On natural cubic splines, with an application to numerical integration formulae Schurer, F.

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1 O atural cubic splies, with a applicati t umerical itegrati frmulae Schurer, F. Published: 0/0/970 Dcumet Versi Publisher s PDF, als kw as Versi f Recrd (icludes fial page, issue ad vlume umbers) Please check the dcumet versi f this publicati: A submitted mauscript is the authr's versi f the article up submissi ad befre peer-review. There ca be imprtat differeces betwee the submitted versi ad the fficial published versi f recrd. Peple iterested i the research are advised t ctact the authr fr the fial versi f the publicati, r visit the DO t the publisher's website. The fial authr versi ad the galley prf are versis f the publicati after peer review. The fial published versi features the fial layut f the paper icludig the vlume, issue ad page umbers. Lik t publicati Citati fr published versi (APA): Schurer, F. (970). O atural cubic splies, with a applicati t umerical itegrati frmulae. (EUT reprt. WSK, Dept. f Mathematics ad Cmputig Sciece; Vl. 70-WSK-04). Eidhve: Techische Hgeschl Eidhve. Geeral rights Cpyright ad mral rights fr the publicatis made accessible i the public prtal are retaied by the authrs ad/r ther cpyright wers ad it is a cditi f accessig publicatis that users recgise ad abide by the legal requiremets assciated with these rights. Users may dwlad ad prit e cpy f ay publicati frm the public prtal fr the purpse f private study r research. Yu may t further distribute the material r use it fr ay prfit-makig activity r cmmercial gai Yu may freely distribute the URL idetifyig the publicati i the public prtal? Take dw plicy f yu believe that this dcumet breaches cpyright please ctact us prvidig details, ad we will remve access t the wrk immediately ad ivestigate yur claim. Dwlad date: 09. Sep. 208

2 TECHNSCHE HOGESCHOOL ENDHOVEN NEDERLAND ONDERAFDELNG DER WSKUNDE Biblitheek Techische Uiverslteit Eidhve =~!:3 tib Telef (040) TECHNOLOGCAL UNVlSTY ENDHOYD THE NETHERLANDS DEPARTMENT OF MATHEMATCS O atural cubic splies, with a applicati t umerical itegrati frmulae by F. Schurer T.H.-Reprt 70-WSK-04 April 970

3 - - trducti ad O. By C[O,l] we dete the set f real-valued ctiuus fuctis defied the iterval [0,]. Let the umbers xo,x"",x be prescribed with = X < Xl < < x _ < x =. The t every divisi f the uit iterval it subitervals [x. l'x,] there crrespds a (+l)-dimesial - subspace S S(xO'x,,x ) f C[O,}] whse members are the atural cubic splie fuctis (hereafter referred t as.c.s.) with des x. S, s E S if ad ly if this fucti satisfies the fllwig three cditis: (i) (U) (iii) S E C 2 [0,], S"(O) = s"() = 0, the restricti f s t a arbitrary subiterval [xi-,x i ] is a plymial f degree at mst three. a umber f papers (see fr istace [6] ad [7]) Scheberg has brught ut the imprtat rle which.c.s. play whe apprximatig liear fuctials. We particularly wat t meti here his fudametal therem J i [7], p. 58. The bject f this reprt is twfld. Always assumig that the des are equally spaed [O,J, we first shw hw explicit frmulae may be give fr the.c.s. This apprach is based up the slutis f a set f liear equatis frm which the s-called cardial atural cubic splie fuctis (c..c.s.) may be calculated. As a applicati f these basic fuctis we preset a ew way t derive sme familiar results f Meyers Sard [4] ad Hlladay [3J. the secd part f the paper we carry research de by Atkis [] ccerig the applicati f.c.s. t umerical itegrati frmulae. We imprve e f his results (therem 7, p. 99) ad establish it i its defiite frm. the curse f the prf use is made f recet wrk f Seveld [8].

