An optimal 3-point quadrature formula of closed type and error bounds

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1 Revist Colombin de Mtemátics Volumen 8), págins 9- An optiml 3-point qudrture formul of closed type nd error bounds Un fórmul de cudrtur óptim de 3 puntos de tipo cerrdo y error de fronter Nend Ujević, Lucij Mijić University of Split, Split, Croti Abstrct. An optiml 3-point qudrture formul of closed type is derived. The obtined optiml qudrture formul hs better estimtions of error thn the well- known Simpson s formul. A few error inequlities for this formul re estblished. Key words nd phrses. Optiml qudrture formul, error inequlities, Ostrowski-like inequlities. Mthemtics Subject Clssifiction. 6D, A55. Resumen. Se estblece un fórmul de cudrtur óptim de 3 puntos de tipo cerrdo. Dich fórmul mejor l estimción de error de l bien conocid fórmul de Simpson. Se estblecen lguns desigulddes de error pr est fórmul. Plbrs y frses clve. Fórmul de cudrtur óptim, desigulddes de error, desigulddes de tipo de Ostrowski.. Introduction nd preliminry results In recent yers number of uthors hve considered error nlyses for qudrture rules of Newton-Cotes type. In prticulr, the mid-point, trpezoid nd Simpson rules hve been investigted more recently [], [3], [], [5], [6], [8]) with the view of obtining bounds on the qudrture rule in terms of vriety of norms involving, t most, the first derivtive. In the mentioned ppers explicit error bounds for the qudrture rules re given. These results re obtined from n inequlities point of view. The uthors use Peno type kernels for obtining specific qudrture rule. We sy tht ft)dt = n k=o w kfx k ) + R, where x < x < < x n b, w k, k =,,..., n, is qudrture 9

2 NENAD UJEVIĆ & LUCIJA MIJIĆ formul of closed type if the end points re included, i.e., = x, b = x n re nodl points. Qudrture formuls cn be formed in mny different wys. For exmple, we cn integrte Lgrnge interpolting polynomil of function f to obtin corresponding qudrture formul Newton-Cotes formuls). We cn lso seek qudrture formul such tht it is exct for polynomils of mximl degree Gussin formuls). Guss-like qudrture formuls re considered in [9]. Here we present new pproch to this topic. Nmely, we give type of qudrture formul. We lso give wy of estimting its error nd ll prmeters which pper in the estimtion. Then we seek qudrture formul of the given type such tht the estimtion of its error is best possible. Let us consider the bove described procedure with more detils. If we define K α, β, γ, δ, t) = t α)t β), t [ ], +b t γ)t δ), t +b, b], then, integrting by prts, we obtin K α, β, γ, δ, t)f t)dt = { f b) b γ) b δ) ) [ + b + b + f ) + b + b γ } f ) α) β) γ + δ α + β ) f ) + b α ) ] δ + α + β ) + b b γ + δ ) fb) + ft)dt. ) β ) f) If we choose α = β = nd γ = δ = b then we get the mid-point qudrture rule. If we choose α = γ = nd β = δ = b then we get the trpezoid rule. If we choose α =, β = +b 3 nd γ = +b 3, δ = then we get Simpson s rule. Volumen, Número, Año 8

3 AN OPTIMAL 3-POINT QUADRATURE FORMULA If we require tht b γ) b δ) = ) ) ) ) + b + b + b + b α β γ δ = α) β) =, then we get qudrture formul of the form K α, β, γ, δ, t)f t)dt = α + β ) γ + δ f) α + β ) f ) + b b γ + δ ) fb) + ft)dt. ) In prctice we cnnot find n exct vlue of the reminder term error) K α, β, γ, δ, t)f t)dt. All we cn do is to estimte the error. It cn be done in different wys. For exmple, K α, β, γ, δ, t)f t)dt mx f t) K α, β, γ, δ, t) dt. ) t [,b] It is nturl question which formul of the type ) is optiml, with respect to given wy of estimtion of the error. The min im of this pper is to give n nswer to this question nd to consider the formul from n inequlities point of view. In fct, we seek qudrture formul of the given type such tht its error bound is miniml. Note tht we cn minimize only the fctor K α, β, γ, δ, t) dt in ). A generl pproch is: we first consider the minimiztion problem nd then we formulte finl results. A few error inequlities for the obtined optiml formul re estblished. Let us mention tht the obtined optiml qudrture formul hs better estimtions of error thn the Simpson s formul see Remrk ). Finlly, we lso mention tht similr optiml qudrture rules re considered in []-[].. Optiml qudrture formul We consider the problem, described in Section, on the intervl [, ]. Let α, β, γ, δ R. We define the mpping K α, β, γ, δ, t) = ], t γ)t δ), t, ]. 3) Revist Colombin de Mtemátics

