INTEGRASI NUMERIS Numerical Differentiation and Integration

Transkripsi

1 INTEGRASI NUMERIS Numericl Dieretitio Itegrtio

2 Itegrsi Numeris q Acu q Chpr, S.C., Cle R.P., 99, Numericl Methos or Egieers, E., McGrw-Hill Book Co., New York. Chpter 5 6, hlm

3 Dieresil, Derivti ( i Δ) y i Δy ( i ) Δy y i Δy y i Δ y i y i Δy Δ i i Δ i i Δ () () (c) ( Δ) ( ) i Δ ierece pproimtio i y lim Δ ( Δ) ( ) i Δ i i

4 Dieresil, Derivti Δy Δ ( Δ) ( ) i Δ i y lim Δ ( Δ) ( ) i Δ i peekt e (higg) ierece pproimtio erivti y yʹ ʹ ( ) erivti lju peruh y terhp

5 Dieresil 5 y y 5 slope y/ y/ 8 y

6 Itegrl y y 5.5 y 6 lus y y kelik ri proses me-ieresil-k lh meg-itegrl-k itegrsi ieresisi

7 Fugsi 7 q Fugsi-ugsi yg i-ieresil-k tu i-itegrl-k pt erup: q q q ugsi kotiu seerh: poliomil, ekspoesil, trigoometri ugsi kotiu kompleks yg tik memugkik iieresilk tu itegrlk secr lgsug ugsi yg ili-iliy isjik lm etuk tel [tulsi t vs ()]

8 8 Cr mecri ili itegrl ( ) cos.5si e.5 () A A A A () lus A i.5.5.5

9 Derivti 9 u u y ( ) ( ) v u y u y uv v u y v u u v y v v u u v y e e l sec t si cos cos si l l log cot csc csc t sec sec csc cot

10 Itegrl ( ) ( ) C C C C u v u u v uv v u > cos si l, l ( ) ( ) ( ) C C e e C e e C C t l l si cos

11 Metoe Itegrsi Newto-Cotes Metoe Trpesium Metoe Simpso Metoe Kurtur Guss

12 Persm Newto-Cotes q Strtegi q meggti ugsi kompleks rumit tu tulsi t eg yg muh utuk iitegrlk I ( ) ( ) ( )... poliomil tigkt

13 Persm Newto-Cotes ( ) ( ) Gris lurus (poliomil tigkt ) sg ugsi peekt. Kurv prol (poliomil tigkt ) sg ugsi peekt.

14 Persm Newto-Cotes ( ) Fugsi yg iitegrlk iekti eg uh gris lurus (poliomil tigkt ). Dpt pul ipki eerp kurv poliomil tigkt yg leih tiggi. Gris lurus (poliomil tigkt ) sg ugsi peekt.

15 5 Metoe Trpesium q Fugsi peekt utuk meghitug itegrl lh poliomil tigkt I ( ) ( ) q Seuh gris lurus pt iytk eg persm ( ) ( ) ( ) ( ) ( )

16 6 Metoe Trpesium I ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) error Metoe Trpesium

17 7 I Metoe Trpesium Peyelesi eksk.8 5 ( )..5 Metoe Trpesium ( ). 5 ( ). (.8). I (.8 ) E t ( 89% ) [error]

18 8 Metoe Trpesium () error q Error tu keslh q etuk trpesium utuk meghitug ili itegrl megik sejumlh esr porsi erh i wh kurv q Kutiiksi error p Metoe Trpesium E ( )( ) t ʹ ʹ ξ ξ lh titik i tr

19 9 Metoe Trpesium () error E ( ) ʹ ʹ ili rt-rt erivti keu:.8 ( ) ʹ ʹ ( ) 6.8 error: (.6)(.8 ). 56 orer o mgitue ili error ii sm eg orer o mgitue ili error terhp ili peyelesi eksk keuy sm t (sm-sm positi)

20 Trpesium multi pis q Peigkt kursi q selg igi meji sejumlh pis eg ler sergm h h

21 Trpesium multi pis () h () h. h

22 Trpesium multi pis h Jik ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) h h h I I ( ) ( ) ( ) i i h I ( ) ( ) ( ) ( )!!!! "!!! $! # $#" rt- rt tiggi ler I i i

23 Trpesium multi pis Error jumlh error p setip pis E t ( ) i ʹ ʹ ( ξ ) i ( ) ( ) ʹ ʹ ξ ʹ ʹ E E i i t ʹ ʹ setip kelipt jumlh pis, error megecil eg ktor kurt peigkt jumlh pis

24 8 6 Trpesium multi pis 5 ( ) I ( ). 65 %E t vs 8 I (ect solutio) ( ).8 (the trpezoil rule) h I E t %E t E % % % %.