4 - 2 - O cardial atural cubic splie fuctis.. t is kw ([2J, lemma ) that with each f E C[O,)] there ca be assciated a uiquely determied elemet s E S with the iterplati prperty, i.e. s(x i ) ::: f(x i ) fr i = O,,.,. f we write f. == f (x.), \..::: s" (x.), the the iterval [x.,x.j the fucti s ca be writte i the fllw- - ig frm: (. ) sex) == f. A. (x) + f. B. (x) + \.. C. (x) + ii. D. (x) - - Here A.(x),,D.(x) are certai cubic plymials with suitable chse l. l. prperties. fact, we have (.2) (.3) (.4) (.5) t is bvius frm these frmulae that Ai(x),Bi(X) ~ 0, whereas Ci(x),Di(X) ~ [x.,x.j. Mrever, -. (. 6) A. (x) l. + B. (x) =<,. X. x. (. 7) J. Ai(x)dx B. (x)dx xi - xi - x. x. l. 2 ' =< =- (.8). C. (x)dx == J D. (x)dx == --- Xi - J xi-l 24 3 view f the cditis (i) ad (ii) the parameters V. have t satisfy sme particular relatis, which ca be fud fr istace i [9J. They take the frm (. 9) (i =,2,,-l),

5 - 3 - with (.0) There exists a uique sluti fr the parameters ll., because the matrix as-. scia~ed with system (.9) is diagally dmiat. Tgether with (.) this establishes a prf f the fact that the iterplatig.c.s. exists ad is uique. Besides (.) there is ather way f represetig the iterplatig i.c.s. sex). f s (x) E S detes the i-th c..c.s. (this fucti is defied by the equatis si(x.) = ~ fr i,j = O,,,), the i terms f these fuctis we have (. ) sex) = r i=o J J i f. s (x). this is a frmula f Lagrage-type..2. As usual we write fr m ~ 0 m (x - t) (x - t)m = + { 0 if x ~ t, if x < t. Mrever, let (.2) L(x,t) = - [(x-t)3 - (-t)x3 - r {(x.-t)3 - (J-t)x~}s.(x) J = the sequel we will eed a result which is due t Atkis. t reads as fllws: THEOREM (Atkis [J) (i) L(x,t) is symmetric, i.e. L(x,t) = L(t,x). Fr each x, it is a.c.s. i t with des xo,xl,,x ' ad x; fr i = O,,,, L(x,x i ) = O. (ii) Assume that f E C 4 [O,lJ. The (. 3) f(x) - i=o with i f. s (x) =. eo(x)f"() + e t (x)f"(o) + JL(X,t) (4)(t)dt

6 - 4 - (.4) e(x) = i- {x3 - x~ le (x) i=o = i- {(l-x)3 - Si(X)}, (l-x.)3 =. 0 Si(X)} As a bvius csequece f (.3) We have COROLLARY (.5) where () f (X)dX - L i=o f. c "() + c f"(o) + J fl (X,t)f(4)(t)dtdx 0 (.6) Uder the assumpti that the des are equally spaced, it is true (cf. frmula (.34» that C = c t ' Relati (l.ts) is f sme iterest because Scheberg [6J has shw that f all umerical itegrati frmulae f type ff(x)dx s::: a w. f., i=o which are exact fr liear fuctis, the best e i the Sese f Sard [5] is btaied by itegratig the.c.s. which iterplates f at the pits Le. (.7) w i := Jsi(X)dX his paper [5] Sard als icluded a shrt table f weights w. i case the des are equally spaced. Actually it is pssible t determie explicit frmulae fr the weights. This was de fr the first time i 950 by Meyers ad Sard [4J withut usig the ccept f splie fucti. We refer the reader t their paper fr mre extesive data the umbers w., this respect e als has t meti a paper by Hlladay [3]. 957 he prved a fudametal result i the thery f.c.s., apparetly withut kwig f

7 the wrk f Scheberg i this area. He als derived frmulae fr the umbers wi usig 0,7), exhibited a table fr the weights ad frmulated sme simple rules t calculate them. Scheberg ted that the data give by lllladay are exactly the same as thse f Meyers-Sard ad this led him t establish a clse cecti betwee the prblems f splie iterplati ad mechaical quadratures, which culmiates i his fudametal therem i [7J. view f Scheberg's therem, it seems t be f sme iterest t get hld f the c..c.s. si(x) (i = O,,.,). They als eable us t calculate the weights w. i a differet way as was de befre by Sard, Meyers- ~ Sard ad bladay. These tw subjects will be dealt with i the ext tw BE'<:t;.s,.3. The calculati f the c..c.s. ca be based up frmulae (.), (.9) ad (.0). Puttig the accrdig t (.) we have the iterval [x. 'x.]: ~-. (.8) sk(x) = k A.(x) + k B (x) + ~~k) C.(x) + ~~k) D (x) i-. i i.- ~. i ' (k = O,,,) rder t cmpute sk(x), we first have t write equatis (.9) i their apprpriate frm. We get (.9) (k) + ].- 2 k 6 (Oi+l 2~ + O~ )..- (~ 0,,,; )... = t 2,, - ' with (.20) (k == O,,2,,) view f (.8) ad tgether with frmulae (.2), (.3), (.4), (.5), the c..c.s. will be cmpletely determied if we ca slve the systems f liear equatis (.9) fr = 2,3, Befre we g it this, we first state a elemetary lemma which will be eeded fr the prf f the ext therem. LEMMA The sluti f the differece equati (l.2) (i = 0,,2,... ),

