Persamaan (1.4) adalah persamaan dari deret Mac Laurin. Persamaan (1.1) biasa dituliskan dengan mensubstitusikan x dengan x-x 0, sehingga :

Transkripsi

1 Fsk Komputs I DERET TAYLOR. Deret Tlor Deret Tlor memegg per g sgt petg dlm lss umerk. Deg deret Tlor kt dpt meetuk l sutu ugs d ttk jk l ugs d ttk 0 g berdekt deg ttk dkethu. Ur deret Tlor dsektr o dtk deg tu : dm, 0 0! d 0 d Deret Mc Lur Deret Mc Lur dlh betuk khusus dr deret Tlor. Dm dggp bhw ttk 0 = 0 sehgg persm. k berubh mejd : dm, 0! 4 d d Persm.4 dlh persm dr deret Mc Lur. Persm. bs dtulsk deg mesubsttusk deg - 0, sehgg : 0 0 7! Sehgg Persm deret Mc Lur k mejd : 0! 8 Cotoh : Htuglh deret Mc Lur utuk S Avd-06

2 Fsk Komputs Jwb : = s ; 0 = 0 = cos ; 0 = = -s ; 0 = 0 = -cos ; 0 = - V = s ; V 0 = 0 V = cos ; V 0 = Dr deret Mc Lur : 0! 0 0 0! 0 0! 0! 0! V 0 4! 4 V 0 5! 5 Sehgg S mejd : !!!.! 0. 4! 4. 5! 5 5 7! 5! 7! Lth :. But deret Mc Lur utuk ugs cos d t. But deret Tlor utuk e d sektr =0. Progrm komputer Progrm komputer dr ugs s utuk deret Mc Lur tersebut jk dtk dlm bhs Vsul C++ dlh : //********************************************************* // Meghtug ugs sus deg deret McLur // Compler : Vsul C++ //********************************************************* #clude <ostrem.h> #clude <mth.h> vod m double eps=e-5; double,s,pem,pe,s,del; t,j,m,k,td; Avd-06

3 Fsk Komputs cout << "Sudut derjt = "; c >> ; =/57.; cout << "s = " << s ; //l sus dr ugs pustk Vsul C++ del=0; cout <<edl<<edl; cout << "Meurut Ur deret Tlor : "<<edl; m=0; whle del > eps m=m+; s=0; j=0; or =;<=*m-;++ % == // tu mod, == j=j+; j % == 0 td = -; else td=; pem=; pe=; or k=;k<=;k++ pem=pem*; pe=pe*k; s=s+td*pem/pe; m> del= bss-s; s=s; cout << "Jumlh suku = "<<m <<edl; cout << "s = " << s <<edl; Avd-06

4 Fsk Komputs Lth :. Butlh progrm komputer dr deret McLur utuk ugs ep : ep!!. Butlh progrm komputer dr deret McLur utuk ugs cos : cos! 4 4! 6 6! Avd-06 4

5 Fsk Komputs II AKAR-AKAR PERSAMAAN Akr sutu persm merupk ttk potog ugs deg sumbu : 0 kr Gmbr.. Akr dr persm Akr dr sutu persm serg dpk utuk meelesk mslh sk sepert : -Megethu volume gs pd sutu tek tertetu P P st P=P berp l V V V Gmbr.. Dgrm P-V gs del -Megethu jrk jtuh peluru g dtembkk deg rh d kecept tertetu Vt st =0 berp l =? Gmbr.. Gerk Peluru. Memperkrk Letk Akr Jk dberk ugs deg letk kr d 0 sepert d gmbr.4. Utuk meetuk letk dr kr ugs dpt dtempuh lgkh - lgkh sbb : Avd-06 5

6 Fsk Komputs 0 Gmbr.4. Akr dr persm -tetuk sebuh ttk pd sumbu dlm derh berlku ugs msl -tetuk ttk l msl, dm = - h -bl > 0, berrt d berd pd phk g sm terhdp 0. -tetuk ttk l msl : = -h -bl > 0, berrt d berd pd phk g sm terhdp 0. h d sm-sm berd dsebelh k 0 0 Gmbr.5. Perkr kr dr persm -peetu ttk-ttk dterusk smp dperoleh : b < 0 egt 0 b d b terletk berseberg terhdp ttk potog 0 berd d sebelh kr 0 b berd d sebelh k 0 Gmbr.6. Perkr kr dr persm -sehgg derh pecr kr-kr k dpersempt dtr d b -lgkh-lgkh dts dulg pd derh : b Avd-06 6

7 Fsk Komputs ctt : Bl kt meetuk sutu rh d smp bts derh tertetu tdk dtemuk ttk potog 0, mk kt cob sekl lg utuk rh g berlw Cotoh : Tetuk ttk potog terhdp sumbu. Dm betuk persm : = utuk 0 6, h h 6 9 5,5 5,5,65 4,5 4,5 -,5 Dr tbel 5,5 4,5 < 0, sehgg ttk potog terletk tr = 4,5 d = 5,5. Metode Newto-Rpsho Peetu kr-kr sutu persm dpt dcr deg memk rumus ters Newto Rpsho g dturuk dr ur deret Tlor. Jk h dlbtk smp suku deg turu pertm mk k ddptk ters Newto-Rpsho orde-,d bl g dlbtk smp suku deg turu kedu mk dsebut ters Newto- Rpsho orde-, d seterus. Ur deret Tlor d sektr 0 dtk : 0 0! 0 0 0! 0 0 0! orde - orde - 0 0!.. Metode Newto-Rpsho orde- Ur deret Tlor dr ugs d sektr 0 smp suku deg turu pertm dtk : = Jk dsubsttusk deg + d 0 dsubsttusk deg mk : + = Bl + dlh kr dr mk + = 0. Sehgg persm. k mejd : Avd-06 7

8 Fsk Komputs ' Persm. dsebut deg persm ters Newto-Rpsho orde-. Nl + dpt dcr dr sutu ttk dpt dcr dr ttk sebelum. Jd bl sutu ttk wl 0 dkethu mk deg persm. dpt dcr,, d seterus smp dtemuk kr dr persm tu ugs g bersgkut l g meebbk = 0. Peetu 0 memegg per g mt petg kre cept lmbt peetu kr dtetuk oleh pegmbl ttk wl 0. Peetu ttk wl 0 hrus dtetuk sedekt mugk deg kr sehgg megurg bk ters. Hl dpt dlkuk deg metode memperkrk letk kr sub bb.. Cotoh : Tetuklh kr = deg metode Newto-Rpsho orde-. Jwb : = + msl kt mbl ttk wl 0 = ,000 9,000 5,000 5,400 5,400,04 7,680 5,00 5,00 0,44 4,80 5,009 5,009 0,06 4,070 5, ,000 0,000 4,000 5,000 Dr tbel dts dpt dlht utuk ketelt gk dbelkg kom 4 = 5 = 5,000. D l = 0,000. Jd kr dr dlh = 5,000.. Metode Newto-Rpsho orde- Jk kt melbtk smp suku deg turu kedu pd ur deret Tlor mk k meghslk ters Newto Rpsho orde-. Peuru persm dpt dtetuk deg cr berkut. Dr persm umum deret Tlor pers.. 0 0! 0 0 0! 0 0 0! 0 0! deg mesubsttus deg +, d 0 deg ddptk Avd-06 8

9 Avd-06 9 Fsk Komputs! Jk + kr dr mk + = 0 Sehgg ' '' ' ' '' '.. Persm. merupk persm ters Newto-Rpsho orde-. Cotoh :. Tetuklh kr = deg metode Newto-Rpsho orde-. Jwb : = + ; = 6 - msl kt mbl ttk wl 0 = 6 N F + 0 6,000 9,000 5,000 4,000 5,67 5,67 4

10 Fsk Komputs. Stu mole gs del suhu 00 0 K, volume dperbesr pd suhu tetp soterms. Berp lter volume gs del tersebut pd st tek 5 tmoser. tetp gs uversl 0,08 l tm/mole 0 K jwb: Persm gs del : PV = R T Dket : N = mole R = 0,08 l tm / mole 0 K T = 00 0 K P = R T / V =. 0,08. 00/ V = 4,69 / V Jk dmslk P = d V = mk = 4,69/ P P =5 =5 V =? V =? Kre g dcr dlh l V st P = 5 mk 4,69 4,69 P 5 tu 5 V 4, sehgg 69 Deg ters Newto-Rpsho orde- ' msl 0 =,5 + 0,500 Avd-06 0

