Numerische Mathematik

Transcription

1 Numer. Mth. 2015) 131: DOI /s Numersche Mthemtk Automtc ntegrton usng symptotclly optml dptve Smpson qudrture Leszek Plskot Receved: 11 September 2013 / Revsed: 30 September 2014 / Publshed onlne: 25 November 2014 The Authors) Ths rtcle s publshed wth open ccess t Sprngerlnk.com Abstrct We present novel theoretcl pproch to the nlyss of dptve qudrtures nd dptve Smpson qudrtures n prtculr whch leds to the constructon of new lgorthm for utomtc ntegrton. For gven functon f C 4 wth f 4) 0 nd possble endpont sngulrtes the lgorthm produces n pproxmton to b f x) dx wthn gven ε symptotclly s ε 0. Moreover, t s optml mong ll dptve Smpson qudrtures,.e., needs the mnml number n f,ε) of functon evlutons to obtn n ε-pproxmton nd runs n tme proportonl to n f,ε). Mthemtcs Subject Clssfcton 65Y20 65D05 41A10 41A25 1 Introducton Consder numercl pproxmton of the ntegrl I f ) = b f x) dx 1) for functon f :[, b] R. Idelly we would lke to hve n utomtc routne tht for gven f nd error tolernce ε produces n pproxmton Q f ) to I f ) such tht t uses s few functon evlutons s possble nd ts error I f ) Q f ) ε. L. Plskot B) Fculty of Mthemtcs, Informtcs, nd Mechncs, Unversty of Wrsw, Bnch 2, Wrsw, Polnd e-ml: leszekp@mmuw.edu.pl

2 174 L. Plskot Ths s usully relzed wth the help of dpton. Recll generl prncple. For gven ntervl two smple qudrture rules re ppled, one more ccurte thn the other. If the dfference between them s suffcently smll, the ntegrl n ths ntervl s pproxmted by the more ccurte qudrture. Otherwse the ntervl s dvded nto smller subntervls nd the bove rule s recursvely ppled for ech of the subntervls. The oldest nd probbly most known exmples of utomtc ntegrton re dptve Smpson qudrtures [8 11], see lso [4] for n ccount on dptve numercl ntegrton. An unquestonble dvntge of dptve qudrtures s tht they try to mntn the error on prescrbed level ε nd smultneously djust the length of the successve subntervls to the underlyng functon. Ths often results n much more effcent fnl subdvson of [, b] thn the nondptve unform subdvson. For those resons dptve qudrtures re now frequently used n computtonl prctce, nd those usng hgher order Guss-Kronrod rules [1,5,15] re stndrd components of numercl pckges nd lbrres such s MATLAB, NAG or QUADPACK [13]. Nevertheless, to the uthor s knowledge, there s no stsfctory nd rgorous nlyss tht would expln good behvor of dptve qudrtures n quntttve wy or dentfy clsses of functons for whch they re superor to nondptve qudrtures. Ths pper s n ttempt to prtlly fll n ths gp. At ths pont we hve to dmt tht there re theoretcl results showng tht dptve qudrtures re not better thn nondptve qudrtures. Ths holds n the worst cse settng over convex nd symmetrc clsses of functons. There re lso correspondng dpton-does-not-help results n other settngs, see, e.g., [12,14,17,18]. On the other hnd, f the clss s not convex nd/or dfferent from the worst cse error crteron s used to compre lgorthms then dpton cn sgnfcntly help, see [2] or[16]. In ths pper we present novel theoretcl pproch to the nlyss of dptve Smpson qudrtures. We wnt to stress tht the restrcton to the Smpson rule s bsc component of composte rules s only for smplcty nd we could eqully well use hgher order qudrtures. The Smpson rule s reltvely smple qudrture nd therefore better enbles cler development of our des. To be more specfc, we nlyze the dptve Smpson qudrtures from the pont of vew of computtonl complexty. Allowng ll possble subdvson strteges our gol s to fnd n optml strtegy for whch the correspondng lgorthm returns n ε-pproxmton to the ntegrl 1) usng the mnml number of ntegrnd evlutons or, equvlently, the mnml number of subntervls. The mn nlyss s symptotc nd done ssumng tht f s four tmes contnuously dfferentble nd ts 4th dervtve s postve. To rech our gol we frst derve formuls for the symptotc error of dptve Smpson qudrtures. Followng [7] we fnd tht the optml subdvson strtegy produces the prtton = x0 < < x m = b such tht x ) 1/5 f 4) x) dx = m b f 4) x)) 1/5 dx, = 0, 1,...,m. Ths prtton s prctclly relzed by mntnng the error on successve subntervls on the sme level. The optml error correspondng to the subdvson nto m subntervls s then proportonl to L opt f ) m 4 where

