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KAEDAH
TERBAHAGI
NEWTON(Newtons Divided Difference Polynomial Method)(Newtons Divided Difference Polynomial Method)
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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
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NEWTONS DIIDED DI!!E"EN#E $ET%OD
Daripada data& diper'leh:
0 0 0
1 0
1 1 1 01 0
1 02 1
( )
( ) ( )[ , ]
( ) ( )( ) ( )
x x b f x
f x f xx x b f x x
x x
f x f xf x f x
= =
= = =
3
2 1 1 02 2 2 1 0
2 0
1 1 0
[ , , ]
[ , , , ]n n n n
x x b f x x xx x
x x b f x x x x
= = =
= =
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Diberi yang mana),,( 00 yx ),,( 11 yx
)()( 0101 xxbbxf +=
)( 00 xfb =
INTERPOLASI NEWTON LINEAR
4
01
01
1)()(
xxxfxfb
=
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CONTO !(")
%ala(u menaik )ebuah r'ket diberikan )ebagai *ung)i ma)a
)eperti dalam +adual 1, #ari hala(u pada t#!$ )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
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,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 +=
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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
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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
=
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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 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
el'ity 2), time data *'r the r'ket e3ample
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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 ++=
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)( 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=
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))(()()( 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 %
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!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
=
==
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Polinomial peringkat ketiga, diberi ),,( 00 yx ),,( 11 yx ),,( 22 yx dan ),,( 33 yx
adalah
))()(](,,,[
))(](,,[)](,[][)(
2100123
1001200103
xxxxxxxxxxf
xxxxxxxfxxxxfxfxf
+
++=
!O"$456 B67I INTE"8O56SI NEWTON 94BI9
14
0b
0x )( 0xf 1b
],[ 01 xxf 2b
1
x )(1
xf ],,[012
xxxf
3
b
],[ 12 xxf ],,,[ 0123 xxxxf
2x )( 2xf ],,[ 123 xxxf
],[ 23 xxf
3x
)( 3xf
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CONTO !(C)
%ala(u menaik )ebuah r'ket diberikan )ebagai *ung)i ma)a
)eperti dalam +adual 1, #ari hala(u pada t#!$ )aat menggunakan
kaedah bea terbahagi Ne.t'n /interp'la)i 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
el'ity 2), time data *'r the r'ket e3ample
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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
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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
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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
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JADUAL PERBANDINGAN
Peringkat
Polinomial
1 2 3
v(t=16) 393.69 392.19 392.06
m s
Ralat MutlakRelatif
---------- 0.38247 % 0.033158 %
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""*""* 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=
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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
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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 =
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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
= =
==
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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
=
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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