25 5 Metoe Simpso The trpezoil rule q Fugsi peekt: poliomil tigkt q Peigkt keteliti pt ilkuk eg meigktk jumlh pis Simpso s rules q Fugsi peekt: poliomil: q tigkt : Simpso / q tigkt : Simpso /8

26 6 Metoe Simpso () () () () Simpso ⅓ Simpso ⅜

27 Metoe Simpso 7 q Poliomil tigkt tu q icri eg Metoe Newto tu Lgrge (liht mteri tetg curve ittig)

28 Simpso ⅓ 8 ( ) ( ) I Jik ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) I () iperoleh eg Metoe Lgrge ( ) ( ) ( ) [ ] h I h ( ) ( ) ( ) ( ) 6 I tu ( ) ( ) ξ 5 8 E t

29 Simpso ⅓ multi pis 9 ( ) ( ) ( ) I... ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) h h h I ( ) ( ) ( ) ( ) ( ) I i i i i,,6,,5 ( ) 5 8 E (estimsi error, rt-rt erivti ke-)

30 Simpso ⅜ ( ) ( ) I Jik ( )( )( ) ( )( )( ) ( ) ( )( )( ) ( )( )( ) ( ) ( )( )( ) ( )( )( ) ( ) ( )( )( ) ( )( )( ) ( ) I () iperoleh eg Metoe Lgrge ( ) ( ) ( ) ( ) [ ] 8 h I h ( ) ( ) ( ) ( ) ( )!!!!! "!!!! $! # $#" rt rt tiggi ler 8 I tu

31 Simpso ⅜ Error E t 8 h 5 ( ξ) tu E t ( ) 5 68 ( ξ)

32 Simpso ⅓ ⅜ 5 ( ) I.8 ( ). 65 (ect solutio) Metoe I E t Simpso ⅓ ( ) (7%) Simpso ⅜ ( ) (7%) Simpso ⅓ ( ) (%)

33 Pis tk sergm: Metoe Trpesium i i ( i ) I ( ) I ihitug eg Metoe Trpesium i setip pis: ( ) ( ) I h.8 h ( ) ( ) ( ) ( ) ( ) h 5

34 Pis tk sergm: Metoe Simpso i i ( i ) I ( ) I ihitug eg Metoe Simpso ⅓ Simpso ⅜: PR, ikumpulk miggu ep 5

35 5 Metoe Itegrsi Numeris Metoe Jumlh pis Ler pis Trpesium Trpesium multi pis > sergm tu tk-sergm Simpso ⅓ sergm Simpso ⅓ mul( pis gep (m, m,, ) sergm Simpso ⅜ sergm Kurtur Guss

36 6 Kurtur Guss ( ) ( ) error terllu esr upy megurgi error

37 Kurtur Guss 7 ( ) ( ) ( ) Kurtur Guss Titik: Guss-Legere ( ) ( ) ( ) c c I ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) c c c c c c c c! c,c,, :!!ukows

38 8 Kurtur Guss ( ) ( ) ( ) ( )

39 Kurtur Guss 9 ( ) ( ) ( ) ( )

40 Kurtur Guss ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) c c c c c c c c ( ) ( ) c c I c c ( ) ( ) I

41 Kurtur Guss Utuk ts itegrsi ri ke : imil sumsi sutu vriel yg pt ihuugk eg vriel sli lm sutu relsi lier jik ts wh,, erkit eg jik ts ts,, erkit eg ( ) ( ) ( ) ( ) ( )

42 Kurtur Guss 5 ( ) I ( ). 65 (ect solutio) Peyelesi eg Metoe Kurtur Guss: (.8 ) (.8 ).8...

43 Kurtur Guss 5 ( ). 9 5(.. ) (.. ) 675(.. ) (.. ) (.. ) 5. ( ).567 ( ( ) I

44