Harmonic Mean Derivative - Based Closed Newton Cotes Quadrature

Transcription

1 IOSR Journl of Mthemtics (IOSR-JM) e-issn: - p-issn: 9-X. Volume Issue Ver. IV (My. - Jun. 0) PP - Hrmonic Men Derivtive - Bsed Closed Newton Cotes Qudrture T. Rmchndrn D.Udykumr nd R.Priml Deprtment of Mthemtics M.V.Muthih Government Arts College for Women Dindigul - 00 Tmil Ndu Indi. Deprtment of Mthemtics Government Arts College (Autonomous) Krur Tmil Ndu Indi. Astrct: A New method of evlution of Numericl integrtion y using Hrmonic Men derivtive - sed closed Newton cotes qudrture rule (HMDCNC) is presented in which the Hrmonic men vlue is used for Computing the function derivtive. It hs shown tht the proposed rule gives increse of single order of precision over the existing closed Newton cotes rule. The error terms re lso otined y using the concept of precision nd re compred with the existing methods. Finlly the ccurcy of the proposed rule is nlyzed using Numericl Exmples nd the results re compred with the existing methods. Keyword: Closed Newton-Cotes formul Error terms Hrmonic Men Derivtive Numericl Exmples Numericl Integrtion. I. Introduction Numericl integrtion is the study of how the numericl vlue of n integrl cn e found. It hs severl pplictions in the field of sttistics. In Sttistics it is used to evlute distriution functions nd other quntities. Mny recent Sttisticl Methods re dependent especilly on multiple integrtion possily in very high dimensions[].in the field of Mthemtics One of the most common method for the evlution of numericl integrtion is qudrture rule given y n w i f x i () i=0 Where there re (n+) distinct points =x 0 < x < < x n = x i =x 0 +ih i=0..n h = nd (n+) weights n w 0 w... w n.these w i cn e otined y using the method sed on the precision of qudrture formul. Select the vlues for w i i=0...n. so tht the error is zero tht is E n f = n w i f x i = 0 for f x = x j j = 0 n () i=0 Definition. An integrtion method of the form ( ) is sid to e of order P if it produces exct results (En[f] = 0) for ll polynomils of degree less thn or equl to P []. The list of Closed Newton cotes formuls (CNC) tht depend on the integer vlue of n re given elow When n= : Trpezoidl rule = f + f() When n =: Simpson's / rd rule f x dx = f + f + When n =: Simpson's / th rule = f + f + () f"(ξ) where ξ () + f + f + 0 f ξ where ξ () + f () 0 f ξ where ξ () DOI: 0.9/-00 Pge

2 When n= : Boole's rule = Hrmonic Men Derivtive - Bsed Closed Newton Cotes Qudrture f + f + + f + + f + + f f ξ where ξ () It is known tht the degree of precision is n+ for even vlue of n nd n for odd vlue of n. There re so mny methods re ville to increse the order of ccurcy in the closed Newton cotes formul. Dehghn et l. incresed the order of ccurcy in the closed Newton cotes formul[] y including the loction of oundries of the intervl s two dditionl prmeter nd rescling the originl integrl to fit the optiml oundry loctions. They hve pplied this technique to open semiopen Guss legendre nd Guss Cheyshev i n t e g r t i o n r ules[]. Burg h s p r o p o s e d d e r i v t i v e s e d c l o s e d o p e n n d M i d p o i n t q u d r t u r e r u l e s [ 9 0 ]. Weijing Zho nd Hongxing Li took different pproch y introducing Midpoint technique t the computtion of derivtive[]. Recently we proposed Midpoint derivtive sed open Newton cotes Qudrture rule[] nd Geometric men derivtive sed closed Newton cotes Qudrture rule[]. In this pper the use of Hrmonic men derivtive t the end points is investigted in closed Newton cotes qudrture formul. This new scheme gives single order of precision thn the existing formul. The error terms re otined nd compred with the existing method. The ccurcy of the proposed rule is oserved y illustrting the Numericl exmples. II. Hrmonic Men derivtive -sed Closed Newton Cotes Qudrture rule To evlute the definite integrl new method of evlution of Hrmonic men derivtive - sed Closed Newton Cotes formuls re derived. In this method the Hrmonic men derivtive is zero if either =0 or =0.Tht is this method is not pplicle if either =0 or =0. Theorem.. Closed T r p e z o i d l R u l e ( n = ) using H r mo n ic The precision of this method is. f + f() () f" + () M e n d erivtive is Since the rule () hs the degree of precision. Now u s e the rule () for f (x) = x. When f x = x x dx = ; n = + () =. It shows tht the solution is Exct. Therefore the precision of Closed T r p e z o i d l R u l e with H r mo n i c M e n d erivtive is. Theorem.. Closed S i m p s o n ' s / rd The precision of this method is. f + f + R u l e with H r mo n i c M e n d erivtive (n=) is () + f() 0 f + () Since the rule () hs the degree of precision.now u s e the rule () for f (x) = x. n = When f x = x x dx = ; () 0 =. It shows tht the solution is Exct. Therefore the precision of Closed S i m p s o n ' s / rd R u l e with H r mo n i c M e n d erivtive is. DOI: 0.9/-00 Pge