8 t with iitial cditis a O = 0, a l =, is give by (.22) a. = i fi(a i - a -i ) (i = 0,,2, ),. where (t 2 + )' J. Mrever, we have (.23) a. a -i a - + a a -2i+, ( ::; i ::; [!!.] ), (J.24) (0 ::; i ::; -l) Sme ther particular slutis are (.25) - b )a i (a + )a- i } (i 0,,2, ) O = b = b. = l "6 {(a + + =,,., (.26) 5 C, c = c. = - {(a, =. 2 l)a i (a - ) a -i} (i 0,,2,... ) (.27) do 2 i -i =, d = 4 d. = a + a (i = 0,,2,... ),.,. PROOF Usig the stadard techique fr slvig differece equatis with cstat cefficiets we easily deduce the frmulae (.22), (.25), (.26), (.27). Relati (.23) ca be prved by usig mathematical iducti. Equality (.24) is a straightfrward calculati. THEOREM 2 Let = 2m, respectively = 2m + (m =,2, ). The fr a arbitrary but fixed umber m the uique sluti f the set f equatis (.9), always assumig that (.20) hlds, takes the frm (. 28) (0) (_l)i- 62 a -i ]. =. a (i =,2,,-l), (.29) (k) ll = (_)i-k a a k a -i (k =,2,,m;. = k+l,k+2,,-l), (J.30) (k) (i) ].t. = llk,. (k,2,...,m; i =,2,...,k-l), (.3) () = - 2(0) + (0) ].tl 2 (.32) (. 33) (k) ].tk = (k-) k-l (-k) = (k) -i. () + 2k- (k = 2,3,,m), (k = O,l,,m; i =,2,..,-l)

9 f/ Here a. is gve by (.22). PROOF t is sufficiet t csider the systems (.9) ly fr k = O,,,m, 'Jecause i case f k = m+,., we ca prceed by replacig k by -k ad i by - i. The we get a system f equatis which we ecuter i case k "" 0,,..,me Therefre we have frmula (.33) which has as a csequece, tgether with (.8) ad (.2), (.3), (.4), (.5), that (. 34) -i i s (x) = s ( - x), (i = 0,,, m) ; this was t be expected because the des are equally spaced. AR fr the prf f therem 2, let m be a arbitrary but fixed psitive iteg'?r.. The k ca be e f the itegers O,,,m ad we have t distiguish betwee several cases. f k = 0, the it fllws frm (.9) that we have t deal with (.35 ) J"~O) + 4"~O) + "~O) = 6 2, ' ~~O) + 4~~0) + ~(O) = 0, l.+ i+2 (i =,2,,-2) Tgether with (.20), it is a immediate csequece f the differece equati (.2) that (.28) is the sluti f system (.35). (f the last elemet f the set thrugh which the idex variable i rus is smaller tha the first elemet, we will always assume that the set f equatis uder csiderat~ (.9) e has fr > 2 is vid.) Nw we tur t the case k =. The i view f () + 4,,() + () _ 22 ~O '"' ~2 =, (.36) () + 4,,() + () 62 ll '"'2 ~3 =,,,() + 4,,() +,,() = 0 '"'i '"'i+l '"'i+2 ' (i = 2,3,,-2) Usig (.20), (.3), (.2) ad that the expressis fr ~~) as the fact that a =, it is easily verified exhibited i (.29), tgether with (.3), are the sluti f system (.36). The case k = 2, whe apprpriate, ca be dealt with i a aalgus way. The mai part f the prf f therem 2 cmes i whe k = 3,4,.,m; these cases ca be csidered all tgether. Let k be a arbitrary but fixed elemet f the set {3,4,,m}. Frm (.9) we btai

