratik0 ,

t v(t)

s m/s

'

0 0

10 227.04

15 362.78

20 517.3522.5 602.97

30 901.67

+adual 1



10/26

200

250

300

350

400

450

500

550517.35

227.04

y s

f range( )

f x desired( )

10

2010 xs

range, xdesired

,

0 0

1 1

2 2

10, ( ) 227.04

15, ( ) 362.78

20, ( ) 517.35

t v t

t v t

t v t

= =

= =

= =

))(()()( 1020102 xxxxbxxbbxf ++=


11/26

)( 00 tvb =

04.227=

01

01

1

)()(

tt

tvtvb

=

1015

04.22778.362

=

148.27=

0112 )()()()( tvtvtvtv

04.22778.36278.36235.517

11

02

01122

ttttttb

=

102010151520

=

10

148.27914.30 =

37660.0=


12/26

))(()()( 102010 ttttbttbbtv ++=

),15)(10(37660.0)10(148.2704.227 ++= ttt 2010 t

Pada ,16=t

)1516)(1016(37660.0)1016(148.2704.227)16( ++=v 19.392= m/s

Ralat mutlak relatif, di antara keputusan polinomial peringkat pertamaa

12

dan peringkat kedua adalah

a 100x

19.392

69.39319.392 =

= 0.38247 %


13/26

!O"$456 B67I INTE"8O56SI NEWTON

946D"6TI9))(()()( 1020102 xxxxbxxbbxf ++=

yang mana

)(][ 000 xfxfb ==

01 )()( xfxf

!3

+ika dituli) )emula& maka

))(](,,[)](,[][)( 1001200102 xxxxxxxfxxxxfxfxf ++=

01

011 ,

xx

xx

==

02

01

01

12

12

02

0112

0122

)()()()(

],[],[],,[

xx

xx

xfxf

xx

xfxf

xx

xxfxxfxxxfb

=

==


14/26

Polinomial peringkat ketiga, diberi ),,( 00 yx ),,( 11 yx ),,( 22 yx dan ),,( 33 yx

adalah

))()(](,,,[

))(](,,[)](,[][)(

2100123

1001200103

xxxxxxxxxxf

xxxxxxxfxxxxfxfxf

+

++=

!O"$456 B67I INTE"8O56SI NEWTON 94BI9

14

0b

0x )( 0xf 1b

],[ 01 xxf 2b

1

x )(1

xf ],,[012

xxxf

3

b

],[ 12 xxf ],,,[ 0123 xxxxf

2x )( 2xf ],,[ 123 xxxf

],[ 23 xxf

3x

)( 3xf


15/26

CONTO !(C)



kaedah bea terbahagi Ne.t'n /interp'la)i kubik0 ,

t v(t)

s m/s

!%

0 010 227.04

15 362.78

20 517.3522.5 602.97

30 901.67

+adual 1



16/26

8r'*il hala(u adalah

))()(())(()()( 2103102010 ttttttbttttbttbbtv +++=

Data yang menghampiri nilai adalah16=t

,100 =t 04.227)( 0 =tv

,151 =t 78.362)( 1 =tv

!$

,2 .2

,5.223 =t 97.602)( 3 =tv


17/26

0b

100 =t 04.227 1b

148.27 2b

,151 =t 78.362 37660.0 3b

914.30 3104347.5 x

,202 =t 35.517 44453.0

248.34

17

b0 = 227.04; b1= 27.148; b2= 0.37660; b3= 5.4347*10-3

,5.223 =t 97.602


18/26

Maka

))()(())(()()( 2103102010 ttttttbttttbttbbtv +++=

)20)(15)(10(10*4347.5

)15)(10(37660.0)10(148.2704.2273

+

++=

ttt

ttt

Pada t = 16

)2016)(1516)(1016(10*4347.5)1516)(1016(37660.0)1016(148.2704.227)16(

3+

++=

v

06.392= m/s

18

Ralat mutlak relatif adalah

a 100x

06.392

19.39206.392 =

= 0.033158 %

a


19/26

JADUAL PERBANDINGAN

Peringkat

Polinomial

1 2 3

v(t=16) 393.69 392.19 392.06

m s

Ralat MutlakRelatif

---------- 0.38247 % 0.033158 %

19


20/26

""*""* D"+P"D"D"+P"D" PPO,+-O,+- "-"."-".

Dapatkan (arak yang dilalui 'leh r'ket ter)ebut pada ma)a

t#!!s hin//a t#!$s

t v(t)

s m/s

20

10 227.04

15 362.78

20 517.35

22.5 602.97

30 901.67

+adual 1


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)20)(15)(10(10*4347.5

)15)(10(37660.0)10(148.2704.227)(

3 +

++=

ttt

ttttv

5.2210 t

32 0054347.013204.0265.212541.4 ttt +++= 5.2210 t

Maka

( ) ( ) ( )=16

11

1116 dttvss

21

dtttt )0054347.013204.0265.212541.4( 32

16

11

+++= 16

11

432

40054347.0

313204.0

2265.212541.4

+++=

tttt

m1605=


22/26

32 0054347.013204.0265.212541.4)( ttttv +++=

( )32 0054347.013204.0265.212541.4)()( tttdt

dtv

dt

dta +++==

P0C.T"N D"+P"D" PO,+- "-".

Dapatkan peutan r'ket pada t1


23/26

BBENTUKENTUK UUMUMMUM

Diberi )1( +n data iaitu ( ) ( ) ( ) ( )nnnn yxyxyxyx ,,,,......,,,, 111100 sebagai

))...()((....)()( 110010 +++= nnn xxxxxxbxxbbxf

yang mana

xb =

],[ 011 xxfb =

],,[ 0122 xxxfb =

],....,,[ 0211 xxxfb nnn =

],....,,[ 01 xxxfb nnn =

23


24/26

NEWTONS DIIDED DI!!E"EN#E $ET%ODdan

( ) ( )[ , ]

( ) ( ) ( ) ( )

[ , ] [ , ]

i j

i j

i j

i j j k

i j j k i j j k

f x f xf x x

x x

f x f x f x f x

x x x x f x x f x x

=

24

1 1 1 2 1 0

1 1 00

, ,

[ , , ] [ , , ]

[ , , , ]

i j k

i k i k

n n n n

n nn

x x x x

f x x x f x x x xf x x x x

x x

= =

==


25/26

NEWTONS DIIDED DI!!E"EN#E $ET%OD

0 1 0 0 2 1 0 0 1

1 1 0 0 1 1

maka diperoleh:

( ) ( ) [ , ]( ) [ , , ]( )( )

[ , , , ]( )( ) ( )

n

n n n

n

f x f x f x x x x f x x x x x x x

f x x x x x x x x x x

= + +

+

2%

0 1 1 0 0 1 1

1

, , ,k k kk

=


26/26

0

1

( )

( )

( )

f x

f x

f x

0

1

x

x 1 0[ , ]f x x

[ , ]f x x x

x [ , ]i jf x x [ , , ]i j kf x x x

26

2

3

( )

( )

( )n

f x

f x

f x

2

3

n

x

x

x

2 1

3 2

1

,

[ , ]

[ , ]n n

f x x

f x x

3 2 1

1 2

[ , ]

[ , , ]n n n

f x x x

f x x x

1 0[ , , ]n nf x x x

Kuliah 4 Interpolasi-NewtonDividedDifference

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