4 NENAD UJEVIĆ & LUCIJA MIJIĆ Let I R be n open intervl such tht [, ] I nd let f : I R be twice differentible function such tht f is bounded nd integrble. We denote Integrting by prts, we obtin K α, β, γ, δ, t)f t)dt = where f = sup ft). ) t [,] t α)t β)f t) + t γ)t δ)f t)dt = αβf ) + γ) δ)f ) 5) + [ ) ) ) )] ) α β γ δ f K α, β, γ, δ, t) = K α, β, γ, δ, t)f t)dt, t α+β, t [ ], t γ+δ, t., ] We require tht the coefficients αβ, [ α) β) γ) δ)] nd γ) δ) be equl to zero. Hence, we require tht α = or β = nd γ = or δ =. If we choose α = nd δ = then we get β + γ =. If we now substitute α =, γ = β nd δ = in 5) then we hve K, β, β,, t)f t)dt = = K, β, β,, t)f t)dt t β ) f t)dt t β ) f t)dt 6) = β f) β)f We lso hve K, β, β,, t)f t)dt f ) β f) + ft)dt. K, β, β,, t) dt, 7) Volumen, Número, Año 8

5 AN OPTIMAL 3-POINT QUADRATURE FORMULA 3 nd K, β, β,, t) dt = We now define gβ) = nd consider the problem t t β dt + t t β dt + t + β t)dt. 8) t + β t)dt, 9) minimize gβ), β R. ) Hence, we should like to find globl minimizer of g. We consider the following cses: i) β, ii) β, iii) β. Cse i). If β then t t β = tt β), for t [, ] nd t + β t = t + β)t ), for t, ]. Thus, gβ) = tt β)dt + = β 8. t + β)t )dt ) Cse iii). If β then t t β = tβ t), for t [, ] nd t + β t = t + β) t), for t, ]. Thus, gβ) = tβ t)dt + = β 8 8. Cse ii). If β then tβ t) t [, β] t t β = tt β) t ], β, t + β) t)dt ) Revist Colombin de Mtemátics

6 NENAD UJEVIĆ & LUCIJA MIJIĆ nd Thus, We hve t + β)t ) t [ t + β t =, β]. t + β) t) t β, ] gβ) = β + tβ t)dt + β = β3 3 β 8 +. t + β)t )dt + β tt β)dt 3) β t + β) t)dt g β) = β nd g β) = β. ) 8 We now solve the eqution g β) =. The solutions of this eqution re β, = ± ). Since g > we conclude tht β = is, t lest, locl minimizer. We hve ) g =. 5) 8 From ), ) nd 5) we conclude tht β = is the globl minimizer. If we now substitute β = in 6) then we get ) K,,,, t f t)dt 6) = ft)dt 8 f) ) f ) 8 f). The bove qudrture formul is optiml in the sense described in Section. From the previous considertions we cn formulte the following result. Theorem. Let I R be n open intervl such tht [, ] I nd let f : I R be twice differentible function such tht f is bounded nd integrble. Then we hve ft)dt 8 f) ) f ) 8 f) f 8. 7) Volumen, Número, Año 8