3 Optml dptve Smpson qudrture 175 b L opt f ) = f 4) x)) 1/5 dx ) 5. For comprson, the errors for the stndrd dptve locl) nd for nondptve usng unform subdvson) qudrtures re respectvely proportonl to L std f ) m 4 nd L non f ) m 4 where b L std f )= b ) ) ) 4 1/4 b f 4) x) dx, L non f )= b ) 4 ) f 4) x) dx. Obvously, L opt f ) L std f ) L non f ). Hence the optml Smpson qudrture s especlly effectve when L opt f ) L std f ). An exmple s 1 δ x 1/2 dx wth smll δ.ifδ = 10 8 then L opt f ), L std f ), L non f ) re correspondngly bout 10 5, 10 8, Even though the optml strtegy s globl t cn be effcently hrnessed to utomtc ntegrton nd mplemented n tme proportonl to m. The only serous problem of how to choose the cceptble error ε 1 for subntervls to obtn the fnl error ε s resolved by splttng the recursve subdvson process nto two phses. In the frst phse the process s run wth the cceptble error set to test level ε 2 = ε. Then the cceptble error s updted to ε 1 = ε m 5/4 2 where m 2 s the number of subntervls obtned from the frst phse. In the second phse, the recursve subdvson s contnued wth the trget error ε 1. As noted erler, the mn nlyss s provded ssumng tht f C 4 [, b]) nd f 4) > 0. It turns out tht usng ddtonl rguments the obtned results cn be extended to functons wth f 4) 0 nd/or possble endpont sngulrtes,.e., when f 4) x) goes to + s x, b. For such ntegrls the optml strtegy works perfectly well whle the other qudrtures my even lose the convergence rte m 4. The contents of the pper s s follows. In Sect. 2 we recll the stndrd locl) Smpson qudrture for utomtc ntegrton. In Sect. 3 we derve formul for the symptotc error of Smpson qudrtures nd fnd the optml subdvson strtegy. In Sect. 4 we show how the optml strtegy cn be used to construct n optml lgorthm for utomtc ntegrton. The fnl Sect. 5 s devoted to the extensons of the mn results. The pper s enrched wth numercl tests where the optml dptve qudrture s compred wth the stndrd dptve nd nondptve qudrtures. We use the followng symptotc notton. For two postve functons of m, we wrte ψ 1 m) ψ 2 m) ff ψ 1 m) lm sup m ψ 2 m) 1, ψ 1 m) ψ 2 m) ff ψ 1 m) lm m ψ 2 m) = 1, ψ 1 m) ψ 2 m) ff lm sup m ψ 1 m) ψ 2 m) <,

4 176 L. Plskot ψ 1 m) ψ 2 m) ff 0 < lm nf m ψ 1 m) ψ 1 m) lm sup ψ 2 m) m ψ 2 m) <. A correspondng notton pples for functons of ε s ε 0. 2 The stndrd dptve Smpson qudrture In ts bsc formulton the locl dptve Smpson qudrture for utomtc ntegrton, whch wll be clled stndrd, cn be wrtten recursvely s follows. Let Smpsonu,v, f ) be the procedure returnng the vlue of the smple three-pont Smpsonruleon[u,v] for the functon f, nd let ε>0 be the error demnd. 0 functon STD, b, f, ε); 1 begn S1 := Smpson, b, f ); 2 S2 := Smpson, +b 2, f ) + Smpson +b 2, b, f ) ; 3 f S1 S2 15 ε then return S2 else 4 return STD, +b 2, f, 2 ε 5 end; ) + STD +b 2, b, f, ε 2 A justfcton of STD tht cn be found n textbooks, e.g., [3,6], s s follows. Denote by S 1 u,v; f ) the three-pont Smpson rule, ) u v S 1 u,v; f ) = f u) + 4 f 6 u + v 2 ) ) ) + f v), nd by S 2 u,v; f ) the composte Smpson rule tht s bsed on subdvson of [u,v] nto two equl subntervls, S 2 u,v; f ) = S 1 u, u + v ) u + v ; f + S ) v u 3u + v = f u) + 4 f 12 4 ) ) u + 3v +4 f + f v). 4 We lso denote I u,v; f ) = v u f x) dx. Suppose tht f C 4 [, b]). ),v; f ) + 2 f ) u + v If the ntervl [u,v] [, b] s smll enough so tht f 4) s lmost constnt, f 4) C nd C = 0, then S 2 u,v; f ) S 1 u,v; f ) = I u,v; f ) S 1 u,v; f )) I u,v; f ) S 2 u,v; f )) ) ) v u)5 = f 4) v u)5 ξ 1 ) f 4) ξ 2 ) 2

5 Optml dptve Smpson qudrture 177 v u) C 15 I u,v; f ) S 2 u,v; f )). 2) Now let = x 0 < < x m = b be the fnl subdvson produced by STD nd S std f ; ε) be the result returned by STD. Then, provded the estmte 2) holds for ny [x 1, x ], we hve m I f ) S std f ; ε) I x 1, x ; f ) S 2 x 1, x ; f ) = m S 1 x 1, x ; f ) S 2 x 1, x ; f ) =1 m =1 15 ε x x 1 b = ε. Ths resonng hs serous defect; nmely, the pproxmte equlty 2) cn be ppled only when the ntervl [u,v] s suffcently smll. Hence STD cn termnte too erly nd return completely flse result. In n extreme cse of [, b] =[0, 4] nd f x) = 4 =0 x ) 2 we hve I f )>0 but STD returns zero ndependently of how smll ε s. Of course, concrete mplementtons of STD cn be equpped wth ddtonl mechnsms to vod or t lest to reduce the probblty of such unwnted occurrences. To rdclly cut the possblty of premture termntons we ssume, n ddton to f C 4 [, b]), tht the fourth dervtve s of constnt sgn, sy, f 4) x) >0 for ll x [, b]. 3) Equvlently, ths obvously mens tht f 4) x) c for some c > 0 tht depends on f. Assumpton 3) ssures tht the mxmum length of the subntervls produced by STD decreses to zero s ε 0 nd the symptotc equlty 2) holds. Indeed, denote by Du,v; f ) the dvded dfference of f correspondng to 5 equspced ponts z j = u + jh/4, 0 j 4, where h = v u,.e., Snce Du,v; f ) = f [z 0, z 1, z 2, z 3, z 4 ] = 32 3h 4 f z 0) 4 f z 1 ) + 6 f z 2 ) 4 f z 3 ) + f z 4 )). the termnton crteron S 2 u,v; f ) S 1 u,v; f ) = h5 Du,v; f ), 27 S 2 u,v; f ) S 1 u,v; f ) 15 ε ) v u b

