COMPOSITE TRAPEZOID RULE FOR THE RIEMANN-STIELTJES INTEGRAL AND ITS RICHARDSON EXTRAPOLATION FORMULA

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1 itli jourl of pure d pplied mthemtics (11 18) 11 COMPOSITE TRAPEZOID RULE FOR THE RIEMANN-STIELTJES INTEGRAL AND ITS RICHARDSON EXTRAPOLATION FORMULA Weijig Zho 1 College of Air Trffic Mgemet Civil Avitio Uiversity of Chi Tiji 0000 P. R. Chi e-mil: @qq.com Zhoig Zhg Zhiji Ye College of Air Trffic Mgemet Civil Avitio Uiversity of Chi Tiji 0000 P. R. Chi Abstrct. I this pper, the composite trpezoid rule for the Riem-Stieltjes itegrlis preseted d its error is ivestigted. Ad the, the rtiolity of the geerliztio of composite trpezoid rule for Riem-Stieltjes itegrl is demostrted. At lst, Richrdso extrpoltio is pplied to the composite trpezoid rule for the Riem- Stieltjes itegrl to obti high ccurcy pproximtios with little computtiol cost. Keywords: composite rule; trpezoid rules; Riem-Stieltjes itegrl; error term. 000 Mthemtics Subject Clssifictio: 65D0; 65D. 1. Itroductio I mthemtics, the Riem-Stieltjes itegrl is kid of geerliztio of the Riem itegrl, med fter Berhrd Riem d Thoms Stieltjes. It is Stieltjes 1 tht first give the defiitio of this itegrl i It serves s istructive d useful precursor of the Lebesgue itegrl. It is kow tht the Riem-Stieltjes itegrl hs wide pplictios i the field of probbility theory -, stochstic process 4 d fuctiol lysis 5, especilly i origil formultio of F. Riesz s theorem, 5 d the spectrl theorem for self-djoit opertors i Hilbert spce, 5. Defiite itegrtio is oe of the most importt d bsic cocepts i mthemtics. Ad it hs umerous pplictios i fields such s physics d egieerig. 1 Correspodig uthor.

2 1 w. zho, z. zhg, z. ye I severl prcticl problems, we eed to clculte itegrls. As is kow to ll, s for I = f(x)dx, oce the primitive fuctio F (x) of itegrd f(x) is kow, the defiite itegrl of f(x) over the itervl, b is give by Newto-Leibiz formul, tht is, (1.1) f(x)dx = F (b) F (). However, the explicit primitive fuctio F (x) is ot vilble or its primitive fuctio is ot esy to obti, such s e ±x, si x si x,, etc. Moreover, the x itegrd f(x) is oly vilble t certi poits x i, i = 1,,...,. How to get high-precisio umericl itegrtio formuls becomes oe of the chlleges i fields of mthemtics 6. I umericl lysis, the trpezoidl rule is the most well kow umericl itegrtio rules for pproximtig the defiite itegrl. Trpezoidl rule with error term for clssicl Riem itegrl is (1.) f(x)dx = b (f() + f(b)) (b ) f (ξ), 1 where ξ (, b). I spite of the my ccurte d efficiet methods for umericl itegrtio beig vilble i 7-9, recetly Mercer hs obtied trpezoid rule for Riem- Stielsjes itegrl which egeder geerliztio of Hdmrd s itegrl iequlity 10. Trpezoidl rule with error term for Riem-Stieltjes itegrl is (1.) (b ) f(t)dg = G g() f() + g(b) G f(b) f (ξ)g (η), 1 where G = 1, ξ, η (, b). b The Mercer develops Midpoit d Simpso s rules for Riem-Stielsjes itegrl 11 by usig the cocept of reltive covexity. Burg 1 hs proposed derivtive-bsed closed Newto-Cotes umericl qudrture which uses both the fuctio vlue d the derivtive vlue o uiformly spced itervls. Zho d Li hve proposed midpoit derivtive-bsed closed Newto-Cotes qudrture 1 d umericl superiority hs bee show. The, the derivtive-bsed trpezoid rule for the Riem-Stieltjes itegrl is preseted by Zho d Zhg 14, which uses derivtive vlues t the edpoits. Recetly, the midpoit derivtive-bsed trpezoid rule for the Riem-Stieltjes itegrl is preseted by Zho, Zhg d Ye 15, which oly uses derivtive vlues t the midpoit. The trpezoid rule or the Riem-Stieltjes itegrl uses oly two poits to pproximte itegrl, certily ot usuitble for use over lrge itegrtio itervls. A differet directio is to subdivide the itervl ito smller itervls d use lower-order schemes, like the trpezoid rule, o these smller itervls. I umericl lysis, Richrdso extrpoltio is sequece ccelertio method,