3 Theorem.. Closed S i m p s o n ' s / rd f + f + The precision of this method is. Hrmonic Men Derivtive - Bsed Closed Newton Cotes Qudrture + f + R u l e with H r mo n i c M e n d erivtive ( n=) is () + f 0 f + (9) Since the rule () hs the degree of precision.now u s e the rule (9) for f (x) = x. When f x = x x dx = ; n = () = 0. It shows tht the solution is Exct. Therefore the precision of Closed S i m p s o n ' s / rd R u l e with H r mo n i c M e n d erivtive is. Theorem.. Closed B o o l e ' s R u l e with H r mo n i c M e n d erivtive (n=) is The precision of this method is. f + f + + f + + f + + f () f + Since the rule () hs the degree of precision.now u s e the rule (0) for f (x) = x. n = + =. When f x = x x dx = ; DOI: 0.9/-00 Pge + + 0( + ) It shows tht the solution is Exct. Therefore the precision of Closed B o o l e ' s R u l e with M e n d erivtive is III. The Error terms of Hrmonic Men derivtive -sed Closed Newton Cotes Qudrture rule The error of pproximtion for the method sed on the precision of qudrture formul is otined y using the difference etween the qudrture formul for the monomil xp + p+! nd the exct result p+! x p+ (0) dx where p is the precision of the qudrture formul. Theorem.. H r mo n i c M e n d erivtive-sed Closed T r p e z o i d l R u l e ( n=)with the error term is f + f f" + where ξ ( ).This is f i f t h order ccurte with the error term E f = ( + ) f (ξ). Let f x = x!! x ( + ) f (ξ) () dx = ; () f + f() f" + = ( + ) ( + )( + ) () () Therefore + ( + )( + ) () = +.

4 Hrmonic Men Derivtive - Bsed Closed Newton Cotes Qudrture Therefore the Error term is E f = ( + ) f (ξ). Theorem.. H r mo n i c M e n d erivtive-sed Closed S i m p s o n ' s / rd the error term is f + f + where ξ ( ).This is s e v e n t h order ccurte with the error term E f = 0( + ) f (ξ). f + f + Let f x = x! + f Therefore 0 Therefore the Error term is 0( + ) R u l e (n=)with () + f() 0 f f ξ ()! x 0 f + = 0( + ) dx = 0 ; ( + ) () ( + ) () = 0( + ). E f = 0 + f ξ. Theorem.. H r mo n i c M e n d erivtive-sed Closed S i m p s o n ' s /th R u l e ( n=) with the error term is f + f + + f + + f Where ξ ( ).This is s e v e n t h order ccurte with the error term E f = ( + ) f (ξ) Therefore f + f + = Let f x = x! + f + + f! x dx = 0 f + 0 ; 0 f + ( + ) f (ξ) () + (( )( + ) ) 0 + (( )( + ) ) = ( + ). Therefore the Error term is E f = ( + ) f (ξ). DOI: 0.9/ Pge