6 178 L. Plskot tht s checked n lne 3 of STD for the current subntervl [u,v], s equvlent to v u) Du,v; f ) ε. 4) b Our concluson bout pplcblty of 2) follows from the nequlty Du,v; f ) c/4! Observe tht ech splttng of subntervl [u,v] results n the symptotc) decrese of the controlled vlue n 4) by the fctor of 2 4. Thus the lgorthm symptotclly returns the pproxmton of the ntegrl wthn ε, s desred. Specfclly, we hve ε 16 Sstd f ; ε) I f ) ε s ε 0. 5) Remrk 1 The nequlty 5) explns why numercl tests often show better performnce of STD thn expected. To vod ths t s suggested to run STD wth lrger nput prmeter, sy 2ε nsted of ε. 3 Optmzng the process of ntervl subdvson The error formul 5) for the stndrd dptve Smpson qudrture does not sy nythng bout how the number m of subntervls depends on ε, or wht s the ctul error fter producng m subntervls. We now study ths queston for dfferent subdvson strteges. In order to be consstent wth STD we ssume tht for gven subdvson = x 0 < x 1 < < x m = b we pply S 2 x 1, x ; f ) for ech of the subntervls [x 1, x ], so tht the fnl pproxmton S m f ) = m S 2 x 1, x ; f ) =1 uses n = 4m + 1 functon evlutons. The gol s to fnd optml strtegy,.e., the one tht for ny functon f C 4 [, b]) stsfyng 3) produces subdvson for whch the error of the correspondng Smpson qudrture S m f ) s symptotclly mnml s m ). We frst nlyze two prtculr strteges, nondptve nd stndrd dptve, nd then derve the optml strtegy. In wht follows, the constnt γ = = = In the nondptve strtegy, the ntervl [, b] s dvded nto m equl subntervls [x 1, x ] wth x = +h, h = b )/m. Let the correspondng Smpson qudrture be denoted by Sm non. Then ) Sm non f ) I f ) = γb 1 m )4 f 4) ξ ) m 4 ξ [x 1, x ]) m =1

7 Optml dptve Smpson qudrture 179 γb ) 4 b ) f 4) x) dx m 4 6) s m. Observe tht for the symptotc equlty 6) to hold we do not need to ssume 3). We now nlyze the stndrd dptve strtegy used by STD. To do ths, we frst need to rewrte STD n n equvlent wy, where the nput prmeter s m nsted of ε. We hve the followng greedy lgorthm. The lgorthm strts wth the ntl subdvson = x 1) 0 < x 1) 1 = b. Inthe k + 1)st step, from the current subdvson = x k) 0 < < x k) k = b subntervl ] s selected wth the hghest vlue [x k) 1, xk) ) x k) x k) 4 ) D 1 x k) 1, xk) ; f, 1 k, 7) nd the mdpont x k) 1 + xk) )/2 s dded to the subdvson. Denote by Sm std f ) the result returned by the correspondng Smpson qudrture when ppled to m subntervls. Then, n vew of 4), the vlues Sm std f ) nd Sstd f ; ε) re relted s follows. Let m = mε) be the mnml number of steps fter whch 4) s stsfed by ech of the subntervls [x m) ]. Then 1, xm) We re redy to show the error formul for S std m S std m f ) = Sstd f ; ε). 8) correspondng to 6). Theorem 1 Let f C 4 [, b]) nd f 4) x) >0 for ll x [, b]. Then S std m f ) I f ) γb ) b f 4) x)) 1/4 dx ) 4 m 4 s m. Proof We fx l nd dvde the ntervl [, b] nto 2 l equl subntervls [z 1, z ] of length b )/2 l. Cll ths prtton corse grd, n contrst to the fne grd produced by S std m.let C = mx f 4) x), c = mn z 1 x z z 1 x z f 4) x). Let m be suffcently lrge, so tht the fne grd contns ll the ponts of the corse grd. Denote by z 1 = x,0 < x,1 < < x,m = z the ponts of the fne grd contned n the th ntervl of the corse grd, nd h, j = x, j x, j 1. Then the error cn be bounded from below s

8 180 L. Plskot m 2l Sm std f ) I f ) = S2 x, j 1, x, j ; f ) I x, j 1 ; x, j ; f ) ) = γ γ =1 j=1 2 l m h, 5 j f 4) ξ, j ) ξ, j [x, j 1, x, j ]) =1 j=1 2 l m h, 5 j c. =1 j=1 Suppose for moment tht for ll, j we hve h, 4 j c = A for some A. Then b )/2 l = m j=1 h, j = m A/c ) 1/4.Usng 2 l =1 m = m we get ) b 4 2 l 4 A = m 4. 2 l Observe now tht ny splttng of subntervl decreses h 4, j c by the fctor of 16. Hence =1 c 1/4 mx, j h 4, j c mn, j h 4, j c 16 nd consequently h 4, j c A/16 for ll, j. Thus S std m f ) I f ) γ 2l m =1 j=1 = 1 16 γb ) b 2 l A h, j 16 = 1 γb ) A 16 ) 4 2 l =1 c 1/4 4 m 4. To obtn the upper bound, we proceed smlrly. Replcng c wth C nd usng the equton h, 4 j C 16A we get tht ) b 4 2 l 4 Sm std f ) I f ) 16γ b ) 2 l C 1/4 m 4. To complete the proof we notce tht both =1 2 l =1 c 1/4 b )2 l nd 2 l =1 C 1/4 b )2 l re Remnn sums tht converge to the ntegrl b f 4) x) ) 1/4 dx s l.

9 Optml dptve Smpson qudrture 181 Remrk 2 From the proof t follows tht the constnts n the notton n Theorem 1 re symptotclly between 1/16 nd 16. The gp between the upper nd lower constnts s certnly much overestmted, see lso Remrk 4. The two strteges, nondptve nd stndrd dptve, wll be used s reference ponts for comprson wth the optml strtegy tht we now derve. We frst llow ll possble subdvsons of [, b] regrdless of the possblty of ther prctcl relzton. Proposton 1 The subdvson determned by ponts where x stsfy = x 0 < x 2 < < x m = b x ) 1/5 f 4) x) dx = m b f 4) x)) 1/5 dx, = 0, 1, 2,...,m, s optml. For the correspondng qudrture Sm we hve S m f ) I f ) γ b f 4) x)) 1/5 dx ) 5 m 4 s m. Proof We frst show the lower bound. Let S m be the Smpson qudrture tht s bsed on n rbtrry subdvson. Proceedng s n the begnnng of the proof of Theorem 1 we get tht for suffcently lrge m the error of S m s lower bounded by 2 l m ) b 5 S m f ) I f ) γ h, 5 j c 2 l γ =1 j=1 2 l c m 4 =1 where m s the number of subntervls of the fne grd n the th subntervl of the corse grd. We ssume wthout loss of generlty tht the corse grd s contned n the fne grd.) Mnmzng ths wth respect to m such tht 2 l =1 m = m we obtn the optml vlues ) 1/5 c m = m, 1 2 l. 2 l =1 c1/5 After substtutng m wth m n the error formul we fnlly get, S m f ) I f ) γ γ ) b 5 2 l 2 l b =1 c 1/5 5 m 4 f 4) x)) 1/5 dx ) 5 m 4. 9)