11 Fsk Komputs. Sutu celh lebr 0,4 mm dsr deg ch deg pjg gelombg 5900 Å. Pol drks g terjd dtgkp lr g jrk 70 cm dr celh. Berp jrk dr tegh-tegh terg pust smp tests ch tggl ½. Jwb : dm, Pol drks pd celh dtetuk deg : I I 0 S I 0 = tests ch g dtg β = ½ k b s θ k = blg gelombg = π/λ ; s θ = δ / D b = lebr celh θ = sudut tr orml pd celh deg rh sr g mejd pust perht Jk I/I 0 dsubsttus mejd d β dsubsttus mejd mk : s kre g dcr dlh l st = ½ I / I 0 = ½ tu I = ½ I 0 Jd persm g dcr solus dlh : s 0,5 =/ 0 =? Utuk medptk ttk potog deg sumbu mk dpk ters Newto- Rpsho orde-. Avd-06

12 Fsk Komputs s 0,5 s s msl 0 =,5 + 0,50 Dr perhtug ddptk ttk potog =,95578 sehgg,9 kbs bs 5,9.0 s,9..0,4. Akr Akr Dr Polom Betuk Umum Polom dlh : cotoh : polomorde 0 8 polomorde -.. Pembg Stetk dlh metode utuk mecr l ugs d l turu ugs. Adpu lgkh-lgkh dlh :.Tulsk koese-koese polom deg urut : 0 bl d suku g tdk hdr tuls 0 utuk koese dr suku tsb..nl dm l ugs k dcr dletkk plg kr dlm deret koesekoese. 0 Avd-06

13 Fsk Komputs.Selg stu brs dr deret koese tersebut but grs lurus sebg grs pejumlh Turuk hgg ke bwh grs pejumlh. Sebut deg b. 0 b + 5.Klk deg b d letkk hsl dts grs pejumlh tept d bwh -. Jumlhk - deg hsl kl td d letkk hsl d bwh grs pejumlh, stu kolom deg -. Ber m hsl pejumlh td deg b - 0 b.b. b b + 6.Klk deg b - d letkk hsl dts grs pejumlh d bwh -. Jumlhk - deg hsl kl td d letkk hsl d bwh grs pejumlh, stu kolom deg -. Ber m hsl pejumlh td deg b - 0 b 7.Lkuk opers g sm utuk koese-koese g l sehgg d bwh grs pejumlh k ddptk deret : b b b b0 8.Nl b 0 merupk l ugs g dcr. Jk pembg stetk dljutk utuk deret b b b b b b0 mk k ddptk l dr turu. Jk lgkh-lgkh tersebut dts drgkum mk k tmpk sepert berkut : b.. b + b b b b b b b + Avd-06

14 Fsk Komputs dm, b b b b b b b Nl b merupk l dr ugs pd g bersgkut. Sedgk utuk meetuk l dr turu dr ugs tersebut dlh deg meerusk pembg stetk dm l,,, dgt deg l b, b, b. b b b dm, c b c c c c c + c c b b c c Nl dr c merupk l turu dr persm pd g bersgkut. Cotoh : Jwb : Mslk kt memlk persm polom orde tu : 9 45 Tetuk l ugs d turu utuk = -Secr Altk : Utuk = mk = - d = 7 -Deg Pembg Stetk mk hsl dlh : Avd-06 4

15 Fsk Komputs Jd ddptk l pd = dlh, sedgk l turu dlh 7. Hsl sm deg hsl g ddptk secr ltk. Sol :. Tetuk kr dr persm : = Tetuk kr dr persm : = - 8 ctt : guk metode Iters Newto Rpsho orde d orde-.. Akr Gd Bl sutu ugs mempu beberp kr mk utuk mecr kr kr g l dpk ters Newto-Rpsho deg ttk wl g l jug. Jk sutu ugs mempu kr gd lebh dr kr pd ttk g sm mk klu h memk metode Newto-Rpsho tdk k dpt meetuk kr gd tersebut. Cotoh : Persm tersebut mempu kr gd d = d kr tuggl d =5. Utuk meelesk mslh kr gd mk kt memk lgkh-lgkh berkut. Utuk sutu polom orde, tu mk lgkh-lgkh mecr kr dlh :.Tetuk kr plg k deg metode Newto-Rpsho. Mslk kr tersebut dlh =.Bg deg -, hsl polom g orde lebh redh tgkt tu : -.Tetuk kr plg k - deg Newto-Rpsho. Mslk kr tersebut dlh = 4.Bg - deg -. Hsl dlh polom g orde lebh redh tgkt dr semul tu - 5.Lgkh dterusk smp ddptk polom orde- cotoh : polom orde -bglh deg -5 hsl : polomorde - -bg deg - hsl : Avd-06 5

16 Fsk Komputs polom orde - Adpu lgkh lgkh membg sutu polom orde- deg polom orde - dlh :.Tulsk koese-koese dr polom semul Lkuk pembg stetk utuk sutu l ddpt dr Newto-Rpsho sehgg ddptk : b b - b - b 0 b 0 = 0 = l ugs.ubh b mejd -, b - mejd -, b mejd 0 Setelh lgkh ke- mk k ddptk koese-koese polom bru g orde tgkt lebh redh tu : Jk lgkh tersebut dljutk smp ddptk koese-koese polom orde-, mk kr terkhr k ddptk dr polom orde. Cotoh : 9 45 orde - Lgkh-lgkh : = orde - sehgg = = -6 0 = 9 Ulg lgkh-lgkh tsb : = orde - sehgg = Avd-06 6

17 Fsk Komputs 0 = - Jd kr-kr dr ugs tsb dlh : = 5 = =.4 Progrm Komputer Progrm komputer ters Newto-Rpsho orde, dm persm g dcr kr dlh : 9 45 //********************************************************************************** // Meghtug Akr dr Fugs deg Metode Newto Rpsho Orde // Iters dbts h smp 5 kl ters // Compler : Vsul C++ //********************************************************************************** #clude <ostrem.h> #clude <mth.h> //heder utuk ugs bs #clude <omp.h> //heder utuk wdht,setoslgs, vod m double [00]; double,,d; t ; cout <<edl; cout << "Ttk wl o : "; c >> [0]; cout <<edl<<edl; //cout <<setprecso; or =; <=5; ++ //h 5 ters =[-]; //***************************** =**-**+9*-45; //persm g k dcr kr d=**-*+9; // Nl turu pertm dr persm //***************************** []=-/d; //rumus Newto Rpsho orde- cout.wdth5; //tmplk 5 dgt cout <<setoslgsos::let; //rt kr Avd-06 7

18 Fsk Komputs cout << ; cout.wdth5; cout <<setoslgsos::let; cout <<[]; cout.wdth5; cout <<setoslgsos::let; cout <<bs[]- ; cout <<edl<<edl; Dlm progrm dts ters dbts h smp 5 ters. Sehgg progrm k berhet begtu 5 ters tercp. Progrm berkut merupk modks dr progrm dts dm ters k berhet jk selsh tr l ters g berturut lebh kecl tu sm deg sutu kostt epslo. Adpu mplemets dr progrm tersebut dlh sepert dbwh : //*********************************************************************** // Meghtug kr dr ugs deg Newto Rpsho orde- // ters dbts oleh selsh tr l ters g berurut // Compler : Vsul C++ //*********************************************************************** #clude <ostrem.h> #clude <mth.h> vod m double eps=e-5; //kostt epslo double [00]; double,,d,selsh; t ; cout <<edl; cout << "Ttk wl o : "; c >> [0]; cout <<edl<<edl; =0; selsh = 0; whle selsh >= eps =+; Avd-06 8

19 Fsk Komputs =[-]; //betuk berkut dpt berubh sesu betuk ugs //***************************** =**-**+9*-45; //Fugs g dcr kr d=**-*+9; //Turu pertm dr ugs //***************************** []=-/d; //pers. Metode Newto-Rpsho orde- selsh=bs[]-; //selsh l ters g berurut cout << "Jumlh ters = "<< <<edl; cout << "Akr = " << [] <<edl; Progrm komputer ters Newto-Rpsho orde, dm persm g dcr kr dlh : 9 45 // ********************************************************************** // Meghtug kr dr ugs deg Newto Rpsho orde- // ters dbts oleh selsh tr l g berurut // Compler : Vsul C++ // ********************************************************************** #clude <ostrem.h> #clude <mth.h> #clude <omp.h> vod m double eps=e-5; double [00]; double,,d,d,pe,selsh; t ; cout <<edl; cout << "Ttk wl o : "; c >> [0]; cout <<edl<<edl; =0; selsh = 0; whle selsh >= eps //ters smp selsh l berurut <= eps Avd-06 9