10 - 8 - jl~k)... 4~k) + p~k) =, (i = 0,,,k-3), (k) + llk-2 4 (k) llk- + ]lk (k) = 6 2, (.37) (k) + 4(k) + (k) = - 22 llk- k llk+l, (k) + 4(k) + ll(k) lk = 6 k+l k+2 2, ll(k)... 4p(k) + jl(k) = 0, i i+l i+2 (i = k+l,k+2,,-2) the first set f equatis f system (.37) we ca make use f (l.30). The e has t shw that ad 'll(l) + (2) = 0 k lk (i) + 4 (i+l) + (i+2) - 0 ~k P k P k -, (i =,2,,k-3) rder t d this we apply (.29).The bth equalities are a csequece f relati (.2) if we take it accut the accmpayig iitial cditis. Usig (.30) the secd equati f system (.37) ca be writte i the frm (.38) view f (.32) ad the applyig (.3), the left-had side f this equality is equal t (k-2) + 4 (k-) _ 2 (0) + (0) + () + + () ~k llk ll ll2 P 3 P2k- Nw the frmulae (.28) ad (.29) ca be used. f we prceed i this way, the the verificati f (.38) amuts t shwig that (.39) ( = 6,7, ; k = 3,4,,m) Ad this is true because f (.23) f lemma. The third ad furth equati f system (.37) ca be dealt with i the same way. We d t carry ut the details f these calculatis, but remark that i bth derivatis it is advatageus t use relati (.39). Fr istace, the validity f the third equati f (.37) the stems frm (.24) f lemma. Fially, it is bvius that i the last set f equatis f system (J.37) the apprpriate frmulae f (.29) ca be applied. The it is sufficiet t refer t (.2) ad the prf f therem 2 is cmplete.

11 - 9 - Takig it accut (.22), the whle set f frmulae f therem 2 is rather cmplicated ad e may have sme dubts whether they are suited fr a rapid calculati f the secd derivatives f the c..c.s. Frtuately, this is the case. We will w first give sme examples by way f illustrati. O tle basis f this we will supply a algrithm which ca be used t calculate the umbers ~~k) recursively by gig frm des t +l des. the fllwig tables we ly exhibit the umbers ~ik) ad i ~ fr k = Otl,,[~J,2,.,-l because f (.20) ad the symmetry relatis (.33). We remark that fr a specific umber all data i the table have t be multiplied by 6 2 where a is give by (.22). This umber, tgether with the a value f, placed i the first rw. As is apparet frm therem 2 the sequece {ail as give i (.22) plays a prmiet rle i ur calculatis. Startig ut with the iitial cditis a O = 0, a l =, the elemets are easily geerated by meas f the recurrece relati (.2). We s btai (.40) { ad = 0, a =, a 2 = 4, a 3 = 5, a 4 = 56, a 5 = 209, a 6 = 780, a 7 = 29, a 8 = 0864, a 9 = 40545, Frmula (.40), tgether with the ctets f therem 2, gives rise t the data fr ~ik) as exhibited i tables,2,,7. ;:: = 5 = ! ~. 0 ~ l Table Table 2 Table

12 j = ,... = ~-..".~ 0 2 ~ ~ i Table Table 5 = ~ f Table ~ 0 J Table 7

13 - - Based the frmulae f therem 2 we will w give a algrithm fr, (k) the calculatl f the data ~i (k ::: 0,""'[2J; i :::,2,,-l) fr a arbitrary umber f des. Fr a Guadt u's i the tables by b ~k) L, specific value f let us dete the (k = O,,.,[~J; i =,2,,-J). The i gig frm t +l e has the fllwig prcedure, which is easy t apply, ad which ca serve t calculate the umbers bi~~). ALGORTHM b(+l) ::: _ b() i+l,k i,k ' (i =,2,,-l; k = O,,,mi(i-,[~J» The e:rt step has ly t be applied whe gig frm is dd t +l is eve. b(+l) i+l,[~j+] "" _ (4b(+l) + b(+l) ) i+l '[] i+l,[~j-l (i = [;)J +,[;J + 2,.,-l) with b (+l) i,i- (+ ) (+ ) ) = - [ 4bi + l,i_ + bi + 2,i- ( ' =, 2,..., [+ -2- J + ), b(3) 2, "" 6, b~~~ "" 24 b(+l) =, +,k, + (k = 0,""'[--2--]) (i =,2,,[~lJ - ; k = i+l,i+2,,[;l J) ~!;~E_2 b (+l) = - 2b (+) + b (+l),.,0 2,0 ~!;~E_~ b ~:l) L,L b(+l) ::: i-,i-l + b(+l) 2i-l, (' = 2,3,, [+l]) ~!~E~Z (k) = 6( + )2 b (+) lli a + i,k (i=,2,.,;k + = 0,,, [-2-]), where a + is give by (.22)