3 composite trpezoid rule for the riem-stieltjes itegrl... 1 used to improve the rte of covergece of sequece. It is used to geerte high-ccurcy results while usig low-order formuls. It is med fter Lewis Fry Richrdso, who itroduced the techique i Oe motivtio for the reserch preseted here lies i costructio of composite trpezoid rule for the Riem-Stieltjes itegrl, which is geerliztio of the results i 10, 11, 14. The other is pply Richrdso extrpoltio to the composite trpezoid rule for the Riem-Stieltjes itegrl to obti high ccurcy pproximtios with little computtiol cost. I this pper, we divide the itervl ito subitervls d pply itegrtio rule to ech subitervl, so the composite trpezoid rule for the Riem-Stieltjes itegrl is preseted. These ew scheme is ivestigted i Sectio. I Sectio, the error term is preseted. I Sectio 4, Richrdso extrpoltio is pplied to the composite trpezoid rule for the Riem-Stieltjes itegrl to obti high ccurcy pproximtios. Filly, coclusios re drw i Sectio 5.. Composite trpezoid rule for the Riem-Stieltjes itegrl I this sectio, by dividig the itervl ito subitervls d pplyig itegrtio rule to ech subitervl, composite trpezoid rule for the Riem-Stieltjes itegrl is preseted. Theorem.1 Suppose tht f(t) d g(t) re cotiuous o, b d g(t) is icresig there. Suppose tht the itervl, b is subdivided ito subitervls, +1 of width h = b by usig the eqully spced odes x k = + kh, for k = 0, 1,...,. The composite trpezoidl rule for subitervls c be expressed s follows: x1 f(t)dg T = g() f() b (.1) + 1 xk+1 xk f( ) + b g(b) b 1 f(b). x 1 Proof. By (1.), the trpezoidl rule for Riem-Stieltjes itegrl is 1 b (.) f(t)dg g() f() + g(b) 1 f(b). b b Applyig formul (.) over ech subitervl, we obti + f(t)dg 1 b x 1 b x1 x 1 g(x 1 ) g() f(x 1 ) + f() + g(x 1 ) 1 b g(x 1 ) 1 b x x 1 x1 f(x ) + f(x 1 )

4 14 w. zho, z. zhg, z. ye + 1 b 1 b xk 1 g( ) f( 1 ) + + g(x 1 ) f(x 1 ) + x 1 x 1 = g() f() + b + x x f(x ) + b x x 1 x 1 g( ) 1 b g(b) 1 xk+1 b x b x 1 f( ) + f(b) x 1 x 1 g(t) dt f(x 1 ) + x f(x 1 ) + g(b) f(b) b x 1 x b x 1 x1 = g() f() + 1 xk+1 xk f( ) b b 1 + g(b) f(b). b x 1 So, we hve the composite trpezoid rule for the Riem-Stieltjes itegrl s desired.. The error term for Riem-Stieltjes composite trpezoid rule I this sectio, the error term of the composite trpezoid rule for the Riem- Stieltjes is ivestigted. Theorem.1 Suppose tht f (t) d g (t) re cotiuous o, b d g(t) is icresig there. Other coditios re the sme s Theorem.1. The composite trpezoid rule for the Riem-Stieltjes itegrl with the error term is x 1 f(t)dg = g() f() b (.1) f( ) + b g(b) b 1 (b ) f(b) f (ξ)g (η), x 1 1 where ξ, η (, b). Ad the error term Rf of this method is (b ) (.) f (ξ)g (η). 1 Proof. We kow from (1.) tht i every subitervl the qudrture error is (.) h 1 f (ξ k )g (η k ), where ξ k, η k ( 1, ), k = 1,...,.