20 Fsk Komputs =+; =[-]; //betuk berkut dpt berubh sesu betuk ugs //***************************** =**-**+9*-45; //Fugs g dcr kr d=**-*+9; //Turu pertm dr ugs d=6*-; //Turu kedu dr ugs //***************************** pe=d-d*/*d; []=-/pe; //pers. Iters Newto-Rpsho orde- cout.wdth0; cout << setoslgsos::let; cout << ; //tmplk ters ke- cout.wdth5; cout << setoslgsos::let; cout << []; //tmplk l ters ke- selsh=bs[]-; //selsh l ters g berurut cout.wdth5; cout << setoslgsos::let; cout << selsh; cout << edl<<edl; cout << "Jumlh ters = "<< <<edl; cout << "Akr = " << [] <<edl; Lth : Butlh progrm komputer utuk meghtug persol gs del. Dm mole gs del g suhu 00 0 K volume dperbesr pd suhu tetp. Berp lter volume gs tersebut ketk tek 5 tmosr tetp gs uversl 0,08 l.tm/mole 0 K. Petujuk : pv = RT Dm, = mole, R=0.08 l.tm/mole 0 K, d T=00 0 K. Sehgg : p = 4.69/V utuk meetuk berp volume gs pd st tek 5 tmosr mk persm tersebut k mejd : p = 4.69/V -5 Atu jk p dsubsttus mejd d V mejd mk : = 4.69/ 5 Avd-06 0

21 Fsk Komputs Deg megguk ters Newto-Rpsho mk k dpt dtetuk solus. Butlh progrm komputer dr mslh. Butlh progrm komputer utuk mslh drks pd celh. Dm sutu celh deg lebr 0.4 mm dsr deg ch deg pjg gelombg 5900 A. Pol drks g terjd dtgkp oleh sebuh lr g jrk 70 cm dr celh tsb. Htug jrk dr tegh-tegh terg pust smp tests ch tggl ½. Petujuk : Pol drks pd celh dtk deg : I I 0 S dm, I 0 tests ch g dtg d kb s. Jk kt mesubsttusk I/I 0 mejd d mejd mk persm tersebut k mejd s utuk medptk l dm I/I 0 = 0.5 mk persm tersebut dubh mejd : s 0.5 Deg memk ters Newto-Rpsho mk solus persm k dpt dcr. Butlh progrm!. Tugs seljut dlh bgm kt membut progrm komputer dr Pembg Stetk. Adpu mplemets dr progrm komputer utuk metode Pembg Stetk dtk sepert berkut : //************************************************************* //Meetuk l ugs d turu deg Metode //pembg Stetk //Compler : Vsul C++ //************************************************************* #clude <ostrem.h> #clude <omp.h> vod pemsdouble [00],t,double,double b[00]; vod m double [00]; Avd-06

22 Fsk Komputs double b[00]; double c[00]; double ; t,orde; cout<<"orde dr polom = "; c >> orde; or =orde; >=0; -- cout<< ""<<<<" = "; c >>[]; //put koese dr msg-msg suku cout <<edl; cout << "tetuk ttk o : "; c >>; cout <<edl<<edl; pems,orde,,b; //pggl ugs pems --->pembg stetk or =orde;>=0;--!=0 cout.wdth5; //pes tempt 5 dgt cout<<setoslgsos::let; //tmplk rt kr cout<<; cout<<setw5; cout<<setoslgsos::let; cout<<b[]; else cout<<setw5; cout<<setoslgsos::let; cout<<; cout<<setw5; cout<<setoslgsos::let; cout<<b[]; cout<<"nl Fugs"; cout <<edl; cout<<edl; pemsb,orde,,c; Avd-06

23 Fsk Komputs or=orde;>=0;--!= cout<<setw5; cout<<setoslgsos::let; cout<<; cout<<setw5; cout<<setoslgsos::let; cout<<c[]; else cout<<setw5; cout<<setoslgsos::let; cout<<; cout<<setw5; cout<<setoslgsos::let; cout<<c[]; cout<<" Nl Turu"; cout <<edl; vod pemsdouble [00],t,double,double b[00] t j; b[]=[]; or j=-; j>=0; j-- b[j]=[j]+*b[j+]; Jk metode Pembg Stetk kt pduk deg ters Newto-Rpsho dlm peetu kr-kr persm Polom mk betuk progrm komputer dr Iters Newto-Rpsho k meglm perubh sepert d bwh : //****************************************************************************** //Meetuk Akr dr Polom deg Newto-Rpsho dm //l ugs d turu dtetuk dg metode Pembg Stetk //Complr : Vsul C++ Avd-06

24 Fsk Komputs //****************************************************************************** #clude <ostrem.h> #clude <omp.h> #clude <mth.h> vod pemsdouble [00],t,double,double b[00]; //deklrs ugs pembg stetk vod m double eps = e-5; double [00]; double b[00]; double c[00]; double [00]; double,,d; t,,orde; cout<<"orde dr polom = "; c >> orde; or =orde; >=0; -- cout<< ""<<<<" = "; c >>[]; //put koese dr msg-msg suku cout <<edl; cout << "tetuk ttk wl o : "; c >>[0]; cout <<edl<<edl; =0; whle bs[]->=eps =+; =[-]; pems,orde,,b; =b[0]; pemsb,orde,,c; d=c[]; []=-/d; ` cout << "Jumlh ters : "<<<<edl; cout << "Akr : "<<[]; cout <<edl; Avd-06 4

25 Fsk Komputs Lth : vod pemsdouble [00],t,double,double b[00] t j; b[]=[]; or j=-; j>=0; j-- b[j]=[j]+*b[j+]; Butlh progrm komputer utuk meelesk mslh gerk peluru. Dm sebuh peluru dtembkk deg kecept wl V 0 = 00 m/s d sudut elevs = Percept grts g = 9,8 m/dt.tetuk berp juh peluru k melg d udr. Petujuk : Persm lts peluru dtk deg : g t v cos tu bs dtuls mejd polom orde- tu : dm, g. 0 v cos 0 t m Deg memk ters Newto-Rpsho g dkombsk deg metode pembg Stetk mk solus permslh dts k dpt dcr. But progrm komputer. Avd-06 5

26 Fsk Komputs III INTEGRAL DAN DIFERENSIAL. INTEGRAL Nl Itegrl I dr sutu ugs metk lus bdg dbwh ugs tr = d =b tu dtuls : b I d. Ad beberp metode g dpt dpk utuk meghtug tegrl dtr : metode perseg pjg, metode trpesum, metode Smpso, metode Romberg dll. Berkut k djelsk beberp metode tersebut... Metode Perseg Pjg Dlm metode lus bdg dbwh kurv tr = d = b dpt dcr deg membg bdg tersebut mejd buh pt g berbetuk perseg pjg, g pjg d lebr, sehgg Pt ke- Pt ke- Pt ke- b Gmbr. Metode Perseg Pjg lus msg-msg pt dpt dtk : L.. Kre tr = d = b terdpt buh pt mk lus seluruh mejd : L. utuk Δ sgt sempt Δ 0 mk persm. dlh persm tegrl sepert g drumusk dlm persm.. Metode utuk meghtug tegrl sepert g dsebutk d ts dkel deg metode Perseg Pjg. Avd-06 6

27 Fsk Komputs.. Metode Trpesum Dlm metode trpesum lus bdg d bwh ugs tr = d = b dpt dcr deg membg bdg tr = d = b mejd buh pt g berbetuk trpesum g msg-msg lebr gmbr.. sepert dperlhtk dlm Pt ke- + Pt ke- Pt ke- + b Dr gmbr. mk lus pt ke- g terletk tr d + dlh : A [ Sehgg utuk buh pt mk lus seluruh mejd : A [ ] ] [ ] [ kre = d + = b mk persm tersebut mejd : cotoh : Gmbr. Metode Trpesum A [ 4 b].4 Jk dkethu ps jes sutu zt sepert tbel berkut : t 0 C c kkl/kg 0 C -00 0, , , , , , ,776 ] Avd-06 7

28 Fsk Komputs Tetuk ps g dperluk utuk memsk kg zt dr 00 0 C C. Jwb: Metode Perseg Pjg LusL 50 0, , , , , ,64 Jumlh Metode Trpesum + LusL 50 0,904 0,486 6, ,486 0,00 6, ,00 0,4046 6, ,4046 0,504 7, ,504 0,64 7, ,64 0,776 8,775 Jumlh 4,7650 Jd ps g dperluk dlh 4,7650 kkl... Progrm Komputer metode Trpesum Membut progrm komputer utuk meghtug tegrl deg metode Trpesum. Dm ugs g k dcr tegrl dlh : = Adpu progrm komputer dlh sepert berkut : //******************************************************* //Meghtug tegrl deg metode Trpesum //Compler : Vsul C++ //******************************************************* #clude <ostrem.h> Avd-06 8