14 - 2 - REMARK The prcess f the algrithm is iitiated with the data f table. f e f the variables i r k rus thrugh a sequece f values the last f which is smaller tha the first e, the it is uderstd that the crrespdig istructis are t t be executed. A examiati f the structure f the algrithm reveals that the first three steps are a csequece f frmulae (.28), (.29). Step 2 is ly t be applied i case is dd, because the i gig frm t +l the value f m (ad thus the umber f clums f the matrix) is icreased by. The steps 4, 5 ad 6 immediately fllw frm (.30), (.3), (.32) respectively. Relati (.33) des t fid its aalgue i the algrithm because we have restricted the values f + k t 0,,. '[-2-J A imprtat csequece is that w, tgether with the frmulae (.2), (.3), (.4), (.5) ad (.8), the c..c.s. are cmpletely at ur dispsal fr a arbitrary umber f (equally spaced) des. By way f illustrati we exhibit these basic fuctis i cases = 2 ad = 3, usig the data frm tables, 2 ad the symmetry relatis (.33) = 2 (x) + s (x) 6C 2 (x) 0 ::; x ::; ) 0 3 tl 6Dl (x), (0 ::; x ::; D, s (x) \ ~(x) B) (x) - 2D (X) (0 ::; x ::; D - 2C 2 (x) 0 ::; x ::; ) 2 s (x) = B2 (x) + 6D (x), (0 ::; x ::; D 6C 2 (x) 0 ::;; x ::;; ).!!-=-~ Al (x) 72 + "5 D) (x), (0 ::; x < -..) 3, 0 72 )8 2 s (x) = 5" C 2 (x) - - D (x) (3::;; x < -) 5 2, - 3, 8 (! < - - C (x) x ::;; )

15 - 3 - Bl (x) Dl (x), ::; x (0 ::;..) 3, - 62 C (x) + ~8 D (x) 2 s (x) == 2 (3 ::; x ::; 3), 2 08 C 3 (x) 2 (3::; x ::; ) D (x), (0 ::; x ::; 3), 2 08 C 2 (x) _ 62 D (x) 2 s (x) = B 2 (x) + (3 ::; x ::; 3), 5 2 A (x) - 62 C (x) (~ < 3 t x ::; ) 8-5" D) (x), (0 ::; x ::; 3), < ~) s (x) == - - C (x) + 5" D 2 (x) (3 ::; x B 3 (x) " C 3 (x) 2 (3 ::; x ::; ]). REMARK Usig varius prperties f the umbers ~~k) (k = O,,.,; i = 0,,..., ) ad takig it accut (.8), e ca shw that the k c..c.s. s (x) have simple zers (k = O,,..,) d t chage sig the subitervals ad at the des. We mit the details f this verificati..4. Besides the derivati f the c..c.s. we recall that at the ed f secti.2 we set urselves the task f calculatig the weights w k (k == O,,...,), which umbers appear i the umerical itegrati frmula (.5). Oe has (cf. (.]7» ] w k == Sk(X)dX (k = O,,,) t where sk(x) is the k-th c..c.s., the des beig equally spaced. These itegrals ca be expressed i terms f the umbers ~~k) (i = O,,,), k the secd derivatives f the fucti s (x) at the des. Because f (.34) it is sufficiet t restrict urselves t values f k fr which 0 ::; k ::; m, where = 2m, respectively = 2m +. t is a csequece f (.8), tgether with (.7), (.8) ad (.20), that e has

16 l (0) w =--- \' 2 23 i:l lli ' (.4) -l wk = i L (k) ll. (k =,2,.,m), (k = 0,,,m) Fr sme explicit frmulae f the uderlyig fuctis sk(x) we refer t the examples give at the ed f the precedig secti. The sums which appear i (.4) ca be dealt with i the fllwig way. Takig it accut (.20) it -l (k) is a easy csequece f (.9) that ll. ca be writte i terms f (k) (k) i= ll} ad!-l' This fact, tgether with the ifrmati supplied by therem 2, actually gives rise t the fllwig expressis: -} L (0) ll. (.42) 6 2 a (_) 6 2 a - l ,.;;.. 2 a a 2 a - 6 = = - {a + a + (-) }, -l L i= () ll () + () ll-l 6 = ;:--- = ( ~ 3) (.43) = = - - {a + (-) } a -l This frmula hlds als i case = 2; i the curse f its derivati use is made f (.2). Whe k = 2,3,.,m we btai (k) + (k) l- () + llk (k) ll-l 6 = 6 = 362 a a l 36 2 a a = (-Ok -k + (_)-2+k k l = 6a 6a (.44) k 6 2 = (-) - {a + (-) ~} a -k.