10 182 L. Plskot Snce for the optml m we hve h, j c1/5 = b 2 l m ) 2 l =1 c 1/5, 10) the lower bound 9) s ttned by the subdvson determned by {x }. Now the queston s whether the optml subdvson nto m subntervls of Proposton1 cn be prctclly relzed,.e., usng 4m + 1 functon evlutons. The nswer s postve, t lest up to n bsolute constnt. The correspondng lgorthm Sm opt runs s Sm std wth the only dfference tht n ech step t hlves the subntervl wth the hghest vlue [nsted of 7)]. ) x k) x k) 5 ) D 1 x k) 1, xk) ; f, 1 k 11) Theorem 2 Let f C 4 [, b]) nd f 4) x) >0 for ll x [, b]. Then S opt m f ) I f ) K γ where K 32. b f 4) x)) 1/5 dx ) 5 m 4 s m Proof The proof goes s the proof of the upper bound of Theorem 1 wth obvous chnges relted to the fcts tht now the lgorthm tres to blnce 11) [nsted of 7)], nd tht mx, j h 5, j C mn, j h 5, j C 32. Remrk 3 The best constnt K of Theorem 2 s certnly much less thn 32, see lso Remrk 4. We summrze the results of ths secton. All the three qudrtures Sm non, Sstd m, Sopt m converge t rte m 4 but the symptotc constnts depend on the ntegrnd f through the multplers b ) L non f ) = b ) 4 f 4) x) dx, b L std f ) = b ) f 4) x)) 1/4 dx ) 4,

11 Optml dptve Smpson qudrture 183 Tble 1 Vlues of L non, L std, L opt for 1 1 δ 2 dx wth x dfferent δ δ L non L std L opt Fg. 1 Error e versus m for 1 1 δ 2 dx wth δ = 10 2 x b L opt f ) = f 4) x)) 1/5 dx ) 5. These multplers ndcte how dffcult functon s to ntegrte usng gven qudrture. Obvously, by Hölder s nequlty we hve L opt f ) L std f ) L non f ). Exmple 1 Consder the ntegrl I δ = 1 δ 1 2 dx x wth 0 <δ<1. In ths cse L non, L std, L opt rpdly ncrese s δ decreses, s shown n Tble 1. Numercl computtons confrm the theory very well. We tested ll the three qudrtures the dptve qudrtures beng mplemented n m log m runnng tme usng hep dt structure) nd rn them for dfferent vlues of δ. Specfc results re s follows.

12 184 L. Plskot Fg. 2 Error e versus m for 1 1 δ 2 dx wth δ = 10 8 x Fg. 3 Km non, K m std, K m opt versus m for 1 1 δ 2 dx wth δ = 10 2 x For δ = 0.5 the qudrtures Sm non, Sstd m, nd Sopt m gve lmost dentcl results ndependently of m. For nstnce, for m = 10 2 the errors re respectvely , , , nd for m = 10 3 we hve , , Note tht the smllest error for the nondptve qudrture s cused by the fct tht Sm non hs lttle better bsolute constnt n the error formul 6) thn the dptve qudrtures. However, the smller δ, the more dfferences between the results. A chrcterstc behvor of the errors for δ = 10 2 nd δ = 10 8 s llustrted by Fgs. 1 nd 2. Observe tht n cse δ = 10 8 the nondptve qudrture needs more thn 10 4 subntervls to rech the rght convergence rte m 4.

13 Optml dptve Smpson qudrture 185 Remrk 4 It s nterestng to see the behvor of K qd m f ) = ) Sm qd f ) I f ) γ L qd f ) m 4, qd {non, std, opt}. By 6) we hve tht lm m Km non f ) = 1. The correspondng lmts for the dptve qudrtures re unknown; however, we rn some numercl tests nd we never obtned more thn 1.5. Ths would men, n prtculr, tht Sm opt s t most 50 % worse thn Sm. Fgure 3 shows the behvor of Km qd f ) for the ntegrl I δ of Exmple 1 wth δ = Automtc ntegrton usng optml subdvson strtegy We wnt to hve n lgorthm tht utomtclly computes n ntegrl wthn gven error tolernce ε. An exmple of such lgorthm s the recursve STD. Recll tht the recursve nture of STD llows to mplement t n tme proportonl to the number m of subntervls usng stck dt structure. However, t does not use the optml subdvson strtegy. On the other hnd, the lgorthm Sm opt uses the optml strtegy, but one does not know n dvnce how lrge m should be to hve the error Sm opt f ) I f ) ε.in ddton, the best mplementton of Sm opt tht uses hep dt structure) runs n tme proportonl to m log m. Thus the queston now s whether there exsts n lgorthm tht runs n tme lner n m nd produces n pproxmton to the ntegrl wthn ε usng the optml subdvson strtegy. Snce the optml subdvson s such tht the errors on subntervls re roughly equl, the suggeston s tht one should run STD wth the only dfference tht t s recursvely clled wth prmeter ε nsted of ε/2. Denote such modfcton by OPT. 0 functon OPT, b, f, ε); 1 begn S1 := Smpson, b, f ); 2 S2 := Smpson, +b 2, f ) + Smpson +b 2, b, f ) ; 3 f S1 S2 15 ε then return S2 else 4 return OPT, +b 2, f,ε) + OPT +b 2, b, f,ε) 5 end; Let S opt f ; ε) be the result returned by OPT. Anlogously to 8) wehve S opt m f ) = S opt f ; ε) f m s the mnml number of steps fter whch 11) s stsfed by ll subntervls.