29 Fsk Komputs #clude <mth.h> #clude <omp.h> vod m double eps=e-5; double trp[00]; double,,,delt,del,pt,; t ; cout << " Bts bwh : "; c >>; cout << " Bts Ats : "; c >>; cout <<edl<<edl; =0; delt=00; //delt = selsh tr hsl tegrlters g berurut whle bsdelt>eps //ters selm delt lebh besr dr eps =+; trp[]=0; =; pt=pow,-; //^- pow = pgkt del=-/pt; //pt= jumlh pt whle < //del=lebr msg-msg pt; //*************************** =6-6*pow,5; //betuk pers. 6-6^5 //*************************** //=bs; //cotoh : pers. grs = //=/ ; //cotoh : pers. lu mget pd kwt berrus == == trp[]=trp[]+; else trp[]=trp[]+*; =+del; trp[]=trp[]*del/; == cout<<setw0; //0 loks dgt cout<<setoslgsos::let; //rt kr cout<< pt; Avd-06 9

30 Fsk Komputs cout<<setw5; cout<<setoslgsos::let; cout<< trp[]; else delt=trp[]-trp[-]; cout<<setw0; cout<<setoslgsos::let; cout<<pt; cout<<setw5; cout<<setoslgsos::let; cout<<trp[]; cout<<setw5; cout<<setoslgsos::let; cout<<delt; cout<<edl; cout <<edl<<edl; cout<<"hrg Itegrs : "<<trp[]; cout <<edl<<edl; Cob jlk progrm tersebut deg bts-bts tegrl =0 d =. Bgmkh hsl.jk progrm ber d k medptk hsl tegrs = Lth : Butlh progrm komputer utuk meghtug tegrl dr ugs : = 4 Eksekus progrm tersebut deg memsukk bts-bts tegrs = 0 d = 5. Jk progrm ber mk k ddptk hsl tegrs = 0. Progrm megdug rt tu kt mecr lus seg empt deg pjg 5 d lebr 4. 4 = Cob but progrm komputer utuk meghtug tegrl dr ugs : Avd-06 0

31 Fsk Komputs = / Cob eksekus progrm tersebut deg bts-bts tegrs dlh =0.0 d =0.09. Jk progrm ber mk d k medptk hrg tegrs = Metode Smpso Keslh keslh g terjd pd metode perseg pjg mupu metode trpesum dlh kre kurv tr = d = b ddekt deg potog potog grs lurus. Keslh dpt dperkecl jk kt tdk megguk grs lurus tetp kurv l. Slh stu pedekt g dpk dlh prbol tu polom orde du. Utuk meuruk rumus tegrs mk dmslk + dtemptk d = 0 sehgg berd d = - sert + berd d =. Jk persm prbol g dpk dlh : 0 5 mk lus psg pt d bwh prbol : A 0 d Nl ugs pd ketg ttk tersebut dlh : Dr persm.8,.9, d.0 mk ddptk koese-koese : Avd-06

32 Fsk Komputs Sehgg persm.7 mejd : A [ 4 ].4 Mk lus semu psg pt dlh : A [ 4 [ 4 4 5] ] [ 4 ]..5 tu A [ ] cotoh : Jk dkethu ps jes sutu zt sepert tbel berkut : t 0 C c kkl/kg 0 C -00 0, , , , , , ,776 Tetuk ps g dperluk utuk memsk kg zt dr 00 0 C smp 00 0 C. Jwb: + + LusL 50 0,904 0,486 0,00, ,00 0,4046 0,504 4, ,504 0,64 0,776 6,560 Jumlh 4,70 Jd ps g dperluk dlh 4,70 kkl Sutu kwt g sgt pjg dlr rus lstrk 0 A. Htug besr lu mget g meembus sutu perseg pjg g ss pjg sejjr d Avd-06

33 Fsk Komputs ss pedek tegk lurus kwt. Pjg perseg pjg 0 cm, lebr 8 cm. Jrk slh stu ss pjg perseg pjg ke kwt dlh cm μ 0 = 4π. 0-7 weber/mp.m cm r 8 cm L 0 cm dr pt = da = Ldr Jwb : Besr lu mget : d B. da Ambl sutu pt sejjr deg kwt berrus pjg L = 0 cm, d lebr dr. Mk lus pt da = L.dr. Sehgg lu mget g meembus pt dlh : 0 B r Mk d 0 0L L dr r dr r 0L 0,09 r 0,0 dr..5 Progrm komputer metode Smpso Membut progrm komputer utuk meghtug tegrl deg metode Smpso. Dm ugs g k dcr tegrl dlh : = Adpu progrm komputer dlh sepert berkut : //******************************************************* //Meghtug tegrl deg metode Smpso //Compler : Vsul C++ //******************************************************* #clude <ostrem.h> #clude <mth.h> #clude <omp.h> vod m double eps=e-5; Avd-06

34 Fsk Komputs double smp[00]; double,,,delt,del,pt,; t,j; cout << " Bts bwh : "; c >>; cout << " Bts Ats : "; c >>; cout <<edl<<edl; =0; delt=00; //delt = selsh tr hsl tegrlters g berurut whle bsdelt>eps //ters selm delt lebh besr dr eps =+; j=0; smp[]=0; =; pt=*pow,-; //^- pow = pgkt del=-/pt; //pt= jumlh pt whle < //del=lebr msg-msg pt; //*************************** =6-6*pow,5; //cotoh : betuk pers. 6-6^5 //*************************** //=bs; //cotoh : pers. grs = //=/ ; //cotoh : pers. lu mget pd kwt berrus j=j+; == == smp[]=smp[]+; else > && < && j % ==0 //suku gep smp[]=smp[]+4*; > && < && j % == //suku gjl smp[]=smp[]+*; =+del; smp[]=smp[]*del/; == cout<<setw0; //0 loks dgt Avd-06 4

35 Fsk Komputs cout<<setoslgsos::let; //rt kr cout<< pt; cout<<setw5; cout<<setoslgsos::let; cout<< smp[]; else delt=smp[]-smp[-]; cout<<setw0; cout<<setoslgsos::let; cout<<pt; cout<<setw5; cout<<setoslgsos::let; cout<<smp[]; cout<<setw5; cout<<setoslgsos::let; cout<<delt; cout<<edl; cout <<edl<<edl; cout<<"hrg Itegrs : "<<smp[]; cout <<edl<<edl; Lth : Cob but progrm g sm sepert pd lth utuk metode Trpesum. Bdgk hsl.. DIFERENSIAL Deresl tu serg dsebut turu dpt dhtug deg memk ur deret Tlor. Jk ur deret Tlor d sektr dtk deg +h d -h. Dm msg-msg dtk deg persm : ' '' ''' h h h h 7 6 h h.. Turu Pertm ' h '' h Avd-06 5 '''

36 Avd-06 6 Fsk Komputs Jk kt megmbl selsh tr kedu persm tersebut.7 d.8 mk k ddptk : 6 ''' ' h h h h..9 Dr persm.9 jk kt megmbl l h g sgt kecl mk suku-suku deg h pgkt tu lebh bs dbk. Sehgg k ddptk persm Turu Pertm dr sutu ugs tu : h h h ' 0.. Turu Kedu Utuk medptk Turu Kedu dr ugs mk kt mejumlhk kedu persm +h d -h dlm persm.7 d.8. Sehgg ddptk persm tu : '' h h h h IV.. Bl h sgt kecl mk suku deg h pgkt tu lebh bs dbk. Sehgg k ddptk persm Turu Kedu dr tu : ' ' h h h. Kedu metode utuk mecr turu dts dsebut deg metode Bed Setrl cetrl derece. Cotoh :.Tetuk turu pertm dr ugs = st = jwb : h h h ' Secr ltk d d bsecr Numerk Metode Bed Setrl

37 Fsk Komputs h = ; h + = h / h +h -h =+h--h/h,00 4,0000 0,0000,00 0,50,500 0,500,00 0,5,565 0,565,00 Jwb :.Tetuk turu pertm d kedu dr ugs = + st = Turu Pertm : Secr ltk d d bsecr Numerk Metode Bed Setrl h = ; h + = h / h +h -h =+h--h/h,00 6,0000 0,0000,00 0,50,7500 0,7500,00 Turu Kedu : Secr ltk d d bsecr Numerk Metode Bed Setrl h = ; h + = h / h +h -h,00 6,0000 0,0000,00 0,50,7500 0,7500,00.. Progrm Komputer Progrm komputer utuk meghtug Turu Pertm dr ugs. Dm ugs g dcr turu dlh : Avd-06 7

38 Fsk Komputs = 5 Adpu mplemets progrm komputer dlh sepert berkut : //*********************************************************************************** //Meghtug Turu pertm sutu ugs deg metode bed setrl //compler : Vsul C++ //*********************************************************************************** #clude <ostrem.h> #clude <mth.h> #clude <omp.h> vod m double eps=e-5; double z[0]; double,del,d,zz,,,h; t,; cout << " Msukk l : "; c >>; cout <<edl<<edl; h=; =0; zz=0; del=0; whle bsdel>=eps //ters selm del lebh besr dr eps +=; or =;<=;++ == =+h; else =-h; //=9.8*68.*-ep-.5*/68.; //=/.5; //***************** =*-5*; //***************** z[]=; d=z[]-z[]/*h; del=zz-d; cout.wdth5; cout<<h; cout.wdth5; cout<<d; cout.wdth5; cout<<del; cout<<edl; zz=d; h=h/0; cout<<edl; cout<<"jumlh ters : "<<<<edl; cout<<"turu pertm dr : "<<d; cout<<edl; //ugs g dcr turu : ^ - 5 Avd-06 8