17 - 5 - As usual, a. (i = 0,, ) is give by (.22). We te that i view f ~ (.43) frmula (.44) ca als be used i case k =. Frmulae (.42) ad (.44), tgether with (.22), ca be used t give - (k) explicit expressis fr the sums u. (k = O,,,m). t turs ut t i= be advatageus t csider the cases is eve ad is dd separately. We mit the smewhat tedius but elemetary calculatis ad state ly the results. case is eve ( = 2m) e has (. 45) (l.46) - U~k) = ~k -!!+k k 2 2 (-) 6 2 {a + a } /2 -/2 a + a (k =,2,,m), whereas i case is dd ( = 2m + ) we btai (.47) -l (-l ) - (0) {a - a } :: l ll -, i= (-) (a - )a + (a - l)a -l (.48) L i= (k) ll - (-) +k k (_)k 62{(a _ )a 2 + (a- _ )a 2 } = ~~----~----~------~--~(--~l)~------~ (a - l)a (a - - )a 2 (k =,2,,m) T give the reader a idea what the sums u. (k = O,,,m) lk like i= fr the first few values f, we use the expressis (.42) ad (.44) istead f (.45)-(J.48). We arrive at the fllwig table; t btai the - value f the sum ll~k) fr a particular ad k, the elemet i the tp ~ rw ad (-J)-th clum has t be multiplied by the etry i the k-th rw ad (-l) -th clum. -l (k)

18 - 6 - ~ i m 989! _" i ~ _... -_._.- -- Table J We te that there are several strikig recurreces i this table. Whe is eve the sequece f umbers {2,4,4,52, } plays a dmiat rle. case is dd the same is true fr the sequece {,5,9,7, }. Bth sequeces are characterized by prperty (.2). Usig the set f frmulae (.45)-(.48) e ca shw that the recurrece features f the etries exhibited i table 8 fr = 2,,] d hld i geeral. Usig these prperties a algrithm ca be deduced fr the cal- culati f the sums l J..l~k). t is apparet frm the data f the table i-. that the cases is eve ad is dd have t be csidered separately. Assume first that = 2m (m =,2, ) ad let the value f m be icreased by ; further let the umbers b. ad d. be defied by (.25), respectively.. (.27). ALGORTHM 2 = 4b - b m m- (m+l) e k = k (-) d m - k +! ' (k =,2,,m+l)

19 - 7-2m+ L i= (0).. = 24(m + )2 b d m + m+' 2m+l L i=l (k) ll. 24(m + )2 (m+) = d + m ek (k =,2,,m+ ) Nw let = 2m + (m =,2, ) ad let the value f m be icreased by ; the umbers a. ad c. are give by (.22), respectively (.26). ALGORTHH 3 = 4c - c m m-' ~~~E_~ (m+l ) k gk = (-) c (k,2,,m+) m - k + =. ~~~E_2 2m+2 (0) 6(2m + 3)2 L = a m + i= C m + ~., 2m+2 (k) 6(2m + 3)2 (m+) ~. = gk (k =,2,,m+). i= C m + REMARK Sice there crrespds a uiquely determied.c.s. t each f E C[O,], ad takig it accut (.), it fllws that L sk(x) = k=o This, tgether with (.6) ad the represetati frmula (.8) fr sk(x), has as a csequece tha~ at a arbitrary de the sum f the secd derivatives f the c..c.s. is zer, i.e. L k=o (k) ~i =, (i=o,,.,), ad thus (cf. table 8)

20 l L l:. k=o i= The ultimate aim f the first part f this paper is the calculati f the weights i the umerical itegrati frmula (.5). This ca w be accmplished by usig (.4) ad the set f frmulae (.45)-(.48). As is t be expected the calculatis ivlved are rather legthy ad therefre mitted here. We btai the fllwig results. Assume first that = 2m. f, as usual, a = 2 + :3 the : (a - a ) (.49) W =-+-- (a + a - 2) (.50) ~-k ~k (- )k+l (a 2 + a 2 ) w =-+ k (a / 2 -/2, 2 + a ) (k =,2,,m) f 2m +, the e has - (l.5) :3 (a + a + 2) W = (a - a ) (.52) ~-k -~k (- )k+ (a 2 - a 2 w ) k =-+ 2 (a / 2 -/2, a ) (k =,2,, m) Apart frm a rmalizati factr, these frmulae ca be prved t be idetical with similar es give by Meyers-Sard ([4J, p. 2-22) ad Hlladay ([3J, p ). t is f sme iterest t gather tgether the weights fr the first few values f. This is de i table 9 (a mre extesive e may be fud i [4J).