14 186 L. Plskot It s cler tht OPT does not return n ε-pproxmton when ε s the nput prmeter. However we re ble to estmte posteror error. Indeed, let m 1 be the number of subntervls produced by OPT for n ε 1. Then m 1 ε 1 32 S opt f ; ε 1 ) I f ) m 1 ε 1 s ε ) We need to fnd ε 1 such tht m 1 ε 1 ε. Snce m 1 depends not only on ε 1 but lso on L opt f ), t seems hopeless to predct ε 1 n dvnce. Surprsngly ths s not true. The de of the lgorthm s s follows. We frst run OPT wth some ε 2 ε obtnng subdvson consstng of m 2 subntervls. Next, usng 12) nd Theorem 2 we estmte L opt f ), nd usng gn Theorem 2 we fnd the rght ε 1. Fnlly OPT s resumed wth the nput ε 1 nd wth subdvson obtned n the prelmnry run of OPT. As we shll see lter, ths de cn be mplemented n tme proportonl to m 1. Concrete clcultons re s follows. From the equlty α 2 m 2 ε 2 = S opt m 2 f ) I f ) = K 2 γ L opt f ) m 4 2 where α 2 nd K 2 depend on ε 2,wehve L opt f ) = α 2 K 2 γ ε 2 m ) We need ε 1 such tht for the correspondng m 1 the error of S opt m 1 f ) s t most ε,.e., α 1 m 1 ε 1 = S opt m 1 f ) I f ) = K 1 γ L opt f ) m 4 1 ε where α 1 nd K 1 depend on ε 1. Substtutng L opt f ) wth the rght hnd sde of 13) we obtn ) K1 α 2 ε 1/5 2 m 1 = m 2, 14) K 2 α 1 ε 1 nd solvng the nequlty α 1 m 1 ε 1 ε wth m 1 gven by 14) we get ε 1 β ε 5 ε 2 m 5 2 ) 1/4 where β = K 2 K 1 α 4 1 α 2 ) 1/4. Recll tht, symptotclly, α 1 nd α 2 re n [1/32, 1] whch mens tht β cn be symptotclly bounded from below by 1. Hence, tkng we hve ε 2 = ε nd ε 1 = ε m 5/4 2 15) S opt f ; ε 1 ) I f ) ε s ε 0.

15 Optml dptve Smpson qudrture 187 The choce of ε 1 gven by 15) s rther conservtve. In prctce, we observe tht the error of S opt f ; ε 1 ) s on verge even 6 or more tmes smller thn ε. Hence we encounter the sme phenomenon s for the stndrd Smpson qudrture, see Remrk 1. Yet, n the ltter cse, the error s usully not so much smller thn ε. As consequence, for ntegrnds f wth L opt f ) = L std f ) the pproxmton S std f ; ε) my use less subntervls thn S opt f ; ε 1 ). To vod n excessve work, we propose to run the optml lgorthm wth the nput B ε 1 nsted of ε 1 where, sy, B = 4 5/4 = 4 2 = Ths corresponds to α 1,α 2 = 1/4.) We stress tht such choce of B s bsed on some heurstcs nd s not justfed by ny rgorous rguments. Exmple 2 We present test results for the ntegrl I δ = 1 δ x 1/2 /2dx of Exmple 1 wth δ = 10 2 nd δ = 10 8, for the stndrd nd optml Smpson qudrtures. In Tbles 2 nd 3 the results re gven correspondngly for S std f ; ε) nd S opt f ; ε 1 ), whle n Tbles 4 nd 5 for S std f ; 2ε) nd S opt f ; 4 2ε 1 ). We end ths secton by presentng rther detled descrpton of the optml lgorthm for utomtc ntegrton tht runs n tme proportonl to m 1. It uses two stcks, Stck1 nd Stck2, correspondng to the two phses of the lgorthm. The elements of the stcks, elt, elt1, elt2, represent subntervls. Ech such element conssts of 6 felds contnng nformton bout: the endponts of the subntervl, functon vlues t the endponts nd t the mdpont, nd the vlue of the three-pont Smpson qudrture for ths subntervl. Such structure enbles evluton of f only once t ech smple pont. Push nd Pop re usul stck commnds for nsertng nd removng elements. Tble 2 Stndrd nd optml qudrtures for 1 1 δ 2 dx wth x δ = 10 2 ε Stndrd B = 1) Optml B = 1) Error m Error m 1.0E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 13 1, E 13 1,757

16 188 L. Plskot Tble 3 Stndrd nd optml qudrtures for 1 1 δ 2 dx wth x δ = 10 8 ε Stndrd B = 1) Optml B = 1) Error m Error m 1.0E E E E E E E E E E E E E E 08 1, E E E 09 1, E E E 10 3, E E E 11 6, E 11 1, E E 12 10, E 12 2, E E 13 19, E 13 4, 945 Tble 4 Stndrd nd optml qudrtures for 1 1 δ 2 dx wth x δ = 10 2 ε Stndrd B = 2) Optml B = 4 2) Error m Error m 1.0E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 13 1, E 13 1,143 Tble 5 Stndrd nd optml qudrtures for 1 1 δ 2 dx wth x δ = 10 8 ε Stndrd B = 2) Optml B = 4 2) Error m Error m 1.0E E E E E E E E E E E E E E E E E 09 1, E E E 10 2, E E E 11 5, E 11 1, E E 12 9, E 12 1, E E 13 16, E 13 3,223

