Error Analysis of the High Order Newton Cotes Formulas

Transcription

1 Itertiol Jourl of Scietific d Reserch Pulictios, Volume 5, Issue, April 5 ISSN Error Alysis of the High Order Newto Cotes Formuls Omr A. AL-Smmrrie*, Mohmmed Ali Bshir** * Omdurm Islmic Uiversity ** Neeli Uiversity Astrct- The importce of umericl itegrtio my e pprecited y otig how frequetly the formultio of prolems i pplied lysis ivolves derivtives. It is the turl to ticipte tht the solutios of such prolems will ivolve itegrls. For most itegrls o represettio i terms of elemetry fuctios is possile, d pproximtio ecomes ecessry. As with y pproximte method, the utility of polyomil iterpoltio cot e stretched too fr. I this pper we shll qutify the errors tht c occur i polyomil iterpoltio d develop techiques to clculte such errors. Idex Terms- Newto s Forwrd Iterpoltio, Numericl itegrtio, Error, Polyomil, Newto Cotes. T I. INTRODUCTION he usul error sources re preset. However, iput errors i the dt vlues y, y,, y re ot mgified y most itegrtio formuls, so this source of error is ot erly so troulesome s it is i umericl differetitio. The tructio error, which is [y(x) P(x)] dx A wide vriety of efforts to estimte this error hve ee mde. A relted questio is tht of covergece. This sks whether, s cotiully higher degree polyomils re used, or s cotiully smller itervls h m etwee dt poits re used with lim h m =, sequece of pproximtios is produced for which the limit of tructio error is zero. I my cses, the trpezoidl d Simpso rules eig excellet exmples, covergece c e proved. Roud off errors lso hve strog effect. A smll itervl h mes susttil computtio d much roudig off. These lgorithm errors ultimtely oscure the covergece which should theoreticlly occur, d it is foud i prctice tht decresig h elow certi level leds to lrger errors rther th smller. As tructio error ecomes egligile, roud off errors ccumulte, limitig the ccurcy otile y give method. This pper costructs of: ) Astrct ) Itroductio 3) PRELIMINARIES ) ERROR ANALYSIS OF NEWTON COTES FORMULAS 5) REPORT 6) Coclusios II. PRELIMINARIES We first itroduce the Weierstrss Approximtio s oe of the motivtios for the use of polyomils. Here re the detils. Let f ε C[, ].d ε >. The there exists polyomil P of sufficietly high degree such tht f(x) P(x) < ε, for ll x ε [, ]. Sttemet ( Newto s Forwrd Iterpoltio Formul ) If x, x, x,, x re give set of oservtios with commo differece h d let y, y, y,, y re their correspodig vlues, where y = f(x) e the give fuctio the f(x) = y + p y + Where p = x x h + + p(p ) y! + p(p )(p ) 3 y 3! p(p )(p ) (p ( )) y! Itegrtio is summig process. Thus virtully ll umericl pproximtios c e represeted y I = f(x)dx = w i f(x i ) + Where : w i = weights, x i = odes, E t = tructio error. If f ε c [, ] d f (+) exists o (, ), the for y x ε [, ]. i= E t f(x) = k! f(k) (x )(x x ) k + [ k= ( + )! f(+) (ξ x )(x x ) + ] error term [ Tylor polyomil ]