5 composite trpezoid rule for the riem-stieltjes itegrl Hece, the overll error is obtied by summig over such terms: (.4) h 1 f (ξ k )g (η k ) = h 1 f (ξ k )g (η k ). 1 Let M = 1 f (ξ k )g (η k ). Clerly, mi {f (x)g (x)} M mx {f (x)g (x)}. Sice f (t) d g (t) x,b x,b re cotiuous o, b, the there exists two poits ξ d η such tht M = f (ξ)g (η). This implies tht the error term is Rf = h 1 f (ξ)g (η). Recllig tht h = b, we obti Rf = (b ) 1 f (ξ)g (η). Corollry. Coditios re the sme s Theorem.1. Whe g(t) = t, equtio (.) reduces to the composite trpezoid rule for the clssicl Riem itegrl. Proof. It is esy to obti ( ) b, b b f(t)dg = f(t)dt b x 1 tdt = b, +1 tdt x 1 tdt = b d g (t) 1. By Theorem.1, 1 tdt = = b f() + ( ) b 1 f( ) + b (b ) f(b) f (ξ) b 1 ( ) = b 1 (b ) f() + f( ) + f(b) f (ξ). 1 Remrk. From Corollry., we kow tht the results i Theorem.1 possess the reducibility. Whe g(t) = t, formul (.1) reduces to the composite trpezoid rule for the clssicl Riem itegrl. So it is resoble geerliztio of composite trpezoid rule for Riem-Stieltjes itegrl. 4. Richrdso extrpoltio formul for the composite trpezoid rule for the Riem-Stieltjes itegrl Extrpoltio c be pplied wheever it is kow tht pproximtio techique hs error term with predictble form, oe tht depeds o prmeter, usully the step size h 8. I this sectio, we will illustrte how Richrdso extrpoltio pplied to results from the composite trpezoid rule for the Riem-Stieltjes itegrl c be used to obti high ccurcy pproximtios with little computtiol cost.

6 16 w. zho, z. zhg, z. ye Theorem 4.1 Suppose tht f (t) d g (t) re cotiuous o, b d g(t) is icresig there. Other coditios re the sme s Theorem.1. The Richrdso extrpoltio formul for the composite trpezoid rule for the Riem-Stieltjes itegrl is (4.1) f(t)dg T = 4 T 1 T. Proof. By equtio (.4), it is esy to obti the error term Rf of composite trpezoid rule for the Riem-Stieltjes itegrl is Rf = h 1 f (ξ k )g (η k ) = h 1 f (ξ k )g (η k ), 1 where ξ k, η k ( 1, ), k = 1,...,. By the defiitio of Riem itegrl, whe is big eough, we obti h 1 f (ξ k )g (η k ) = h h f (ξ k )g (η k ) h f (x)g (x)dx It mes (4.) Similrly, f(t)dg T h 1 f (x)g (x)dx. (4.) h f(t)dg T = ( 1 ) f (x)g (x)dx. Therefore, (4.4) tht is, f(t)dg T f(t)dg T 1 4. Thus, f(t)dg T 1 4 ( f(t)dg 4 T 1 T. f(t)dg T ), Deote T = 4 T 1 T, so we hve the Richrdso extrpoltio formul s desired. Corollry 4. Coditios re the sme s Theorem 4.1. The Richrdso extrpoltio formul for the composite trpezoid rule for the Riem-Stieltjes itegrl is 4th-order ccurte. Proof. We kow tht the composite trpezoid rule for the Riem-Stieltjes itegrl is secod-order ccurte (see equtio (.)).