17 Optml dptve Smpson qudrture functon OPTIMAL, b, f, ε); 01 begn elt.left := ; elt.rght := b; c := + b)/2; 02 fleft := f ); fcntr := f c); frght := f b); 03 elt.fl := fleft; elt.fc := fcntr; elt.fr := frght; 04 elt.smps := fleft + 4 fcntr + frght) b )/6; 05 PushStck1, elt); 06 m := 0; 07 repet elt := PopStck1); 08 l := elt.left; r := elt.rght; c := l + r)/2; 09 fleft := elt.fl; fcntr := elt.fc; frght := elt.fr; 10 cl := l + c)/2; cr := c + r)/2; 11 fcl := f cl); fcr := f cr); 12 Sleft := fleft + 4 fcl + fcntr) c l)/6; 13 Srght := fcntr + 4 fcr + frght) r c)/6; 14 S1 := elt.smps; S2 := Sleft + Srght; 15 elt1.left := l; elt1.rght := c; elt1.smps := Sleft; 16 elt1.fl := fleft; elt1.fc := fcl; elt1.fr := fcntr; 17 elt2.left := c; elt2.rght := r; elt2.smps := Srght; 18 elt2.fl := fcl; elt2.fc := fcr; elt2.fr := frght; 19 f S2 S1 <= 15 ε then 20 begn m := m + 1; 21 PushStck2, elt1, elt2) 22 end else 23 PushStck1, elt1, elt2) 24 untl StckEmptyStck1); 25 ε1 := B ε m 5/4 ; Result := 0.0; 26 repet elt := PopStck2); 27 l := elt.left; r := elt.rght; c := l + r)/2; 28 fleft := elt.fl; fcntr := elt.fc; frght := elt.fr; 29 cl := l + c)/2; cr := c + r)/2; 30 fcl := f cl); fcr := f cr); 31 Sleft := fleft + 4 fcl + fcntr) c l)/6; 32 Srght := fcntr + 4 fcr + frght) r c)/6; 33 S1 := elt.smps; S2 := Sleft + Srght; 34 f S2 S1 <= 15 ε1 then Result := Result + S2 else 35 begn elt1.left := l; elt1.rght := c; elt1.smps := Sleft; 36 elt1.fl := fleft; elt1.fc := fcl; elt1.fr := fcntr; 37 elt2.left := c; elt2.rght := r; elt2.smps := Srght; 38 elt2.fl := fcl; elt2.fc := fcr; elt2.fr := frght; 39 PushStck2, elt1, elt2) 40 end 41 untl StckEmptyStck2); 42 return Result 43 end;

18 190 L. Plskot 5 Extensons: f 4) 0 nd endpont sngulrtes We hve nlyzed dptve Smpson qudrtures ssumng tht f C 4 [, b]) nd f 4) > 0. It turns out tht the obtned results hold nd utomtc ntegrton cn be successfully ppled lso for functons wth f 4) 0 nd functons wth endpont sngulrtes. An observed good behvor of dptve qudrtures for such functons cnnot be explned usng drectly prevous tools. Wht we need s non-symptotc error bound for S 2 u,v; f ). Such bound, together wth the correspondng result for S 1 u,v; f ), s provded by the followng lemm. Lemm 1 Suppose tht f C[u,v]) nd f C 4 [u 1,v 1 ]) for ll u < u 1 <v 1 < v. If, n ddton, f 4) x) 0 for ll x u,v), then nd wth conventon tht 0/0 = 1). 1 S 1u,v; f ) I u,v; f ) S 1 u,v; f ) S 2 u,v; f ) 2 0 S 2u,v; f ) I u,v; f ) S 1 u,v; f ) S 2 u,v; f ) 1 Proof Gven c u,v), we hve tht for ny x [u,v] x x t) 3 f x) = T c x) + f 4) t) dt, 16) c 3! where T c s Tylor polynoml for f of degree 3 t c. The formul s obvous for x, b) nd by contnuty of f t extends to x = u,v.) Furthermore, ntegrtng 16) wth respect to x we get tht v v c u t) 4 v f x) dx = T c x) dx + f 4) v t) 4 t) dt + f 4) t) dt. u u u 4! c 4! 17) Usng 16) forz j = u + jh/4, 0 j 4, h = v u, we then obtn S 1 u,v; f ) S 2 u,v; f ) = h5 2 7 Du,v; f ) = v wth the Peno kernel ψ 0 u,v; t) = h 4 0 t u)/h) where t 3 /72, 0 t 1/4, 0 t) = t 3 4t 1/4) 3 )/72, 1/4 < t 1/2, 0 1 t), 1/2 < t 1. u ψ 0 u,v; t) f 4) t) dt 18)

19 Optml dptve Smpson qudrture 191 For the error of S 1 we smlrly fnd tht S 1u,v; f ) I u,v; f )) = where ψ 1 u,v; t) = h 4 1 t u)/h), Snce 1 t) = v u ψ 1 u,v; t) f 4) t) dt, { 5t 3 1/3 t/2)/64, 0 t 1/2, 1 1 t), 1/2 < t t) 0 t) 15, t 0, 1) 8 nd both bounds re shrp), we get the desred bounds. For the error of S 2 u,v; f ) we nlogously fnd tht 15 S 2 u,v; f ) I u,v; f )) = v where the kernel ψ 2 u,v; t) = h 4 2 t u)/h), u ψ 2 u,v; t) f 4) t) dt, 5t 3 1/3 t)/8, 0 t 1/4, 2 t) = 2 1/2 t), 1/4 < t 1/2, 2 t 1/2), 1/2 < t 1. The remnng bound follows from the nequlty 2 t) 15, t 0, 1). 0 t) The Peno kernels 0, 1, nd 2 re presented n Fg. 4. In wht follows we concentrte on generlzng Theorem 2 bout Sm opt snce the other results Theorem 1 nd Proposton 1) cn be treted n smlr fshon. Frst we prove tht the ssumpton f 4) > 0 n Theorem 2 cn be replced by f 4) x) 0 x [, b]. 19) Proof Suppose wthout loss of generlty tht f 4) s not everywhere zero n [, b]. We frst produce course grd {z } =1 2l of length b )/2l nd remove from t ll the ponts z 1 2 l 1) such tht f 4) x) = 0 x [z 1, z +1 ].