2 Itertiol Jourl of Scietific d Reserch Pulictios, Volume 5, Issue, April 5 ISSN Where ξ x is sme poit etwee x d x. Let f(x) e rel-vlued fuctio defied o [, ] d + times differetile o (, ).If P (x) is the polyomil of degree which iterpoltes f(x) t the ( + ) distict poits x, x,. x ε[, ], the for ll x ε [, ], there exists poit ξ = ξ(x) ε (, ) such tht. E (x) = f(x) P (x) = f+ ( + )! ( x x i ) E (x) is clled the Iterpoltio error. Exmple If P(x) is the polyomil tht iterpoltes the fuctio f(x) = si (x) t poits o the itervl [, ], wht is the gretest possile error? I this exmple, we hve + = d f (+) (x) = f () (x) = si (x) So the lrgest possile error would e the mximl vlue of i= mx f ()! ( x x i ) For x, x,. x, ξ [, ].Clerly, o the itervl [, ] mx x x i = mx f (+) = mx si = Ad the mximl error would e! ()() = Propositio Let [, ] e rel itervl, <, d let I e the qudrture rule sed o iterpoltig polyomils for the distict poits x, x,, x ε [, ].If ω + (x) = i= (x x i ) hs oly oe sig o [, ], (i. e. ω + or ω + o [, ]), the, for ech f ε C + [, ], there exists τ ε[, ] such tht i= f(x)dx I (f) = f(+) (τ) ( + )! ω +(x) dx I prticulr, if f (+) hs just oe sig s well, the oe c ifer the sig of the error term (i.e. of the right-hd side ). III. ERROR ANALYSIS OF NEWTON COTES FORMULAS Suppose tht i= { i f(x i )} is the + poit closed Newto- Cotes formul with = x, = x d h =. There exists ξ ε (, ) for which is odd we get : f(x)dx i f(x i ) i= + h+ f (+) t(t ) (t )dt ( + )! The we get the Estimtio error At N : E = h+ f (+) ( + )! t(t ) (t )dt. Estimtio error At = ( Trpezoidl Rule ) Form equtio we get E = h3 f () t(t )dt =! h 3 f () (t t)dt =! h 3 f () [( t3! 3 t )] = h3 f () Exmple Fid the pproximte vlue of e x dx. e x dx e x dx = h [f() + f()] h3 f () ( ) h = = = [ + e ] = f() Here, f(x) = e x oud the secod derivtive, otiig f (x) = xe x is prticulrly simple, we c clculte d f (x) = ( + x )e x To fid the vlue of ξ we eed to get f = mx x f (x). We hve f (x) = ( + 8x + 6x )e x The we eed to fid the vlue of x, whe f (x) = ( + 8x + 6x )e x = Give us x = {, ± 6 6, ± }. The oly vlue tht`s i (, ) is 6 6, Tht is ξ = 6 6 The f () = ( + ξ )e ξ f() ( 6 6 ) = Ad y Trpezoidl Rule we get e x dx = [ + e ] = e x dx = = Estimtio error At = ( Simpso rule ) 3 Form equtio we get

3 Itertiol Jourl of Scietific d Reserch Pulictios, Volume 5, Issue, April 5 3 ISSN E = h f (3) t (t )(t )dt = o (3)! This does't me tht the error is zero. It simply mes tht the cuic term is ideticlly zero. The error term c e otied from the ext term i the Newto Polyomil, otiig E = h f (3) t (t )(t )dt. (3)! ecmes E = h5 f () ()! E = h5 f () ()! = h5 f () ()! t (t )(t )(t 3)dt t(t )(t )(t 3)dt (t 6t 3 + t 6t) dt = h5 f () [( t5 ()! 5 3t + t 3t )] E = h5 f () 9 Estimtio error of Newto Cotes formul of degree Form eq. we get E = h7 f (6) t(t )(t )(t 3)(t )dt = (6)! Agi, tht does't me tht the error is zero. The error term c e otied from the ext term i the Newto Polyomil. i.e. E = h7 f (6) t(t )(t )(t 3)(t )(t 5)dt = (6)! h 7 f (6) (t 6 5t t 5t 3 + 7t t)dt (6)! = h7 f (6) [( t7 (6)! 7 5t t5 55t 7t3 5t )] 3 = 8 95 h7 f (6) Estimtio error of Newto Cotes formul of degree Form eq. we get E = h5 f () t(t )(t )(t 3)(t ) ( (t 5)(t 6)(t 7)(t 8) ) dt = ()! (t 9)(t )(t )(t ) Ad, tht does't me tht the error is zero. The error term c e otied from the ext term i the Newto Polyomil. i.e. E = h5 f () t(t )(t )(t 3)(t )(t 5) ( (t 6)(t 7)(t 8)(t 9) ) ()! (t )(t )(t )(t 3) = h5 f () ()! (t 9 t t 99 t t t t t t t t t t 678 t)dt = h5 f () [( t5 ()! 5 t t3 99 t + 33 t t t9 t t t t t t t )] = h5 f () 5 ()! E = h5 f () Estimtio error of Newto Cotes formul of degree 3 E 3 = h5 f () 3 t(t )(t )(t 3)(t (t 5) ( (t 6)(t 7)(t 8)(t 9) ) dt ()! (t )(t )(t )(t 3) = h5 f () 3 (t 9 t t 99 t ()! t t t t t t t t t 678 t)dt = h5 f () [( t5 ()! 5 t t3 99 t + 33 t t t t t t6 t t t t )] 3 E 3 = h5 f () Note tht we use the sme polyomil i the lst two equtios. Ad we c see tht, oth of them hve the sme mout of h 5 d the sme umer of derivtives f (). Here we clerly fid out tht E < E 3.i.e.

