PERBAIKAN ATURAN KUADRATUR NEWTON-COTES TERTUTUP. Dina Oktavieny 1, Bustami 2 ABSTRACT

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1 PERBAIKAN ATURAN KUADRATUR NEWTON-COTES TERTUTUP Din Otvieny 1, Bustmi 2 1 Mhsisw Progrm Studi S1 Mtemti 2 Dosen Jurusn Mtemti Fults Mtemti dn Ilmu Pengethun Alm Universits Riu Kmpus Binwidy Penbru (28293), Indonesi otvienydin@gmil.com ABSTRACT This pper discusses the improvement of closed Newton-Cotes qudrture rules. The ide is bsed on deriving weights of closed Newton-Cotes qudrture rules hving the sme length of intervls using degree of ccurcy. By including the lower limit of the integrl nd the width of the intervl s the two new dditionl vribles, we obtin weights, the lower limit nd the width of the intervl used to form the improvement of closed Newton-Cotes qudrture rules. Computtionl tests using exmples show tht the ccurcy of improvement formul is better thn closed Newton-Cotes qudrture rules. Keywords: closed Newton-Cotes qudrture formuls, numericl integrtion methods, degree of ccurcy ABSTRAK Artiel ini membhs tentng perbin turn udrtur Newton-Cotes tertutup, yng beriden pd penentun bobot dri penurunn turn udrtur Newton- Cotes tertutup dengn lebr intervl yng sm menggunn derjt eteptn. Dengn mengiutn bts bwh integrl dn lebr intervl sebgi du vribel tmbhn diperoleh bobot bru, bts bwh dn lebr intervl yng digunn untu membentu formul perbin turn udrtur Newton-Cotes tertutup. Contoh omputsi yng dilun menunjun etelitin formul perbin yng diperoleh lebih bi dri turn udrtur Newton-Cotes tertutup. Kt unci: turn udrtur Newton-Cotes tertutup, integrsi numeri, derjt eteptn JOM FMIPA Volume 1 No. 2 Otober

2 1. PENDAHULUAN Integrsi numeri merupn sutu teni numeri untu mengprosimsi nili dri f(x)dx. Teni numeri yng sering dilun dlh dengn mengprosimsi f(x) menggunn polinomil interpolsi p n [5, h.12], yitu f(x)dx p n (x)dx. (1) Bil diberin bsis yng berbed x, x 1,..., x n pd [, b], formul integrsi numeri bertipe interpolsi dpt ditulis dlm bentu f(x)dx w i f(x i ), (2) dengn w i menytn bobot. Ad du cr [2, h. 74],[4, h. 79] untu menghitung oefisien bobot w i pd persmn (2). Cr pertm dlh dengn menginterpolsin f(x) di n + 1 titi x, x 1,, x n menggunn polinomil Lgrnge p n, dn emudin mengintegrsin polinomil tersebut untu memperoleh hsil dlm bentu hubungn persmn (2). Cr edu dlh dengn memilih onstnt w, w 1,, w n sedemiin hingg error E n (f) = (x)f(x)dx w i f(x i ), (3) bernili nol untu semu f(x) = x j, j =, 1, n. Kedu cr ini memperoleh hsil yng sm. Jdi, oefisien bobot w, w 1, w n dlh tunggl. Pd rtiel ini direview sebgin rtiel Deghn [3] untu mendptn formul integrsi berdsrn polinomil interpolsi, yitu perbin turn udrtur Newton-Cotes tertutup yng didsrn pd penentun bobot dri turn udrtur Newton-Cotes tertutup. Untu itu, dibgin du diberin turn udrtur Newton- Cotes tertutup, emudin dilnjutn di bgin tig tentng perbin turn udrtur Newton-Cotes dn di bgin empt dilun uji omputsi. Misln 2. ATURAN KUADRATUR NEWTON-COTES TERTUTUP p n (x) = f(x i ) w (x i )(x x i ), = (x x ) (x x n ), (4) dlh polinomil Lgrnge yng menginterpolsin f(x) di x, x 1, x n. ren itu formul integrsi pd persmn (2) menjdi [1, h.19] f(x)dx f(x i ) w (x i ) Oleh dx. (5) (x x i ) JOM FMIPA Volume 1 No. 2 Otober

3 yng dienl dengn formul integrsi Newton-Cotes. Bil intervl [, b] dibgi menjdi n subintervl dengn pnjng yng sm dengn titi-titi x i = + ih, i =, 1,, n, h = b n, m persmn rus nn (5) dpt dinytn dengn f(x i ) w (x i ) (x x i ) dx = f(x ) w (x ) dengn = (x x )(x x 1 ) (x x n ). menjdi f(x i ) b w (x i ) (x x i ) dx = f(x ) w (x ) + f(x 1) w (x 1 ) + f(x n) w (x n ) (x x ) dx + + f(x n) w (x n ) (x x )(x x 1 ) (x x n ) dx (x x ) (x x n ) dx. (6) Jdi persmn (6) dpt ditulis (x x )(x x 1 ) (x x n ) dx + (x x 1 ) (x x )(x x 1 ) (x x n ) dx. (7) (x x n ) Selnjutny perhtin suu e- pd persmn (7) tnp f(x ), yitu (x x ) (x x 1 )(x x )(x x +1 ) (x x n ) dx w (x )(x x ) = (x x ) (x x 1 )(x x +1 ) (x x n ) dx. (8) (x x ) (x x 1 )(x x +1 ) (x x n ) Bil dilun trnsformsi x = + th dn ren w (x ) = (x x ) (x x 1 )(x x +1 ) (x x n ), m persmn (8) menjdi (x x ) (x x 1 )(x x +1 ) (x x n ) (x x ) (x x 1 )(x x +1 ) (x x n ) dx n ( ) (t ) (t ( 1))(t ( + 1)) (t n) = h dt. (9) ( ) ( ( 1))( ( + 1)) ( n) Perhtin penyebut pd persmn (9), yitu ( )( 1)( 2) ( ( 2))( ( 1)) =!( 1) n (n )!. (1) Dengn demiin persmn (9) dpt dinytn dengn n ( ) (t ) (t ( 1))(t ( + 1)) (t n) h dt = ( ) ( ( 1))( ( + 1)) ( n) (b ) B (n) n, (11) JOM FMIPA Volume 1 No. 2 Otober

4 dengn B (n) = ( 1)n!(n )! n (t ) (t ( 1))(t ( + 1)) (t n)dt. (12) Dri persmn (11) dn persmn (12), persmn (7) dpt dinytn dengn f(x i ) b (b ) dx = w (x i ) (x x i ) n = Jdi diperoleh formul Newton-Cotes tertutup sebgi f(x)dx = = f( + h)b (n). (13) w (n) f( + h), (14) dengn w (n) (b ) = B (n) n. (15) Ad beberp subels dri formul Newton-Cotes yng bergntung pd nili bilngn bult n. Mislny, turn trpesium untu n = 1 dn turn Simpson untu n = 2 merupn du turn pertm dlm fmili Newton-Cotes. Untu n = 3 diperoleh formul Simpson 3/8 dn untu n = 4 formul Boole. Untu n = 3 diperoleh turn Simpson 3/8, yitu f(x)dx 3h 8 dengn error [4, h. 96] [ ( 2 + b ) ( + 2b ) f() + 3f + 3f 3 3 ] + f(b), (16) (b )5 E 3 (f) = 648 f (4) (ξ). (17) 3. PERBAIKAN ATURAN KUADRATUR NEWTON-COTES Aturn udrtur Newton-Cotes n + 1 titi yng diberin persmn (13) dpt dinytn dengn +nh f(x)dx w (n) f() + w (n) 1 f( + h) w (n) n f( + nh). (18) Formul ini dlh es untu polinomil berderjt n, yitu formul (13) bernili es untu fungsi dsr f(x) = x j, j =, 1,..., n. Jdi bobot w (n) i dpt ditentun dengn metode oefisien t tentu, yni dengn mengmbil +nh x j dx = w (n) i ( + ih) j, j =, 1, 2,... n. (19) JOM FMIPA Volume 1 No. 2 Otober

5 Persmn (19) dlh sistem persmn liner berorde (n+1) (n+1) dlm w (n) i, i =, 1,..., n. Ji dn h diethui dengn nili w (n) i dpt ditentun dengn menyelesin sistem liner pd persmn (19). Ide perbin turn Newton Cotes bermul dri sini, yitu ji dijdin dn h sebgi du vribel tmbhn dn sistem yng d pd persmn (19) dips hrus es untu polinomil dsr berderjt n + 2, m persmn (19) berubh menjdi +nh x j dx = ( + nh)j+1 () j+1 j + 1 ( + nh) j+1 () j+1 = w i ( + ih) j, j + 1 j =, 1,..., n + 2 (2) dimn, h, w, w 1, w 2,..., w n dlh n + 3 vribel yng hrus ditentun. Bil sistem (2) diselesin dengn Mple dengn tolernsi , diperoleh nili, h, w, w 1, w 2,..., w n untu n = 3,..., 9 sebgimn diberin Tbel 1[3]. Nili E n pd Tbel 1 dlh nili msimum error, yitu ji m e (j) n = ( + nh) j+1 j+1 j + 1 w i ( + ih) j, j =, 1, 2,..., n + 2, (21) E n = mx e (j) n, j n + 2. (22) Untu menerpn perbin turn udrtur Newton-Cotes tertutup dlm menghitung d c ϕ(t)dt, (23) terlebih dhulu dilun trnsformsi bts integrsi dri intervl [c, d] e intervl [, + nh = b] sehingg dengn f(x) = d c ϕ(t)dt = =+nh f(x)dx, (24) ( ) ( ) d c d c bc d ϕ x +. (25) b b b 4. UJI KOMPUTASI Beriut ini dilun uji omputsi untu membndingn turn udrtur Newton- Cotes dengn perbin turn udrtur Newton-Cotes. JOM FMIPA Volume 1 No. 2 Otober