21 ----~ ~ 8. O 28 38ii r ~ ~ i Table 9 Fr a arbitrary value f ad k (0 ~ k ~ m) the weight w k is equal t the prduct f the elemet i the tp rw f the -th clum ad the etry i the k-th rw ad the -th clum. We d t hesitate t say that the structure f table 9 is beautiful. As Meyers-Sard remark ([4J, p. 2) there are several kids f strikig regularities. Fr istace, mius ay etry plus fur times the etry tw places its right equals the etry fur places its right. This is, i fac~relati (.2) ad shws agai that it is apprpriate t distiguish betwee is eve ad is dd. Mrever, if is dd, say, the the differece betwee the etry i the secd rw ad the etry i the first rw is just 7a, where a is defied by (.22). A similar statemet hlds whe m m is eve. O the basis f frmulae (.49), (.50), (.5) ad (.52) it ca be shw that the varius recurreces which may be bserved whe = 2,3,, hld i geeral. This makes it pssible t cstruct a easily applicable algrithm fr the calculati f the weights (cf. als Hlladay's paper). We first assume that is eve ( = 2m). Usig (.25) ad (.27) f lemma we have ALGORTHM 4

22 a(m+l) b L 7b == m+ 2 '.,.. m+ l' fj (k ==,2,,m). W == 4(m + l)d ' m+ a (m+l) k k 4(m + l)d m +), w == ~--~~~-- (k ==,2,, m+ ) Let w be dd ( = 2m + ). f a. ad c. are defied as i lemma, ~ ~ tte OT" has t apply ALGORTHM 5 y(m+l) = y(m+l ) k+l (m+l) k == Yk + (-) 6a m + _ k, (k ==,2,,m) (m+l ) Yk 2(2m + 3)c m + ' = 2(2m + 3)c m + ' (k ==,2,,m+ J) REMARK Because (.5) is exact fr liear fuctis it fllws that i=o w. ==, ~ i=o iw. = /2 ~ Als it ca be shw that fr a arbitrary value f the weights with eve subscripts frm a icreasig sequece, whereas the ther had W > w3 > ; mrever, m~x w 2i < m~ w 2i + We mit the verificati f this result. ~ ~

23 - 2 - A applicati t umerical itegrati frmulae 2.. Whereas i the first part f the paper we were maily ccered with the calculati f the c..c.s. ad the weights wi appearig i the itegrati frmula (.5), we w fix the atteti t the right-had side f (.5). The purpse f this secd part is a imprvemet i its defiite frm f the fllwig result due t Atkis. THEO~M 3 ([J, p. 99) give by (.6). The Let the des be equally spaced ad let C (- c ) be (2, ).02 < -c < Furthermre, if f E C 4 [0,J, the e has ff(x)dx - w f ~ where f (4). 0 ~ ~.= = max f (4) (x). The lwer bud f (2.) implies that O~x~l greater a rder tha is pssible whe 3 f"(o) + fll() ~ 0 The derivati f this therem (ad als ur imprvemet f it) is based up crllary f secti.2. Frm this crllary we cclude that jf(x)dx - w. i=o ~ cllf"(o) + ffl(l) + f Lex,t)f(4)(t)dt dxl ~ 0 (2.2) ~ cllf(o) + f"(o + (4)[ f fl(x,t)dtldx 0 T calculate the cstat C we may te that it is a immediate csequece f (.5), (.4) ad (.7) that we have {l c =- --- l ~=