4 Itertiol Jourl of Scietific d Reserch Pulictios, Volume 5, Issue, April 5 ISSN E E 3 = = Both equtios A & B, re deferet. As we see it i the grph. IV. REPORT I Newto Cotes formul, whe is eve the we hve. Report )- Exmple t(t ) (t )dt =. At = t(t )(t )dt At = = (t 3 3t + t)dt t(t )(t )(t 3)(t )dt = (t 5 t + 35t 3 5t + t)dt = At = = t(t )(t )(t 3)(t (t 5)(t 6)(t 7) (t 8)(t 9)(t )(t )(t )dt (t 3 78 t + 77 t 5577 t t t t 7 65 t t 5 888t t t t)dt = Report ) We c fid the we c fid error term of E, from two differet mys. Although we hve two deferet grphs 6t. A) f(t) = t 6t 3 + t B) f(t) = t 3t 3 + t. But they oth gives the sme mout of re form t = to t =. At = the error coefficiet will e From A). t(t )(t )(t 3)(t )(t 5)dt = (t 6 5t t 5t 3 + 7t + t)dt = 8. From B). t (t )(t )(t 3)(t )dt = (t 6 t t 5t 3 + t )dt = 8. As we see it i the grph A). From equtio to e otied from the ext term i the Newto Polyomil. i.e. E = h+ f (+) t(t ) (t )(t ( + ))dt. ( + )! B). From this equtio E = h+3 f (+) t (t ) (t )dt ( + )! Exmple At = the error coefficiet will e From A). t(t )(t )(t 3)dt = (t 6t 3 + t 6t)dt From B). t (t )(t )dt Note tht = (t 3t 3 + t )dt = 5 = 5 A) f(t) = t 6 5t t 5t 3 + 7t t B) f(t) = t 6 t t 5t 3 + t. Ech oe hve deferet grphs, ut they oth gives the sme mout of re form t = to t =.