20 192 L. Plskot Fg. 4 The Peno kernels 0 one hump, thck lne), 1 one hump, thn lne), nd 2 two humps). The curves ntersect t 1/3nd2/3 Denote the successve ponts of the modfed grd by {ẑ } k =1, k 2l.Let C = mx f 4) x), c = mn f 4) x), ẑ 1 x ẑ ẑ 1 x ẑ J 0 ={ : c = 0}, J 1 ={ : c > 0}, nd P t = J t [ẑ 1, ẑ ], t = 0, 1. From 18) t follows tht subntervl s further subdvded f nd only f f 4) 0 n ths subntervl. Hence for suffcently lrge m the corse grd s contned n the fne grd produced by Sm opt nd the subntervls [ẑ 1, ẑ ] wth C > 0 hve been subdvded t lest once. Let ẑ 1 = x,0 < < x,k = ẑ be the ponts of the fne grd contned n [ẑ 1, ẑ ], nd h, j = x, j x, j 1. Defne β = mx, j S 1 x, j 1, x, j ; f ) S 2 x, j 1, x, j ; f ). 20) We now mke n mportnt observton tht for ny J 0 wth C > 0 nd ny 1 j k β 15 γ2h, j ) 5 C. 21) Indeed, f ths were not stsfed by subntervl [x, j 1, x, j ] then ts predecessor, whose length s 2h, j nd belongs to the th subntervl of the corse grd, would not be subdvded.

21 Optml dptve Smpson qudrture 193 Hence, denotng by m 0 the number of subntervls of the fne grd n P 0,wehve m 0 β = m 0 β 1/5) 5 m γ M 5 0 m 4 0 wth M 0 = k J 0 j=1 h, j C 1/5. 22) Ths mples m γ) 1/5 M 0 β 1/5. Denotng by m 1 the number of subntervls of the fne grd n P 1,wehve m 1 β = m 1 β 1/5 ) 5 m γ M 5 1 m 4 1 wth M 1 = k J 1 j=1 h, j c 1/5, 23) whch mples m 1 15γ) 1/5 M 1 β 1/5. Hence m 0 /m 1 2M 0 /M 1 nd ths bound s ndependent of m. However t depends on l. Tkng l lrge enough we cn mke m 0 /m 1 rbtrrly smll. From Lemm 1 t follows tht the ntegrton error n P 0 s upper bounded by m 0 β. Snce f 4) s postve n P 1, the error n P 1 s symptotclly s m )lower bounded by m 1 β/15 32). Hence for l lrge enough the error n P 0 s rbtrrly smll compred to tht n P 1. In ddton, the error n P 1 follows the upper bound of Theorem 2. The proof s complete. We now pss to functons wth endpont sngulrtes. To fx the settng, we ssume tht f s contnuous n the closed ntervl [, b] nd f C 4 [ 1, b]) for ll < 1 < b. Moreover, lm f 4) x) =+, x nd ths dvergence s symptotclly monotonc,.e., there s δ>0 such tht f 4) x 1 ) f 4) x 2 )>0 for ll < x 1 x 2 + δ. As before, we prove tht for such functons the upper error bound for Sm opt 2 s stll vld. n Theorem Proof Frst, we observe tht the dfference S 1, +h; f ) S 2, +h; f ) converges to zero fster thn h. Indeed, n vew of 18) wehve S 1, + h; f ) S 2, + h; f ) = h h +h +h h 3 0 t )/h) f 4) t) dt 0 t ) f 4) t) dt. 24) Ths ssures tht the prtton s denser nd denser n the whole [, b] nd the ntegrton error goes to zero.

22 194 L. Plskot Second, we hve tht L opt f )<. Indeed, by Hölder s nequlty b f 4) x)) 1/5 dx = b ) 1/5 x ) 3/5 x ) 3 f 4) x) b ) 4/5 b 1/5 x ) 3/4 dx x ) 3 f 4) x) dx), whch s fnte due to 16). Now, let l be such tht b )2 l δ, nd let {z } k = wth k = 2l 1bethe nfnte) corse grd defned s { + b )2 z = l+, 1, + b )2 l + 1), 0 k. Denote, s before, C = mx z 1 x z f 4) x). We obvously hve C = f 4) z 1 )> 0 for ll 0. For smplcty, we lso ssume C > 0for1 k. Let m be suffcently lrge so tht the fne grd produced by Sm opt contns ll the ponts z for 0. Moreover, we cn ssume tht ech subntervl [z 1, z ] wth 1 hs been subdvded t lest once. Let [, z s ] be the frst subntervl of the fne grd. Let us further denote P 0 =[, z 0 ] nd P 1 =[z 0, b]. Then P 0 = P 0,0 P 0,1 where P 0,0 conssts of [0, z s ] nd ll subntervls of the course grd tht hve not been subdvded by Sm opt.letm 0,0, m 0,1, m 1 be the numbers of subntervls of the fne grd n P 0,0, P 0,1, P 1, respectvely. Defne β s n 20). In vew of 24), the dstnce z s ) decreses slower thn β s m, nd therefore m 0,0 s t most proportonl to log 2 1/β). Snce 21) holds for the subntervls n P 0,1, the number m 0,1 cn be estmted s m 0 n 22) wth M 0 0 = s+1 z z 1 )C 1/5 2 z 1 0 f 4) x)) 1/5 dx, where the lst nequlty follows from monotoncty of f 4). Snce m 1 cn be estmted s n 23) we obtn, nlogously to the prevous proof, tht the number of subntervls n P 1 nd the error n P 1 domnte the scene. The proof s complete. We stress tht for contnuous functons wth endpont sngulrtes we lwys hve L opt f )<, whch s not true for L std f ). An exmple s provded by f x) = 1 x t x) 3 f 4) t) dt, 0 t 1, 3! wth f 4) x) = t ln t) 4. Indeed, snce f 0) = 1 0 3! t ln4 t) 1 dt <, the functon s well defned nd L opt f )<, but