39 Fsk Komputs Cob jlk eksekus progrm tersebut. Jk d msukk l = mk k ddptk hsl turu pertm dlh. Coblh utuk memsukk l g l d lht hsl. Bdgk deg perhtug secr mul. Membut progrm komputer utuk meghtug Turu kedu dr ugs. Dm ugs g dcr turu dlh : = 5 Adpu mplemets progrm komputer dlh sepert berkut : //****************************************************************** //Meghtug Turu pertm sutu ugs deg metode bed setrl //compler : Vsul C++ //****************************************************************** #clude <ostrem.h> #clude <mth.h> #clude <omp.h> vod m double eps=e-5; double z[0]; double,del,d,zz,,,h; t,; cout << " Msukk l : "; c >>; cout <<edl<<edl; h=; =0; zz=0; del=0; whle bsdel>=eps //ters selm del lebh besr dr eps +=; or =;<=;++ swtch cse : =+h; brek; cse: =; brek; cse: =-h; brek; //=9.8*68.*-ep-.5*/68.; //=/.5; //***************** =**-5*; //ugs g dcr turu : ^ - 5 //***************** Avd-06 9

40 Fsk Komputs z[]=; d=z[]-*z[]+z[]/h*h; del=zz-d; cout.wdth5; cout<<h; cout.wdth5; cout<<d; cout.wdth5; cout<<del; cout<<edl; zz=d; h=h/0; cout<<edl; cout<<"jumlh ters = "<<<<edl; cout<<"turu kedu dr = "<<d; cout<<edl; Cob jlk eksekus progrm tersebut. Jk d msukk l = mk k ddptk hsl turu pertm dlh. Coblh utuk memsukk l g l d lht hsl. D bdgk deg perhtug secr mul. Lth : Cob but progrm komputer utuk meetuk turu pertm dr sutu mslh peerju pug Dm dkethu persm lju peerju pug setelh pug megembg dlh : v t gm [ e c ct / m ] dm g = 9.8 ms -, mss peerju m = 68. kg, hmbt oleh udr c =.5 kg.s -. Dtk berp percept peerju pug pd st t = 0 s. Cob but progrm komputer utuk meetuk turu kedu dr sutu mslh peerju pug Dm dkethu persm lju peerju pug setelh pug megembg dlh : v t gm [ e c ct / m ] dm g = 9.8 ms -, mss peerju m = 68. kg, hmbt oleh udr c =.5 kg.s -. Dtk berp percept peerju pug pd st t = 0 s. Avd-06 40

41 Fsk Komputs IV MENYELESAIKAN PERSAMAAN DIFERENSIAL BIASA 4. Solus Persm Deresl deg srt wl Mslk kt memlk persm deresl sepert d bwh : d m r dt d dt km 0.4. secr lss mk solus dr persm tersebut dlh : Ae t s t 0, utuk r 4km t B B t e, utuk r 4km t t Ce Ce, utuk r 4km Sedgk secr umerk kt tdk k medptk solus sepert dts, tetp kt k mecr solus pd sutu wktu tertetu. Sehgg dperluk sutu srt wl gr solus dpt dcr tu mslk pd wktu mul-mul solus mellu sutu ttk tertetu. Srt semcm dperluk kre solus persm deresl dpt berbed-bed kre d sutu kostt. Adk kt mempu persm deresl : ' 4 5 solus secr lss mtemtk dr persm dlh : d 4 5 d 4 5d 5 C dpt dlht bhw solus tdk h stu tetp tergtug dr l C. Sedgk solus secr umerk kt meetuk sutu srt wl terhdp ugs tersebut msl bhw solus persm tersebut mellu ttk,. Sehgg k ddptk h sebuh solus tu : 5 6 msl kt msukk l = mk solus dr persm deresl tersebut dlh : ' Jd dpt dsmpulk bhw solus dr persm deresl 4 5 d = d srt wl, dlh. Ad beberp metode g dpt dterpk utuk mecr solus dr persm deresl tu : metode Euler, Ruge-Kutt dll. Avd-06 4

42 Fsk Komputs 4. Metode Euler Utuk mecr solus persm deresl bs deg srt wl dpt dlkuk deg metode Euler. Metode Euler terdr dr metode tu : orde d orde. Berkut kt k membhs stu perstu dr ke du metode tersebut. 4.. Metode Euler orde +h tu : Metode dturuk dr ur deret Tlor dsektr dtk deg h ' h '' h ''' h Jk dmbl smp suku h pgkt mk persm tersebut mejd : ' h h ' dm dlm Persm Deresl Bs dpt dtuls mejd : ', sehgg persm dts mejd : h, h persm bs dtuls dlm betuk ters :, h4. persm serg dsebut deg ters Euler orde. Jk kt meetuk srt wl dlh ttk 0, 0 d kt meggk solus persm deresl d ttk p mk kt hrus membg selg tr 0 d p mejd buh pt g msgmsg lebr h sehgg dperoleh ttk-ttk 0,,, p. Dr srt wl tu ttk 0, 0 mk deg rumus ters Euler kt dpt meetuk deg bss = 0 + h. Seljut dr ttk, kt dpt meetuk deg bss = + h. Demk seterus smp ddptk p g bss dlh p. Deg demk l p merupk solus dr persm deresl pd ttk p. Cotoh : Jwb : Htuglh solus persm deresl berkut pd = : d d 4 5,dm srt wl 0, 0 =, Jk kt membg tr = d = mejd 0 pt mk lebr msg-msg pt h dlh : Avd-06 4

43 Fsk Komputs h pt 0. 0, h h , h , hsl selegkp dpt dlht dlm tbel berkut., Jd solus persm dresl pd = dlh.8 Cotoh : Sebuh bed bergerk lurus sepjg sumbu. Lju bed Vt setp setegh detk dlh : t Vt Berp pjg jl g dtempuh smp deg detk ke 5. Jk mul-mul t 0 bed berd pd poss =.7 Jwb : d dt vt Iters Euler orde : v t h ; h 0.5 detk Avd-06 4

44 Fsk Komputs Awl t 0 = 0.5, 0 =.7 St = 0, v t h hsl seljut dpt dlht dlm tbel d bwh : t vt Lth : Sehgg jrk g dtempuh smp deg detk ke 5 dlh 5.5 m.tetuk solus persm deresl : d d pd st = d srt wl dlh 0,.Tetuk solus persm deresl : = + st = d srt wl, 4.. Progrm Komputer metode Euler orde Membut progrm komputer utuk mecr solus persm deresl bs deg ters Euler orde-. Dm betuk persm deresl g dcr solus dlh : = 4+5 Adpu mplemets progrm komputer dlh sepert berkut : //**************************************************************************** //Meghtug Persm Deresl deg metode Euler orde stu //compler : Vsul C++ //**************************************************************************** #clude <ostrem.h> Avd-06 44

45 Fsk Komputs #clude <mth.h> #clude <omp.h> vod m double eps=e-; double,,,,,,,del,delt,pt; t ; cout << " Msukk ttk wl o,o : "; //msukk l o <sps> o c >>>>; cout << " Msukk ttk khr : "; c >>; cout <<edl<<edl; =0; delt=00; whle bsdelt>eps //ters selm delt lebh besr dr eps +=; pt=pow,; //^ del=-/pt; =; =; whle < //************************* //=; =4*+5; //************************* =+*del; //ters Euler orde- =+del; == cout.wdth5; cout<<pt; cout.wdth5; cout<<; else delt=-; cout.wdth5; cout<<pt; cout.wdth5; cout<<; cout.wdth5; cout<<delt; cout<<edl; =; cout<<"nl Fugs d "<<<<" dlh : "<<; cout<<edl; Cob jlk progrm tersebut d berk msuk srt wl 0, 0 =, sert berk msuk utuk l =. Jk progrm berjl deg ber mk d k medptk output berup l ugs d = dlh Metode Euler orde Avd-06 45