5 Itertiol Jourl of Scietific d Reserch Pulictios, Volume 5, Issue, April 5 5 ISSN At = the error coefficiet will e From A) t(t )(t )(t 3)(t )(t 5) ( (t 6)(t 7)(t 8)(t 9) (t )(t )(t )(t 3) ) = (t 9 t t 99 t t t t t t t t t t 678 t)dt = From B) t (t )(t )(t 3)(t ) ( (t 5)(t 6)(t 7)(t 8) ) (t 9)(t )(t )(t ) = (t 78 t t 5577 t t t t t t t t 8688 t t )dt = V. CONCLUSION For ll N the estimtio error of the Newto Cotes formul will e h + f (+) ( + )! t(t ) (t )dt ; ( : odd) E = h +3 f (+) t(t ) (t )(t ( + ))dt ; ( : eve) { ( + )! Kowig tht t(t ) (t )dt. llwys =, whe is eve Whe is eve, we c fid coefficiet of E, from two differet equtios. h +3 f (+) ( + )! t (t ) (t )dt E = h +3 f (+) t(t ) (t )(t ( + ))dt { ( + )! 3 I Newto Cotes formuls of high degree, lso for low oce, The eve degree formuls re slightly more ccurte th the ext higher odd degree formuls. For exmple, lthough oth hve O(h ) ccurcy, the d degree formul (Simpso's rule) is slightly more ccurte th the 3rd degree formul (Simpso's 3/8 rule). I the mtter of the coefficiet of the estimted E. The ccurcy of the high order Newto Cotes, is much more etter th the low oce. ( E E = E < E ) ( E E 6 = E 6 < E ) ( E 6 E = E < E 6 ). REFERENCES [] Alkis Costtiides & Nvid Mostoufi, Numericl Methods for Chemicl Egieers with MATLAB Applictios, New Jersey, Pretice Hll,, Ch. 3 &. [] Prem K. Kythe, Michel R. Schäferkotter, Hdook of computtiol methods for itegrtio, Uited Sttes of Americ, Chpm & Hll/CRC Press, 5, P 5 6. [3] Peter Philip, Numericl Alysis I, Lecture Notes, LMU Muich 3, Ch. [] Todd Youg, Mrti J, Itroductio to Numericl Methods d Mtl Progrmmig for Egieers, Mohlekmp,Athes, OH,, Ch 3. [5] Doro Levy, Itroductio to Numericl Alysis, (CSCAMM) Uiversity of Mryld,, Ch. 6. [6] Christopher J. Zrowski, A itroductio to umericl lysis for electricl d computer egieers, A JOHN WILEY & SONS, INC. PUBLICATION,, Ch. 6 & 9. [7] Peter Arez, Numericl Methods for Computtiol Sciece d Egieerig. Computer Sciece Deprtmet, ETH Zürich, 3, Lecture. [8] Wrd Cheey, Dvid Kicid, Numericl Mthemtics d Computig, Sixth editio, Thomso Brooks/Cole, USA. 8, Ch., 5 & 6. [9] Richrd L. Burde, J. Dougls Fires, Numericl Alysis, Brooks/ Cole., Ch. 3 &. [] T. Gmill, Defiite Itegrls Newto Cotes, Deprtmet of Computer Sciece Uiversity of Illiois t Ur-Chmpig,, P -38. [] Uri M.Ascher, Che Greif, A First Course i Numericl Methods, The Society for Idustril d Applied Mthemtics, Cd,,Ch. & 5. [] D. CRUZ-URIBE, C. J. NEUGEBAUER., A Elemetry Proof of Error Estimtes for the Trpezoidl Rule, MATHEMATICS MAGAZINE, VOL 76. NO.. OCTOBER 3. [3] T. E. Simos, New Stle Closed Newto-Cotes Trigoometriclly Fitted Formule forlog-time Itegrtio. Hidwi Pulishig Corportio Astrct d Applied Alysis Volume. Article ID 8536, 5 pges. [] S. Ismil Azd, Itzr Husi, Kleem A. Qurishi d Mohd. Sdiq, A Numericl Algorithm for Newto-Cotes Ope d Closed Itegrtio Formule Associted with Eleve Eqully Spced Poits, Advces i Computer Sciece d its Applictios (ACSA) 369, Vol., No.,. [5] Omr A. AL-Smmrrie, Mohmmed Ali Bshir, Geerliztio of Newto s Forwrd Iterpoltio Formul, Itertiol Jourl of Scietific d Reserch Pulictios, Volume 5, Issue 3, Mrch 5. [6] N. S. Kmo, Error of the Newto-Cotes d Guss-Legedre Qudrture Formuls, MATHEMATICS OF COMPUTATION, VOLUME, NUMBER, APRIL, 97. [7] Nsri Akter Rip, Alysis of Newto s Forwrd Iterpoltio Formul, Itertiol Jourl of Computer Sciece & Emergig Techologies (E-ISSN: -6) Volume, Issue, Decemer. [8] J. H. Hetherigto, Note o Error Bouds for Numericl Itegrtio, MATHEMATICSO F COMPUTATION,V OLUME 7, NUMBER, APRIL, 973. [9] D. CRUZ-URIBE, S F & C. J. NEUGEBAUER, A Elemetry Proof of Error Estimtes for the Trpezoidl Rule, Triity College Hrtford,& Purdue Uiversity West Lfyette VOL. 76. NO.. OCTOBER 3. [] J. Wg, Fudmetls of erium-doped fier mplifiers rrys (Periodicl style Sumitted for pulictio), IEEE J. Qutum Electro., sumitted for pulictio. AUTHORS First Author Omr A. AL - Smmrrie, M.Sc., Omdurm Islmic Uiversity - Sud, ooomr55@yhoo.com. Secod Author Mohmmed Ali Bshir, Prof., Neeli Uiversity - Sud, mshir9@gmil.com.

6 Itertiol Jourl of Scietific d Reserch Pulictios, Volume 5, Issue, April 5 6 ISSN 5-353