6 Tbel 1: Nili numeri dri, h, w, w 1,..., w n n I n = [, + nh] w i E n = w = h = w 1 = w 2 = w 3 = = w = h = w 1 = w 2 = w 3 = w 4 = = w = h = w 1 = w 2 = w 3 = w 4 = w 5 = = w = h = w 1 = w 2 = w 3 = w 4 = w 5 = w 6 = w 7 = w 8 = = w = h = w 1 = w 2 = w 3 = w 4 = w 5 = w 6 = w 7 = w 8 = w 9 = Contoh 1 Hitunglh e x cos xdx, (26) secr nliti, dengn menggunn turn udrtur Newton-Cotes untu n = 3 JOM FMIPA Volume 1 No. 2 Otober

7 dn perbin turn udrtur Newton-Cotes. Solusi. Nili nliti dri ex cos xdx dlh Nili integrl (26) dengn menggunn Newton-Cotes untu n = 3 menggunn formul (16), dengn =, b = π, n = 3 dn h = π 3 dlh e t cos(t)dt = (27) Untu menentun (26) menggunn formul (19), pertm-tm ditrnsformsin intervl [, π] e [, +nh = b], dimn, h dlh nili-nili yng terdpt pd Tbel 1. Jdi dengn menggunn persmn (24) dn persmn (25) didpt dengn e t cos(t)dt = f(x)dx, (28) ( ( π π ) π f(x) = e nh nh) x+ ( π nh cos nh x + π ). (29) nh Dri persmn (29) dn Tbel 1 didpt f() = , f( + h) = , f( + 2h) = , dn f( + 3h) = , m diperoleh nili prosimsi untu (26) dengn perbin turn udrtur Newton-Cotes dlh e t cos(t)dt =(w f() + w 1 f( + h) + w 2 f( + 2h) + w 3 f( + 3h) e t cos(t)dt = (3) Dengn demiin error turn udrtur Newton-Cotes n = 3 dlh E NC4 = dn error perbin Newton-Cotes n = 3 dlh E INC4 = Ji omputsi di ts dilun untu n = 4, 5, 8, 9, m hsilny sebgimn disjin Tbel 2. Pd Tbel 2 terliht bhw hsil dri nili prosimsi dn error perbin turn udrtur Newton-Cotes lebih ecil dri turn udrtur Newton-Cotes bis. JOM FMIPA Volume 1 No. 2 Otober

8 Tbel 2: Hsil omputsi pengprosimsin ex cos(x)dx untu n = 4, 5, 8, 9. n Aturn Nili prosimsi Error 4 NC INC NC INC NC INC NC INC UCAPAN TERIMA KASIH Ungpn terim sih penulis ucpn epd Bp Dr. Imrn M., M.Sc. yng telh membimbing penulis dlm menyelesin rtiel ini. DAFTAR PUSTAKA [1] Burden, R. L. & Fires, J.D. 21. Numericl Anlysis, 7 th. Broos/ Cole Pub Co., New Yor. [2] Dvis, P. J. & Rbinowitz, P Methods of Numericl Integrtion, 2 nd Ed. Acdemic Press, Inc., London. [3] Dehghn, M., Jmie, M. M,. & Eslhchi, M. R. 25. On Numericl Improvement of Close Newton- Cotes Qudrture Rules. Applied Mthemtics nd Computtion. 165: [4] Krylov, V. I Approximte Clcultion of Integrl. McMiln Co., New Yor. [5] Phillips, G. M. 23. Interpoltion nd Approximtion by Polynomils. Springers-Verlg New Yor, Inc., New Yor. JOM FMIPA Volume 1 No. 2 Otober

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