46 Fsk Komputs Dr ur deret Tlor utuk +h, jk kt meertk suku-suku deg h pgkt mk k ddptk persm : dm h ', ' h '' h 4. '' ',,,, persm 4. dpt dtk dlm betuk ters :, h ', h.4.4 Persm dsebut sebg ters Euler orde Progrm komputer Metode Euler orde Progrm komputer utuk mecr solus persm deresl bs deg ters Euler orde-, dm betuk persm deresl g dcr solus dlh : = 4+5 Adpu mplemets progrm komputer dlh sepert berkut : //*************************************************************************** //Meghtug Persm Dresl deg metode Euler orde du //compler : Vsul C++ //*************************************************************************** #clude <ostrem.h> #clude <mth.h> #clude <omp.h> vod m double eps=e-; double,,,,,,,d,del,delt,pt; t ; cout << " Msukk ttk wl o,o : "; //msukk l o <sps> o c >>>>; cout << " Msukk ttk khr : "; c >>; cout <<edl<<edl; =0; delt=00; whle bsdelt>eps //ters selm delt lebh besr dr eps +=; pt=pow,; //^ del=-/pt; =; =; whle < //************************ //=; //betuk ugs //d=; //turu =4*+5; Avd-06 46

47 Fsk Komputs d=4; //************************* =+*del+d*powdel,/; //ters Euler orde =+del; == cout.wdth5; cout<<pt; cout.wdth5; cout<<; else delt=-; cout.wdth5; cout<<pt; cout.wdth5; cout<<; cout.wdth5; cout<<delt; cout<<edl; =; cout<<"nl Fugs d = "<<<<" dlh : "<<; cout<<edl; Cob jlk progrm tersebut d berk msuk srt wl 0, 0 =, sert berk msuk utuk l =. Jk progrm berjl deg ber mk d k medptk output berup l ugs d = dlh Metode Ruge-Kutt Persm umum ters Ruge-Kutt dlh : dm h deg,, h4.5,, h k k k h, k k h ph, q k h ph, qk q k m k m k m h pm h, qm,k qm,k qm, mk m 4.. Ruge-Kutt orde Avd-06 47

48 Avd Fsk Komputs Jk dmbl m = mk k meghslk ters Ruge-Kutt orde tu :..4.6 k 4.7, h kre,, k h d, h k. Deg megmbl mk persm 4.7 sm deg persm ters Euler orde- tu :, h 4.. Ruge-Kutt orde Jk dmbl m = mk k meghslk ters Ruge-Kutt orde tu : 4.8 k k dm, h k, k q h p h k jd dlm hl kt perlu meetuk l-l dr d,,, q p. Utuk medptk l dr kostt-kostt tersebut mk pertm-tm kt lht ur deret Tlor smp orde tu : 4.8, '' ' h h h dm, ',,,, ' ' ' sehgg persm 4.7 mejd :..4.9,,,, h h h Jk dtk deg rumus ters mk persm 4.8 mejd :.4.0,,,, h h utuk k jk durk dlm deret Tlor k mejd : h k q h p k,,, sehgg rumus ters persm 4.0 mejd :,,, h p h h,, h q,,,,, p h.4. h q

49 Fsk Komputs dr persm 4.0 d 4. ddptk : p q Metode Heu Dlm metode dmbl l dr kostt : p q Sehgg persm ters Ruge-Kutt orde mejd : dm k k k h, k h h, k b Metode Rltso.4.5 Dlm metode dmbl l dr kostt : p q 4 Sehgg persm ters Ruge-Kutt orde mejd : dm k k h, k k 4.6 h, 4 h 4 k c Metode Polgo Dlm metode dmbl l dr kostt : 0 p q Sehgg persm ters Ruge-Kutt orde mejd : dm k 4.7 k h, k Cotoh : h, h k Avd-06 49

50 Fsk Komputs Jwb : Htuglh solus persm dresl berkut pd = deg metode Heu: d d 4 5 dm srt wl 0, 0 =, Jk kt membg tr = d = mejd 0 pt mk lebr msg-msg pt h dlh : dm h pt k k k h, k sehgg : h h, k 0 k k 0. 0 k jd k h k k hsl selegkp dpt dlht dlm tbel berkut. k k Avd-06 50

51 Fsk Komputs Jd solus persm dresl pd = dlh Progrm komputer Metode Ruge Kutt orde - Heu Membut progrm komputer utuk mecr solus persm deresl bs deg ters Ruge-Kutt orde berdsrk metode Heu. Dm betuk persm deresl g dcr solus dlh : = 4+5 Adpu mplemets progrm komputer dlh sepert berkut : //******************************************************************************************** //Meghtug Persm Deresl deg metode Ruge-kutt orde du-heu //compler : Vsul C++ //******************************************************************************************** #clude <ostrem.h> #clude <mth.h> #clude <omp.h> vod m double eps=e-; double k[4]; double,,,,,,,,,del,delt,pt; t,; cout << " Msukk ttk wl o,o : "; //msukk l o <sps> o c >>>>; cout << " Msukk ttk khr : "; c >>; cout <<edl<<edl; =0; delt=00; whle bsdelt>eps //ters selm delt lebh besr dr eps +=; pt=pow,; //^ del=-/pt; =; =; whle < =; =; or =;<=;++ //************************ //=; //betuk ugs =4*+5; //=-ep-.5*/68.; //=*9.8*68./.5; //************************* k[]=del*; == =+del; =+k[]; Avd-06 5

52 Fsk Komputs //or =+del; =+k[]+k[]/; //whle == cout.wdth5; cout<<pt; cout.wdth5; cout<<; else delt=-; cout.wdth5; cout<<pt; cout.wdth5; cout<<; cout.wdth5; cout<<delt; cout<<edl; =; //whle cout<<"nl Fugs d = "<<<<" cout <<edl; cout<<"bk ters = "<<; cout<<edl; dlh : "<<; Cob jlk progrm tersebut d berk msuk srt wl 0, 0 =, sert berk msuk utuk l =. Jk progrm berjl deg ber mk d k medptk output berup l ugs d = dlh. Membut progrm komputer utuk mecr solus persm deresl bs deg ters Ruge-Kutt orde berdsrk metode Rltso. Dm betuk persm deresl g dcr solus dlh : = 4+5 Adpu mplemets progrm komputer dlh sepert berkut : //********************************************************************************************** //Meghtug Persm Deresl deg metode Ruge-kutt orde du-rltso //compler : Vsul C++ //********************************************************************************************** #clude <ostrem.h> #clude <mth.h> #clude <omp.h> vod m double eps=e-; double k[4]; double,,,,,,,,,del,delt,pt; t,; cout << " Msukk ttk wl o,o : "; //msukk l o <sps> o c >>>>; Avd-06 5

53 Fsk Komputs cout << " Msukk ttk khr : "; c >>; cout <<edl<<edl; =0; delt=00; whle bsdelt>eps //ters selm delt lebh besr dr eps +=; pt=pow,; //^ del=-/pt; =; =; whle < =; =; or =;<=;++ //************************ //=; //betuk ugs =4*+5; //=-ep-.5*/68.; //betuk ugs //=*9.8*68./.5; //************************* k[]=del*; == =+*del/4; =+*k[]/4; //or =+del; =+k[]+*k[]/; //ters ruge-kutt orde - Rltso //whle == cout.wdth5; cout<<pt; cout.wdth5; cout<<; else delt=-; cout.wdth5; cout<<pt; cout.wdth5; cout<<; cout.wdth5; cout<<delt; cout<<edl; =; //whle cout<<"nl Fugs d = "<<<<" dlh : "<<; cout <<edl; cout<<"bk ters = "<<; cout<<edl; Avd-06 5

54 Fsk Komputs Cob jlk progrm tersebut d berk msuk srt wl 0, 0 =, sert berk msuk utuk l =. Jk progrm berjl deg ber mk d k medptk output berup l ugs d = dlh. Membut progrm komputer utuk mecr solus persm deresl bs deg ters Ruge-Kutt orde berdsrk metode Polgo. Dm betuk persm deresl g dcr solus dlh : = 4+5 Adpu mplemets progrm komputer dlh sepert berkut : //****************************************************************** //Meghtug Persm Deresl deg metode Ruge-kutt orde du-polgo //compler : Vsul C++ //****************************************************************** #clude <ostrem.h> #clude <mth.h> #clude <omp.h> vod m double eps=e-; double k[4]; double,,,,,,,,,del,delt,pt; t,; cout << " Msukk ttk wl o,o : "; //msukk l o <sps> o c >>>>; cout << " Msukk ttk khr : "; c >>; cout <<edl<<edl; =0; delt=00; whle bsdelt>eps //ters selm delt lebh besr dr eps +=; pt=pow,; //^ del=-/pt; =; =; whle < =; =; or =;<=;++ //************************ //=; //betuk ugs =4*+5; //=-ep-.5*/68.; //betuk ugs //=*9.8*68./.5; //************************* k[]=del*; == Avd-06 54