7 AN OPTIMAL 3-POINT QUADRATURE FORMULA 5 Remrk. If we set β = 3 in 6) then we get Simpson s rule: We hve ft)dt 6 [ f) + f ) ] + f) = K, 3, 3,, t ) f t)dt. 8) ft)dt 6 f) ) 3 f 6 f) f. 9) 8 It is obvious tht 7) is better estimte thn 9). Note tht 6) nd 8) re 3-point qudrture rules of the sme closed) type. 3. Error inequlities On the spce of squre integrble functions, L, b), we introduce the stndrd inner product f, g) = ft)gt)dt, ) with the corresponding norm written f. The resulting spce is Hilbert spce. We lso define the Chebyshev functionl T f, g) = f, g f, e g, e, ) where f, g L, b) nd e =. This functionl stisfies the pre-grüss inequlity T f, g) T f, f) T g, g). ) We now define σf) = σf;, b) = b )T f, f). 3) More bout the bove mentioned quntities cn be found, for exmple, in [7] nd []. Finlly, we define the functionl Qf) = Qf;, b) ) [ ) ) ] + b = ft)dt 8 f) + f + 8 fb) b ). We need the following lemm. Lemm 3. Let f t), t [, x ] ft) = f t), t x, b], 5) where x [, b], f C, x ), f C x, b). If f x ) = f x ) then f is n bsolutely continuous function. Revist Colombin de Mtemátics

8 6 NENAD UJEVIĆ & LUCIJA MIJIĆ A proof of this lemm cn be found in []. We now define b ) P f;, b) = 3 ) [f b) f )]. 6) 96 Note tht Qf;, b) P f;, b) = Rf;, b) is corrected qudrture formul with the reminder Rf;, b)) which is similr to the corrected trpezoid nd corrected mid-point qudrture formuls. It hs similr properties s the lst two mentioned formuls which re known nd cn be found in the literture. Here we only mention tht the corrected formul improves the originl formul. For the simplicity, in this pper we choose [, b] = [, ]. How we cn obtin corresponding results in the rbitrry intervl [, b] it is described, for exmple, in [3]. In this book we cn lso find how to write corresponding compound formuls. Theorem. Let f : [, ] R be n bsolutely continuous function such tht f L, ) nd there exist rel numbers γ, Γ such tht γ f t) Γ, t [, ]. Then Qf;, ) P f;, ) Γ γ ) , 7) 96 3 where Qf;, ) nd P f;, ) re defined by ) nd 6), respectively. If there exists rel number γ such tht γ f t), t [, ] then ) Qf;, ) P f;, ) S γ ), 8) 3 where S = f ) f ). If there exists rel number Γ such tht f t) Γ, t [, ] then ) Qf;, ) P f;, ) Γ S ). 9) 3 Proof. We define the function p t) = Let p be defined by Then we hve t ) p t) = t t ) t + + 3, t [ ], ) + 3, t, ]. 3) t 8, t [ ], t + 8, t. 3), ] p, f ) = p, f ) P f;, ) = Qf;, ) P f;, ) 3) Volumen, Número, Año 8

9 AN OPTIMAL 3-POINT QUADRATURE FORMULA 7 since p, f ) = Qf;, ), 33) holds. On the other hnd, we hve f Γ ) + γ, p = f, p ), 3) since p, e) =. From ) we get f Γ ) + γ, p since f Γ + γ Γ γ nd p = From 3)-35) we see tht 7) holds. We now prove tht 8) holds. We hve since f γ, p ) f γ p = f γ = f Γ + γ p 35) ) 5 9 Γ γ 6 3, 96 3, ) S γ ), 3 f t) γ ) dt = f ) f ) γ, nd p = 3. In similr wy we cn prove tht 9) holds. Remrk 5. Note tht we cn pply the estimte 7) only if the second derivtive f is bounded. It mens tht we cnnot use 7) to estimte directly the error when pproximting the integrl of such well-behved function s ft) = t 3 on [, ], since f t) = 3/ t) is unbounded on [, ]). On the other hnd, we cn use the estimtion 8), since γ = 3/ on [, ] for the given function). Theorem 6. Let f : [, ] R be n bsolutely continuous function such tht f L, ). Then 7 Qf;, ) P f;, ) σ f ;, ), 36) Revist Colombin de Mtemátics