23 Optml dptve Smpson qudrture 195 Fg. 5 Error e versus m for x dx L std f ) = dt =. t ln t For such functons, the subdvson process of Sm std wll not collpse [whch follows from 24)] nd the error wll converge to zero; however, the convergence rte m 4 wll be lost. On the other hnd, f L std f )< then the error bounds of Theorem 1 hold true. Exmple 3 Consder the ntegrl 1 0 p + 1) x p dx. The ntegrnd s contnuous t 0 only f p 0. Then both, L opt f ) nd L std f ), re fnte. However, L non f )< only f p s non-negtve nteger or p > 3. Fgures 5 nd 6, where the results for p = 1/2 nd p = 1/20 re presented, show tht, ndeed, the dptve qudrtures Sm std nd Sopt m converge s m 4, nd Sm non converges t very poor rte. We end ths pper by showng the mportnce of contnuty of f. Exmple 4 Consder the ntegrl b f x) dx wth = 1/2, b = 1, { 0, 1/2 x 0, f x) = x 1/2 /2, 0 < x 1. In ths cse L opt f )< but L std f ) =. Fgure 7 shows tht Sm opt enjoys the rght convergence m 4 but Sm std completely fls. Ths s becuse the crtcl vlue

24 196 L. Plskot Fg. 6 Error e versus m for x dx Fg. 7 Error e versus m for functon f of Exmple 4 mx S 1 x m) 1, xm) ) ; f S 2 x m) 1, xm) ) ; f does not converge fster thn h; the lgorthm keeps dvdng the subntervl contnng 0. As result, the stndrd dptve lgorthm s symptotclly even worse thn the nondptve lgorthm. Eqully strkng s the dfference between OPTIMAL nd STD. Whle OPTIMAL works perfectly well, see Tble 6, STD wll never rech the stoppng crteron for ε 10 3, nd wll loop forever. s consequence of our lck of bd luck rther thn rule. Indeed, t s enough to chnge the vlue of f n [ 1/2, 0] from 0 to 7/3 to see tht then S 1, +b)/2; f ) Unfortuntely, ths exmple s msledng. The very good behvor of S opt m

25 Optml dptve Smpson qudrture 197 Tble 6 Optml qudrture for functon f of Exmple 4 ε Optml B = 4 2) Error m 1.0E E E E E E E E E E E E E E E E 11 1, E E 12 2, E E 13 4,131 S 2, + b)/2; f ) = 0 lthough +b)/2 f x) dx = 19/12 > 0. As consequence, lm m Sm opt f ) = 13/6 whle the ntegrl equls 25/12. Acknowledgments The uthor would lke to thnk Grzegorz Wslkowsk nd Henryk Woźnkowsk for dscussons on the results of ths pper, nd n nonymous referee for constructve comments. Ths reserch ws prtlly supported by the Ntonl Scence Centre, Polnd, bsed on the decson DEC- 2013/09/B/ST1/ Open Access Ths rtcle s dstrbuted under the terms of the Cretve Commons Attrbuton Lcense whch permts ny use, dstrbuton, nd reproducton n ny medum, provded the orgnl uthors) nd the source re credted. References 1. Clvett, D., Golub, G.H., Grgg, W.B., Rechel, L.: Computton of Guss-Kronrod qudrture rules. Mth. Comput. 69, ) 2. Clncy, N., Dng, Y., Hmlton, C., Hckernell, F.J., Zhng, Y.: The cost of determnstc, dptve, utomtc lgorthms: cones, not blls. J. Complex. 30, ) 3. Conte, S.D., de Boor, C.: Elementry numercl nlyss n lgorthmc pproch, 3rd edn. McGrw- Hll, New York 1980) 4. Dvs, P.J., Rbnowtz, P.: Methods of numercl ntegrton, 2nd edn. Acdemc Press, Orlndo 1984) 5. Gnder, W., Gutsch, W.: Adptve qudrture revsted. BIT 40, ) 6. Kncd, D., Cheney, W.: Numercl nlyss: mthemtcs of scentfc computng, 3rd edn. AMS, Provdence, RI 2002) 7. Kruk, A.: Is the dptve Smpson qudrture optml? Fculty of Mthemtcs, Informtcs nd Mechncs, Unversty of Wrsw, Mster Thess n Polsh) 2012) 8. Lyness, J.N.: Notes on the dptve Smpson qudrture routne. J. Assoc. Comput. Mch. 16, ) 9. Lyness, J.N.: When not to use n utomtc qudrture routne? SIAM Rev. 25, ) 10. McKeemn, W.M.: Algorthm 145: dptve numercl ntegrton by Smpson s rule. Commun. ACM 5, ) 11. Mlcolm, M.A., Smpson, R.B.: Locl versus globl strteges for dptve qudrture. ACM Trns. Mth. Softw. 1, )

26 198 L. Plskot 12. Novk, E.: On the power of dpton. J. Complex. 12, ) 13. Pessens, R., de Doncker-Kpeng, E., Uberhuber, C.W., Khner, D.K.: QUADPACK. A subroutne pckge for utomtc ntegrton. Sprnger, Berln 1983) 14. Plskot, L.: Nosy Informton nd computtonl complexty. Cmbrdge Unversty Press, Cmbrdge 1996) 15. Press, W.H., Teukolsky, S.A., Vetterlng, W.T., Flnnery, B.P.: Numercl recpes: the rt of scentfc computng, 3rd edn. Cmbrdge Unversty Press, New York 2007) 16. Plskot, L., Wslkowsk, G.W.: Adpton llows effcent ntegrton of functons wth unknown sngulrtes. Numersche Mthemtk 102, ) 17. Trub, J.F., Wslkowsk, G.W., Woźnkowsk, H.: Informton-bsed complexty. Acdemc Press, New York 1988) 18. Wslkowsk, G.W.: Informton of vryng crdnlty. J. Complex. 1, )