55 Fsk Komputs =+del/; =+k[]/; //or =+del; =+k[]; //ters ruge-kutt orde - Polgo //whle == cout.wdth5; cout<<pt; cout.wdth5; cout<<; else delt=-; cout.wdth5; cout<<pt; cout.wdth5; cout<<; cout.wdth5; cout<<delt; cout<<edl; =; //whle cout<<"nl Fugs d = "<<<<" dlh : "<<; cout <<edl; cout<<"bk ters = "<<; cout<<edl; Cob jlk progrm tersebut d berk msuk srt wl 0, 0 =, sert berk msuk utuk l =. Jk progrm berjl deg ber mk d k medptk output berup l ugs d = dlh Ruge-Kutt orde Rumus ters dr metode ddptk deg meertk suku-suku orde- dlm ur deret Tlor. Jk dmbl m= dlm persm 4. mk ddpt rumus ters Ruge-Kutt orde dlh : deg 6 k k h, k k k 4k.4.8 h, h k h h, k k 4..5 Progrm Komputer Ruge-Kutt orde Avd-06 55

56 Fsk Komputs Membut progrm komputer utuk mecr solus persm deresl bs deg ters Ruge-Kutt orde. Dm betuk persm deresl g dcr solus dlh : = 4+5 Adpu mplemets progrm komputer dlh sepert berkut : //************************************************************************************** //Meghtug Persm Deresl deg metode Ruge-Kutt orde Tg //compler : Vsul C++ //************************************************************************************** #clude <ostrem.h> #clude <mth.h> #clude <omp.h> vod m double eps=e-; double k[4]; double,,,,,,,,,del,delt,pt; t,; cout << " Msukk ttk wl o,o : "; //msukk l o <sps> o c >>>>; cout << " Msukk ttk khr : "; c >>; cout <<edl<<edl; =0; delt=00; whle bsdelt>eps //ters selm delt lebh besr dr eps +=; pt=pow,; //^ del=-/pt; =; =; whle < =; =; or =;<=;++ //************************ //=; //betuk ugs =4*+5; //=-ep-.5*/68.; //betuk ugs //=*9.8*68./.5; //************************* k[]=del*; swtch cse : =+del/; =+k[]/; brek; cse: Avd-06 56

57 Fsk Komputs =+del/; =-k[]+*k[]; brek; //or =+del; =+k[]+4*k[]+k[]/6; //ters Ruge-Kutt orde //whle == cout.wdth5; cout<<pt; cout.wdth5; cout<<; else delt=-; cout.wdth5; cout<<pt; cout.wdth5; cout<<; cout.wdth5; cout<<delt; cout<<edl; =; //whle cout<<"nl Fugs d = "<<<<" dlh : "<<; cout <<edl; cout<<"bk ters = "<<; cout<<edl; Cob jlk progrm tersebut d berk msuk srt wl 0, 0 =, sert berk msuk utuk l =. Jk progrm berjl deg ber mk d k medptk output berup l ugs d = dlh Ruge-Kutt orde 4 Metode dtk deg meertk suku-suku orde-4 dlm ur deret Tlor. Adpu rumus ters Ruge-Kutt orde 4 dlh : deg 6 k k h, k k k k k k 44.9 h, h k h, h k h h, 4 k Avd-06 57

58 Fsk Komputs 4..7 Progrm Komputer Ruge-Kutt orde 4 Membut progrm komputer utuk mecr solus persm deresl bs deg ters Ruge-Kutt orde 4. Dm betuk persm deresl g dcr solus dlh : = 4+5 Adpu mplemets progrm komputer dlh sepert berkut : //****************************************************************** //Meghtug Persm Deresl deg metode Ruge-Kutt orde Empt //compler : Vsul C++ //****************************************************************** #clude <ostrem.h> #clude <mth.h> #clude <omp.h> vod m double eps=e-; double k[4]; double,,,,,,,,,del,delt,pt; t,; cout << " Msukk ttk wl o,o : "; //msukk l o <sps> o c >>>>; cout << " Msukk ttk khr : "; c >>; cout <<edl<<edl; =0; delt=00; whle bsdelt>eps //ters selm delt lebh besr dr eps +=; //cch ters pt=pow,; //^ del=-/pt; =; =; whle < =; =; or =;<=4;++ //************************ //=; //betuk ugs =4*+5; //=-ep-.5*/68.; //betuk ugs //=*9.8*68./.5; //************************* k[]=del*; == == =+del/; =+k[]/; else == Avd-06 58

59 Fsk Komputs =+del; =+k[]; //or =+del; =+k[]+*k[]+k[]+k[4]/6; //ters Ruge-Kutt orde 4 //whle == cout.wdth5; cout<<pt; cout.wdth5; cout<<; else delt=-; cout.wdth5; cout<<pt; cout.wdth5; cout<<; cout.wdth5; cout<<delt; cout<<edl; =; //whle cout<<"nl Fugs d = "<<<<" dlh : "<<; cout <<edl; cout<<"bk ters = "<<; cout<<edl; Cob jlk progrm tersebut d berk msuk srt wl 0, 0 =, sert berk msuk utuk l =. Jk progrm berjl deg ber mk d k medptk output berup l ugs d = dlh. Avd-06 59

60 Fsk Komputs V MENYELESAIKAN SISTEM PERSAMAAN LINEAR 5. Sstem Persm Ler Persm Ler dlh sutu ugs tu persm dm pgkt tertgg dr vrbel-vrbel dlh, msl : Dm,,z,b,c k b cz k = vrbel = koese = kostt Sedgk g dmksud deg Sstem persm Ler dlh beberp persm ler g slg berhubug dm bk koese, vrbel, d kostt dlh sm. Agr sstem persm ler tersebut dpt dcr solus dpt dhtug l vrbel mk bk persm dlm sstem persm ler hrus sm deg bk vrbel. Cotoh sstem persm ler : z z z 5 Atu persm dts bs dtuls : Tmpk dlm sstem persm ler dts bk persm d vrbel dlh sm tu. Secr umum sstem persm ler dtk deg : b b b..5.. m m m m b Sstem persm ler tersebut terdr dr m persm d vrbel. Persm ler tersebut k dpt dselesk jk jumlh m =. Peeles persm ler m Avd-06 60

61 Fsk Komputs dlh mecr l =,,,. Sstem persm ler 5. dpt dtk dlm betuk mtrk tu : m m Atu bs dtk deg : Dm : A = B b b.5. m bm A = m m b, =, d B = b m b m Mtrk A dsebut mtrk koese tu mtrk Jcob. Vektor dsebut sebg vektor vrbel d vektor B dsebut sebg vektor Kostt. Augmeted Mtr mtrk perlus dlh perlus dr mtrk koese A deg membhk vektor B pd kolom terkhr dr mtrk A. Augmeted A = [A B] Sehgg Augmeted Mtrk dr persm ler smult 5. dpt dtulsk : m m m b b.5. bm Cotoh permslh g dpt dtk dlm betuk sstem persm ler msl : Cotoh : Seorg pembut boek g membut mcm boek tu boek A d boek B. Kedu boek tersebut dbut deg megguk mcm bh tu k d kcg. Adk boek A memerluk 0 meter k d 6 kcg sedgk boek B memerluk 8 meter k d 8 kcg mk htuglh berp buh boek A d boek B g dpt dhslk jk kt memlk 8 meter k d 6 kcg. Mslh dpt dmodelk deg metk : = jumlh boek A Avd-06 6

62 Fsk Komputs = jumlh boek B Sehgg utuk setp bh k d kcg dpt dtk persm : 0 utuk boek A + 8 utuk boek B = > k 6 utuk boek A + 8 utuk boek B = > kcg Persm dts dpt dtk deg : = = 6 Peeles dr persm tersebut dlh l d g memeuh kedu persm dts. Ad beberp metode g dpt dpk utuk mecr solus dr sstem persm ler. Metode metode k djelsk pd sub bb berkut. Teorem 5. Sutu sstem persm ler mempu peeles tuggl bl memeuh srt-srt sebg berkut : - betuk sstem persm ler bujursgkr, dm jumlh persm ler sm deg jumlh vrbel bebs - sstem persm ler berst o-homoge, tu mml d stu l vektor kostt B tdk ol tu d b 0 - determ dr mtrk koese A tdk sm deg ol Utuk meelesk permslh-permslh sstem persm ler dpt dlkuk deg megguk metode-metode ltk sepert pemk metode grs, tur Crmmer, tu vers mtrk. Metode-metode tersebut dpt dlkuk deg mudh pbl jumlh vrbel d jumlh persm d bwh 4, tetp bl ukur besr mk metode-metode dts mejd sult dlkuk, sehgg pemk metode umerk mejd sutu ltert g bk dguk. Metode umerk g dpt dguk utuk meelesk permslh sstem persm ler tr l : - metode Elms Guss - metode Elms Guss-Jord - Metode Iters Guss-Sedl 5. Metode Elms Guss Metode Elms Guss merupk metode g dkembgk dr metode elms, tu meghlgk tu megurg jumlh vrbel sehgg dpt dperoleh l dr sutu vrbel. Utuk megguk metode Elms Guss, terlebh dhulu betuk mtrk dubh mejd Augmeted Mtrk sebg berkut : Avd-06 6