7 composite trpezoid rule for the riem-stieltjes itegrl A more detiled study of the qudrture error revels tht the differece betwee f(t)dg d T c be writte s (4.5) f(t)dg = T + c 1 h + c h 4 + c k h k + o ( h k+). The exct vlues of the coefficiets, c k, re of o iterest to us s log s they do ot deped o h. We c ow write similr qudrture tht is bsed o double the umber of poits, i.e., T. Therefore, (4.6) f(t)dg = T + c 1 ( h ) + c ( h ) c k ( h ) k + o ( ( h 4*(4.6)-(4.5), this ebles us to elimite the h error term, f(t)dg = 4T T + ĉ h ĉ k h k + o ( h k+), where ĉ k = ( k 1 ) c k, k =,,... Tht is, ) k+ ). (4.7) f(t)dg = 4 T 1 T + c h c k h k + o ( h k+) = T + c h c k h k + o ( h k+). where c k = 1 ( k 1 ) c k, k =,,... By equtio (4.7), it is esy to obti tht the Richrdso extrpoltio formul for the composite trpezoid rule for the Riem-Stieltjes itegrl is 4thorder ccurte. 5. Coclusios We briefly summrize our mi coclusios i this pper s follows. 1) The composite trpezoid rule for the Riem-Stieltjes itegrl is preseted d its error term is ivestigted. ) The rtiolity of the geerliztio of composite trpezoid rule for Riem- Stieltjes itegrl is demostrted. ) Richrdso extrpoltio formul for the composite trpezoid rule for the Riem-Stieltjes itegrl is obtied. Ackowledgemets. This work is supported by the Scietific Reserch Foudtio of Civil Avitio Uiversity of Chi (No. 01QD01X), the Fudmetl Reserch Fuds for the Cetrl Uiversities (No. 1014C0).

8 18 w. zho, z. zhg, z. ye Refereces 1 Gordo, R.A., The itegrls of Lebesgue, Dejoy, Perro d Hestock, Americ Mthemticl Society, Providece, Billigsley, P., Probbility d Mesures, Joh Wiley d Sos, Ic., New York, Egghe, L., Costructio of cocetrtio mesures for geerl Lorez curves usig Riem-Stieltjes itegrl, Mthemticl d Computer Modellig, 5 (00), Kopp, P.E., Mrtigles d Stochstic Itegrls, Cmbridge Uiversity Press, Cmbridge, Rudi, W., Fuctiol Alysis, McGrw Hill Sciece, McGrw, Biley, D.H., Borwei, J.M., High-precisio umericl itegrtio: Progress d chlleges, Jourl of Symbolic Computtio, 46 (011), Atkiso, K., A Itroductio to Numericl Alysis, secod ed., Wiley, Burde, R.L., Fires, J.D., Numericl Alysis, Brooks/Cole, Bosto, Mss, USA, 9th editio, Iscso, E., Keller, H.B., Alysis of Numericl Methods, Joh Wiley d Sos, New York, Mercer, P.R., Hdmrd s iequlity d Trpezoid rules for the Riem- Stieltjes itegrl, Jourl of Mthemticl Alysis d Applictios, 44 (008), Mercer, P.R., Reltive covexity d qudrture rules for the Riem- Stieltjes itegrl, Jourl of Mthemticl iequlity, 6 (01), Burg, O.E., Derivtive-bsed closed Newto-Cotes umericl qudrture, Applied Mthemtics d Computtio, 18 (01), Zho, W.J., Li, H.X., Midpoit Derivtive-Bsed Closed Newto-Cotes Qudrture, Abstrct d Applied Alysis, Article ID 49507, Zho, W.J., Zhg, Z.N., Derivtive-Bsed Trpezoid Rule For The Riem-Stieltjes Itegrl, Mthemticl Problems i Egieerig, Article ID , Zho, W.J., Zhg, Z.N., Ye, Z.J., Midpoit Derivtive-Bsed Trpezoid Rule for the Riem-Stieltjes Itegrl, Itli Jourl of Pure d Applied Mthemtics, (014), Richrdso, L.F., Gut, J.A., The deferred pproch to the limit, Philosophicl Trsctios of the Royl Society, A 6 (197), Accepted: