derivatif numerik
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8/16/2019 Derivatif Numerik
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Rabu, 9 November
2015
Derivatif Fungsi Secara Numerik
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Derivatif SecaraNumerik 1. Forward difference
(Selisih Maju)Deret Talor !
( ) ( ) ( ) +′+=+ h x f x f x f iii 1
3
i1i2i3i
i
///
h
x f x f 3 x f 3 x f x f
)()()()()(
−+−= +++
2
i1i2i
i1ii
//
h
x f x f 2 x f h
x f x f x f
)()()(
)()()(
//
+−=
−=
++
+
h
x f x f x f i1i
i
/ )()()( −= +
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Derivatif SecaraNumerik
". #ackward difference(Selisih Mundur)
h
x f x f x f 1ii
i
/ )()()( −−=
3
3i2i1iii
///
h
x f x f 3 x f 3 x f x f
)()()()()( −−− −+−=
2
2i1ii
1iii
//
h
x f x f 2 x f
h
x f x f x f
)()()(
)()()(
//
−−
−
+−=
−=
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Derivatif Secara Numerik
$. %entered difference
(Selisih Tengah)
h2
x f x f x f 1i1i
i
/ )()()( −+
−=
3
2i1i1i2ii
///
h2
x f x f 2 x f 2 x f x f
)()()()()( −−++ −+−=
2
1ii1ii
//
h
x f x f 2 x f x f
)()()()( −+
+−=
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• Derivatif numerik baik metode selisih majumaupun mundur mempunyai error yangsebanding dengan derajat h. Artinya bahwaerror akan menurun secara linear denganmenurunnya h.
• Derivatif Numerik metode pusat mempunyaierror yang berbanding kuadrat denganmenurunnya h2, artinya bahwa error akan
menurun secara kuadratik terhadapmenurunnya h.
• Notasi Oh! and Oh2! menyatakan errorberderajat h dan h2.
Derivatif Secara Numerik
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High AccuracyDierentiation ormu!a"
"igh#accuracy $nite#di%erence formulas canbe generated by including additional termsfrom the &aylor series e'pansion.
An e'ample( "igh#accuracy forward#di%erence formula for the $rst derivative.
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Derivation# High$accuracy for%ar&$&ierence formu!a for f'()*
&aylor series e'pansion
)olve for f*'!
High$accuracyfor%ar&$&ierence
formu!a
)ubstitute the forward#di%erence appro'. of
f+'!
( ) ( ) ( ) ( )
+′′
+′+=+2
1!2
h x f
h x f x f x f iiii
( ) ( ) ( ) ( ) ( )212
2
34hO
h
x f x f x f x f iii
i +−+−
=′ ++
( ) ( ) ( ) ( )
( )hO
h
x f x f x f x f iii
i ++−
=′′ ++2
12 2
( ) ( ) ( ) ( ) ( )21
!2hOh
x f
h
x f x f x f iii
i +′′
−−
=′ +
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)imilar improved versions can be
developed for the backward and centered
formulas as well as for the appro'imations
of the higher derivatives.
Derivation# High$accuracy back%ar&$&ierence formu!a for f'()*
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Higher +r&er or%ar& Divi&e&Dierence
3
i1i2i3i4i
i
///
h2
x f 5 x f 18 x f 24 x f 14 x f 3 x f
)()()()()()(
−+−+−=
++++
2
i1i2i3ii
//
h
x f 2 x f 5 x f 4 x f x f
)()()()()(
+−+−=
+++
h2
x f 3 x f 4 x f x f i1i2i
i
/ )()()()(
−+−= ++
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Higher +r&er ack%ar& Divi&e&Dierence
h2
x f x f 4 x f 3 x f
2i1ii
i
/ )()()()(
−− +−=
3
4i3i2i1ii
i
///
h2
x f 3 x f 14 x f 24 x f 18 x f 5 x f
)()()()()()(
−−−− +−+−
=
2
3i2i1iii
//
h
x f x f 4 x f 5 x f 2 x f )()()()()( −−− −+−=
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Higher +r&er -entra! Divi&e& Dierence
h12
x f x f 8 x f 8 x f x f 2i1i1i2i
i
/ )()()()()( −−++ +−+−=
3
3i2i1i1i2i3i
i ///
h8
x f x f 8 x f 13 x f 13 x f 8 x f
x f
)()()()()()(
)(
+−−+++ +−+−+−=
2
2i1ii1i2i
i
//
h12
x f x f 16 x f 30 x f 16 x f x f
)()()()()()(
−−+= −+−+−=
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.unakan meto&e "e!i"ih ma/u, mun&ur, &an
tengah untuk memerkirakan &erivatif or&e"atu &ari#
&i ) 05 &engan h 05 &an 025 ("o!u"i
ek"ak $09125* or%ar& Dierence
ack%ar& Dierence
-ontoh# Derivatif +r&e Satu
2 .1 x 25 .0 x 5 .0 x 15 .0 x 1 .0 ) x ( f 234
=−=−
=
−
−=′=
=−=−
=−
−=′=
%5.26 ,155.1
25.0
925.063632813.0
5.075.0
)5.0()75.0()5.0( ,25.0
%9.58 ,45.15.0
925.02.0
5.01
)5.0()1()5.0( ,5.0
t
t
ε
ε
f f f h
f f f h
=−=−
=−
−=′=
=−=−
=−
−=′=
%7.21 ,714.025.0
10351563.1925.0
25.05.0
)25.0()5.0(
)5.0( ,25.0
%7.39 ,55.05.0
2.1925.0
05.0
)0()5.0()5.0( ,5.0
t
t
ε
ε
f f
f h
f f f h
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-ontoh# Derivatif +r&eSatu #
-entra!Dierence
%4.2 ,934.05.0
10351563.163632813.0
25.075.0
)25.0()75.0()5.0( ,25.0
%6.9 ,0.1
1
2.12.0
01
)0()1()5.0( ,5.0
t
t
=−=−
=−
−=′=
=−=−
=
−
−=′=
ε
ε
f f f h
f f f h
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or%ar& inite$&ivi&e& &ierence"
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ack%ar& 3nite$&ivi&e& &ierence"
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-entere& inite$Divi&e& Dierence"
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-ontoh# Derivatif +r&e Satu #
4enggunakan formu!a akura"i tinggi
(h025*#
xi-2= 0.0 f(0.0) = 1.2
xi-1= 0.25 f(0.25) = 1.103516
xi = 0.5 f(0.5) = 0.925 xi+1 = 0.75 f(0.75) = 0.63633
xi+2 = 1.0 f(1.0) = 0.2
or%ar& Dierence( ) ( ) ( ) ( )
( ) 8594.0)25.0(2
)925.0(3)6363281.0(42.05.0
234 12
−=−+−
=′
−+−=′ ++
f
h x f x f x f x f iii
i
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-ontoh# Derivatif +r&e Satu #
#ackward Difference
%entral Difference
( ) 8781.0)25.0(2
2.1)035156.1(4)925.0(35.0 −=
+−=′ f
( ) 9125.0)25.0(12
)035156.1(8)636328.0(82.05.0 −=
−+−=′ f
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e"imu!an
h & '." Forward(h") #ackward(h") %entered(h*)
+stimate -0.859375 -0.878125 -0.9125
, t, 5.82% 3.77% 0%
ormu!a&a"ar
h & '." Forward(h)
#ackward(h)
%entered(h")
+stimate -1.155 -0.714 -0.934
, t, 26.5% 21.7% 2.4%
ormu!aakura"itinggi
6rue va!ue# f'(05* $09125