63 Fsk Komputs m m m b b.5.4 bm Metode Elms Guss dlh sutu metode dm Augmeted mtrk 5.4 pd bg kr dubh mejd mtrk segtg ts tu segtg bwh deg meerpk Opers Brs Elemeter OBE. b b b b c c c 0 0 c c c 0 c c c d d d d Sehgg peeles dlh : d c d c, c, c d c c44 c d c c c c Opers Brs Elemeter dlh OBE dlh sutu opers perubh l eleme mtrk berdsrk brs, tp megubh mtrk. OBE pd brs ke +k deg dsr brs ke dpt dtuls : k, j k, j c.,j Dm c dlh kostt pegl g dmbl dr perbdg tr l d k,, Cotoh : Selesk sstem persm ler berkut : 6 Avd-06 6

64 Fsk Komputs 0 Jwb : Augmeted mtrk dr sstem persm ler tersebut dlh : 6 0 Lkuk OBE sebg berkut : B B B B B B B B B Sehgg ddpt peeles tu : Algortm Metode Elms Guss Adpu lgortm dr metode Elms Guss dlh :.Msukk mtrk A d vektor B besert ukur.but Augmeted Mtrk [A B], mk deg A.Utuk brs ke- dm = smp, perhtk pkh l, sm deg ol. Jk Y : Tukrk brs ke- deg brs +k, dm +k, tdk sm deg ol, bl tdk d berrt perhtug tdk bs dljutk d proses dhetk tp peeles Jk tdk : ljutk 4.Utuk brs ke-j, dm j = + smp, Lkuk Opers Brs Elemeter OBE : - htug c j,, -utuk kolom k dm k = smp + htug j, k j,k c.,k Avd-06 64

65 Fsk Komputs 5.Htug l, utuk = smp bergerk dr brs ke- smp brs ke-, b dm l +k,j,j, 5.. Progrm komputer Metode Elms Guss Berdsrk lgortm g dtk dlm sub bb sebelum mk kt dpt meulsk tu megmplemetsk lgortm tersebut mejd kode progrm source code deg bhs Vsul C++ sepert berkut : //*****Metode Elms Guss - solus sstem pers. Ler**** #clude <ostrem.h> vod m double A[0][0]; double B[0],X[0]; double c; t,,j,k; cout << " Ukur Mtrk : "; c >>; cout<<"mtrk A : "<<edl; or =;<=;++ or j=;j<=;j++ cout<<"a["<<<<j<<"] = "; c>>a[][j]; cout<<edl; cout<<" Vektor B : "<<edl; or =;<=;++ cout<<"b["<<<<"] = "; c>>b[]; cout <<edl; //Augmeted Mtrk or =;<=;++ A[][+]=B[]; or =;<=;++ A[][]==0 //tukr brs or k=+;k<=;k++ else Avd-06 65

66 Fsk Komputs or j=+;j<=;j++ //opers brs elemeter c= A[j][]/A[][]; or k=;k<=+;k++ A[j][k]=A[j][k]-c*A[][k]; //tmplk mtrk hsl opers brs elemeter or =;<=;++ or j=;j<=+;j++ cout<<"a["<<<<j<<"] = "<<A[][j]; cout <<edl; cout<<edl; //htug vrbel X or =;>=;-- == X[]=A[][+]/A[][]; else X[]=A[][+]; or j=+;j<=;j++ X[]=X[]-A[][j]*X[j]; X[]=X[]/A[][]; //tmplk X or =;<=;++ cout<<"x["<<<<"] = "<<X[]<<edl; Coblh jlk progrm tersebut d kemud coblh utuk meelesk mslh sstem persm ler berkut : 6 0 Berkut dlh hsl eksekus progrm utuk mslh tersebut dts : Ukur Mtrk : Avd-06 66

67 Fsk Komputs Mtrk A : A[] = A[] = A[] = A[] = A[] = A[] = - A[] = A[] = A[] = Vektor B : B[] = 6 B[] = B[] = 0 A[] = A[] = A[] = A[4] = 6 A[] = 0 A[] = A[] = - A[4] = -4 A[] = 0 A[] = 0 A[] = - A[4] = -6 X[] = X[] = X[] = 5. Metode Elms Guss-Jord Metode Elms Guss Jord hmpr mrp deg metode Guss, h sj pd metode, d bg sebelh kr dr Augmeted mtrk dubh mejd mtrk dgol sepert dperlhtk berkut : b b b b d d d d Peeles dr sstem persm ler dlh l d,d,d,,d d tu : = d, = d, = d,, = d Cr g dguk utuk medptk mtrk dgol dlm metode Guss Jord dlh sm sepert metode Guss tu deg meerpk Opers Brs Elemeter OBE. Cotoh : Selesk sstem persm ler berkut : Avd-06 67

68 Fsk Komputs Jwb : 4 8 Augmeted mtrk dr sstem persm ler tersebut dlh : 4 8 Lkuk OBE sebg berkut : B B B B B B 0 B B Sehgg ddpt peeles tu : 5.. Algortm Metode Elms Guss Jord berkut : Dr ur dts mk dpt dbut lgortm dr metode Guss Jord sepert.msukk mtrk A d vektor B besert ukur.but Augmeted Mtrk [A B], mk deg A.Utuk brs ke- dm = smp, perhtk pkh l, sm deg ol. Jk Y : Tukrk brs ke- deg brs +k, dm +k, tdk sm deg ol, bl tdk d berrt perhtug tdk bs dljutk d proses dhetk tp peeles Jk tdk : ljutk Jdk l dgol mejd stu, deg cr utuk setp kolom k dm k = smp +, htug, k, k, 4.Utuk brs ke-j, dm j = + smp, Lkuk Opers Brs Elemeter OBE : - htug c j, -utuk kolom k dm k = smp + htug j, k j, k c., k Avd-06 68

69 Fsk Komputs 5.Htug l, utuk = smp, 5.. Progrm komputer Metode Elms Guss Jord Dr lgortm Metode Guss Jord g sudh djbrk dlm sub bb sebelum mk seljut kt dpt megmplemetsk dlm betuk progrm komputer Compler Vsul C++ : //Metode Guss - Jord : solus sstem pers. ler #clude <ostrem.h> vod m double A[0][0]; double B[0],X[0]; double c; t,,j,k,brs,kol; cout << " Ukur Mtrk : "; c >>; cout<<"mtrk A : "<<edl; or =;<=;++ or j=;j<=;j++ cout<<"a["<<<<j<<"] = "; c>>a[][j]; cout<<edl; cout<<" Vektor B : "<<edl; or =;<=;++ cout<<"b["<<<<"] = "; c>>b[]; cout <<edl; //Augmeted Mtrk or =;<=;++ A[][+]=B[]; cout<<"augmeted Mtrk"<<edl; or =;<=;++ or j=;j<=+;j++ cout<<"a["<<<<j<<"] = "<<A[][j]; cout<<edl; cout<<edl; or =;<=;++ A[][]==0 //tukr brs or k=+;k<=;k++ Avd-06 69

70 Fsk Komputs else c=a[][]; or kol=;kol<=+;kol++ //eleme dgol dbut A[][kol]=A[][kol]/c; or brs=;brs<=;brs++ //opers brs elemeter brs!= c=a[brs][]; or kol=;kol<=+;kol++ A[brs][kol]=A[brs][kol]-c*A[][kol]; //tmplk mtrk hsl opers brs elemeter or =;<=;++ or j=;j<=+;j++ cout<<"a["<<<<j<<"] = "<<A[][j]; cout <<edl; cout<<edl; //htug vrbel X or =;<=;++ X[]=A[][+]; //tmplk X or =;<=;++ cout<<"x["<<<<"] = "<<X[]<<edl; Eksekuslh progrm tersebut d kemud coblh utuk meelesk mslh sstem persm ler berkut : 4 8 Berkut dlh hsl eksekus progrm utuk mslh tersebut dts : Ukur Mtrk : Mtrk A : A[] = A[] = A[] = Avd-06 70

71 Fsk Komputs A[] = 4 Vektor B : B[] = B[] = 8 Augmeted Mtrk A[] = A[] = A[] = A[] = A[] = 4 A[] = 8 A[] = A[] = 0 A[] = A[] = 0 A[] = A[] = X[] = X[] = 5.4 Metode Iters Guss Sedl Berbed deg metode sebelum g megguk Opers Brs Elemeter OBE mk dlm Metode Guss Sedl kt megguk proses ters hgg dperoleh l g koverge bertmbh kecl smp lebh kecl dr sutu kostt. Bl dberk sstem persm ler : b b b. m m m m b Berlh l wl utuk setp = smp kemud sstem persm ler dts dtuls mejd : b b b Avd-06 7 m Deg meghtug l-l = smp deg memk persm dts secr terus meerus hgg dperoleh sutu l g sm medekt deg