10 8 NENAD UJEVIĆ & LUCIJA MIJIĆ where σf;, ) is defined by 3). The inequlity 36) is shrp in the sense tht the constnt cnnot be replced by smller one Proof. We define the function p t) = Then we hve t ) t ) t + ) t t [ ], t. 37), ] p, f = p, f p, e f, e, 38) since p = p p, e. From 3) nd 38) it follows T p, f ) = Qf;, ) P f;, ), 39) since p, f = p, f ) if [, b] = [, ]. From ) we get T p, f ) T p, p ) T f, f 7 ) = σf ;, ), ) since T p, p ) = From 39) nd ) we see tht 36) holds. We now prove tht 36) is shrp. For tht purpose we define the function ft) = such tht f t) = t ) t t 3 6 ) 6 t + ) t 6 ) 8 t t 8 t3, t [ ], ) t +, t t , ], ) 6 t, t [, ), t, ], ) nd f t) = p t). From Lemm 3 we see tht the function f, defined by ), is n bsolutely continuous function. For the function defined by ) the left-hnd side of 36) becomes L.H.S.36) = The right-hnd side of 36) becomes R.H.S.36) = We see tht L.H.S.36) = R.H.S.36). Thus, 36) is shrp. ] Volumen, Número, Año 8

11 AN OPTIMAL 3-POINT QUADRATURE FORMULA 9 Remrk 7. The estimtion 7) is better thn the estimtion 36). However, note tht we cn pply the estimte 7) only if the second derivtive f is bounded. It mens tht we cnnot use 7) to estimte directly the error when pproximting the integrl of such well-behved function s ft) = 3 t 5 on [, ], since f t) = / 9 3 t ) is unbounded on [, ]). On the other hnd, we cn use the estimtion 36) since f = 7 for the given function). Note lso tht the estimtion 36) is expressed by mens of the quntity σf ;, ). This is better estimtion thn n estimtion expressed by mens of the norm f, since σf ;, ) = f f ) f )) f. Furthermore, the term f ) f )) cn be esily clculted. References [] Atkinson, K., nd Hn, W. Theoreticl numericl nlysis: A functionl nlysis frmework. Springer-Verlg, New York-Berlin-Heidelberg,. [] Cerone, P. Three points rules in numericl integrtion. Nonliner Anl. Theory Methods Appl. 7, ), [3] Cruz-Uribe, D., nd Neugebuer, C. J. Shrp error bounds for the trpezoidl rule nd Simpson s rule. J. Inequl. Pure Appl. Mth. 3, ),. [] Drgomir, S. S., Agrwl, R. P., nd Cerone, P. On Simpson s inequlity nd pplictions. J. Inequl. Appl. 5 ), [5] Drgomir, S. S., Cerone, P., nd Roumeliotis, J. A new generliztion of Ostrowski s integrl inequlity for mppings whose derivtives re bounded nd pplictions in numericl integrtion nd for specil mens. Appl. Mth. Lett. 3 ), 9 5. [6] Drgomir, S. S., Pečrić, J., nd Wng, S. The unified tretment of trpezoid, Simpson nd Ostrowski type inequlities for monotonic mppings nd pplictions. Mth. Comput. Modelling 3 ), 6 7. [7] Mitrinović, D. S., Pečrić, J., nd Fink, A. M. Clssicl nd new inequlities in nlysis. Kluwer Acd. Publ., Dordrecht-Boston-London, 993. [8] Perce, C. E. M., Pečrić, J., Ujević, N., nd Vrošnec, S. Generliztions of some inequlities of Ostrowski-Grüss type. Mth. Inequl. Appl. 3, ), 5 3. [9] Ujević, N. Inequlities of Ostrowski-Grüss type nd pplictions. Appl. Mth. 9, ), [] Ujević, N. An optiml qudrture formul of open type. Yokohm Mth. J. 5 3), [] Ujević, N. Error inequlities for qudrture formul nd pplictions. Comput. Mth. Appl. 8, - ), [] Ujević, N. Two shrp Ostrowski-like inequlities nd pplictions. Meth. Appl. Anlysis, 3 ), [3] Volkov, E. A. Numericl methods. Mir Publishers, Moscow, 986. Recibido en mrzo de 8. Aceptdo en julio de 8) Revist Colombin de Mtemátics

12 NENAD UJEVIĆ & LUCIJA MIJIĆ Deprtment of Mthemtics University of Split Teslin /III, Split, Croti e-mil: Deprtment of Mthemtics University of Split Teslin /III, Split, Croti e-mil: Volumen, Número, Año 8