derivatif numerik

8/16/2019 Derivatif Numerik

http://slidepdf.com/reader/full/derivatif-numerik 1/19

Rabu, 9 November

2015

Derivatif Fungsi Secara Numerik



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

/ )()()( −= +



Derivatif SecaraNumerik

". #ackward difference(Selisih Mundur)

h

x f x f x f 1ii

i

/ )()()( −−=

3

3i2i1iii

///

h


)()()()()( −−− −+−=

2

2i1ii

1iii

//

h

x f x f 2 x f

h

x f x f x f

)()()(

)()()(

//

−−

−

+−=

−=



Derivatif Secara Numerik

$. %entered difference

(Selisih Tengah)

h2

x f x f x f 1i1i

i

/ )()()( −+

−=

3

2i1i1i2ii

///

h2


)()()()()( −−++ −+−=

2

1ii1ii

//

h

x f x f 2 x f x f

)()()()( −+

+−=



• 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



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.



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 +′′

−−

=′ +



)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'()*



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

/ )()()()(

−+−= ++



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 )()()()()( −−− −+−=



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

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

−−+= −+−+−=



.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



-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



or%ar& inite$&ivi&e& &ierence"



ack%ar& 3nite$&ivi&e& &ierence"



-entere& inite$Divi&e& Dierence"



-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



-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



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

derivatif numerik

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