5 Hrmonic Men Derivtive - Bsed Closed Newton Cotes Qudrture Theorem.. H r mo n i c M e n d erivtive-sed Closed B o o l e ' s r u l e ( n=) with the error term is f + f + + f + + f + + f f f ξ () Where ξ ( ).This is n i n t h order ccurte with the error term 9 E f = 00 + f ξ. + f + f = Therefore 00 Let f x = x! + f +! x + f + dx = 00 ; + f f +!. (( )( + )!. (( )( + ) () ) () ) 9 = Therefore the Error term is 9 E f = 00 + f ξ. The summry of Precision the orders nd the error terms for H r m o n i c men derivtive sed Closed Newton- Cotes Qudrture re shown in Tle. Tle : Comprison of Error terms Rules Precision Order Error terms Trpezoidl rule (n=) ( + ) f (ξ) Simpson's / rd rule (n=) 0 + f ξ Simpson's / th rule (n=) ( + ) f (ξ). Boole's rule (n=) f ξ IV. Numericl Results In this section The vlues of e x dx dx nd re estimted using the Hrmonic Men derivtive - sed closed Newton cotes formul nd the results re compred with the existing closed Newton-Cotes qudrture formul. The comprisons re shown in Tle nd. we know tht Error = Exct vlue - Approximte vlue +x Exmple.:solve e x dx nd compre the solutions with the CNC nd HMDCNC rules. DOI: 0.9/ Pge

6 Hrmonic Men Derivtive - Bsed Closed Newton Cotes Qudrture Solution: Exct vlue of e x dx=.0 Tle : Comprison of CNC nd HMDCNC rules vlue CNC HMDCNC of n Approximte vlue Error Approximte vlue Error n= n= n= n= Exmple : solve dx +x nd compre the solutions with the CNC nd HMDCNC rules. +x Solution: Exct vlue of dx =0.00 Tle : Comprison of CNC nd HMDCNC rules Vlue of CNC HMDCNC n Approximte vlue Error Approximte vlue Error n= n= n= n= V. Conclusion In this pper new scheme of Hrmonic men derivtive - sed Closed Newton- Cotes qudrture formuls were presented long with their error terms. This Hrmonic men derivtive vlue is included in the existing formul nd the error terms were derived y using the difference etween the qudrture formul for the monomils nd the exct results. This proposed scheme gives single order of precision thn the existing formul. Finlly the ccurcy of the proposed scheme is illustrted with Numericl exmples. References [] M.Evn nd T.Swrtz "Methods for pproximting integrls in sttistics with specil Emphsis on Byesin integrtion prolem"sttisticl Science vol.0 pp [] M.K.JinS.R.K.Iyengr nd R.K.Jin Numericl methods for Scientific nd Computtion New Age Interntionl (P) limited Fifth Edition 00. [] M.Dehghn M.Msjed-Jmei nd M.R.Eslhchi On numericl improvement of closed Newton-Cotes qudrture rules Applied Mthemtics nd Computtions vol. pp [] M.Dehghn M.Msjed-Jmei nd M.R.Eslhchi On numericl improvement of open Newton-Cotes qudrture rules Applied Mthemtics nd Computtions vol. pp [] M.Dehghn M.Msjed-Jmei nd M.R.Eslhchi The semi-open Newton-Cotes qudrture rule nd its numericl improvement Applied Mthemtics nd Computtions vol. pp [] E.Bolin M.Msjed-Jmei nd M.R.Eslhchi On numericl improvement of Guss - legendre Qudrture rules Applied Mthemtics nd Computtions vol.0 pp [] M.R.Eslhchi M.Dehghn nd M.Msjed-Jmei On numericl improvement of the first kind Guss-Cheyshev Qudrture rules Applied Mthemtics nd Computtionsvol. pp [ ] Clrence O.E.Burg Derivtive-sed closed Newton-cotes numericl qudrture Applied Mthemtics nd Computtions vol. pp [9] Fiz ZfrSir Sleem nd Clrence O.E.Burg New Derivive sed open Newton- cotes qudrture rules Astrct nd Applied AnlysisVolume 0 Article ID 09 pges0. [ 0 ] Clrence O.E.Burg nd Ezechiel Degny Derivtive-sed midpoint qudrture rule Applied Mthemtics nd Computtions vol. pp [] Weijing Zho nd Hongxing Midpoint Derivtive-Bsed Closed Newton-Cotes Qudrture Astrct nd Applied Anlysis vol.0 Article ID 0 pges0. [] T.Rmchndrn nd R.Priml"Open Newton cotes qudrture with midpoint derivtive for integrtion of Algeric furnctions"interntionl Journl of Reserch in Engineering nd TechnologyVol. pp [] T.RmchndrnD.Udykumr nd R.Priml"Geometric men derivtive -sed closed Newton cotes qudrture " Interntionl Journl of Pure & Engineering. MthemticsVol. pp DOI: 0.9/-00 Pge