24 (2.3) { } c :: L i 3 w., 064 3' 0.= where the weights w. are give by frmulae (.49)-(.52). But the expressis. *) just exhibited fr C d t seem t be s sultable fr calculatl. Therefre 'ile prceed i a differet way. Usig frmula (.2) fr L(x,t) e has L(X,t)dt But = i J [(X-t)! - (-t)x 3 - i~o{(xi-t)! - (l-t)xi}si(x)]dt = i=o J (-t)dt + xi si(x) (-t)dt (x.-t)!. 0 = L i=o ] sl(x) J(Xi-t)! dt = Xl = sl(x) J (x t -t)3dt + x2 s2(x) J (x 2 -t)3dt + + s(x) (-t)3dt \' 4 i =! l.. x. s (x) i=o. *) Atkis shws that (i) c = _ i= k... k.- -l where the cstats. (i = 0,,2, ) ad k. (i = 0,,2, ) are defied. recursively by the relatis (it) ad (iii) 3 + O. \5 i + = 4 4" + \5 ~,. ] ki+l "" 4 - k:- ' Usig this he arrives at iequality (2.). (Actually, the lwer bud f (2.) is t valid fr = 2, cf. (2.5).) t is pssible t calculate the exact value f C the basis f the expressis (i), (ii), (iii). This was pited ut t me by F. Gbel ad F.W. Steutel (Twete Uiversity f Techlgy).

25 Usig this it is the easily verified that (2.4) JL(X,t)dt = ~4 {X4-2x 3 - i~o(x~ - 2X~)Si(X)} rder t fid the cstat C it turs ut t be wrthwhile t csider the itegral fl(x,t)dx' Usig (.7) we btai fl(x,t)dx = i {i- (l-t)4 -! (-t) + a which ca be writte i the frm (2.5) because f (2.3). fl(x,t)dx =! {- 6c O (-t) (-t) r w. x~ - r w.(x.-t)!}, = = +.,.. (l-t)4 - w. (x.-t)+3 } '+ 0 = At this jucti we eed e f the results f therem ] which says that the fucti L(x,t) is symmetric, i.e. L(x,t) = L(t,x). This prperty has as a imprtat csequece that the right-had side f (2.5) is equivalet t the right-had side f (2.4) if the variable t is replaced by x. The expressi i (2.4) ca be regarded as the differece f the fucti ~4 (x 4-2x 3 ) ad its iterplatig.c.s. Takig it accut (2.5) the cstat C will be determied if we kw the first derivative f the fucti at the pit x =. We will cme t this later Returig t (2.2), e is led t csider the itegral J J~(x,t)dtldx 0 fr psitive iteger values f ~ 2. rder t calculate ad estimate these umbers accurately e eeds ifrmati abut the fucti i (2.4), especially where it chages sig [0,]. As we already ticed the right-

26 had side f (2.4) ca be see as the differece f the fucti (x4-2x3) ad its iterplatig.c.s. Because f the fact that the 4 3 secd derivative f 24 (x - 2x ) vaishes at the ed pits f the uit iterval, the crrespdig iterplatig.c.s. is equivalet t a type f splie fuctis as csidered by Seveld i his paper [8]. Therefre his results are applicable ad we will use them t examie the behaviur f the fucti JL(X,t)dt. Fllwig Seveld we dete by Yl/(f;x) the uique cubic splie assciated with f(x) ad havig the prperties f Y/(f;xi ) == f. t (0 ::;; i S ), yj' / (f ;x.) == f'.' (i == 0, i = ) ~. ~, the subscript li meas that we assume the des t be equally spaced [O,J. his paper Seveld establishes a relati betwee cubic splie iterplati ad cubic Hermite iterplati.':assumig f(x) E C[O,tJ, this classical apprximati fucti yh(fjx) satisfies the fllwig cditis: (ii) yh(fjx) is a plymial f degree at mst three each subiterval [x. l,x.j,.-. (iii) (i == O,l,,), (iiii) (i=o,,,) Furthermre, if f C 4 [0,J, the it is well kw that e has view f this we ca write [xi,x i + ] (2.6) (x. < ~. < x'+])...

27 Let us m., take i particular f(x) = ~4 (x 4-2x 3 ) [O,lJ. The the splie. fuctis -- L (x~ - 2x~)sl(x) ad Y!(f;x) are idetical ad it fllws 2'f 0 l= frm (2. f, that we have Usig Seveld's results (i particular (2.25.b) ad the set f frmulae p. 3 f [8J) the differece f the cubics yh(f;x) ad YJ!(f;x) ca be evaluated. Prceedig i this way we get [xi,xi+jj (2.7) where the umbers z! (i = O,,,) are the sluti f a system f liear equatis f the frm 2z' + z' ::::-!z' + 2z' +!z' = 0 t 02 (2.8) Z + 2z' + z' :::: 0 ~ -2 -l ~ ' z' + 2z' :::: - 3 Fr the ivestigati f (2.7) we eed the sluti f the set (2.8). THEOREM 4 Let be eve ( = 2m) ad a. :::: The the uique sluti f the system f liear equatis (2.8) is give by (2.9) (i = O,,..,m), z'. :::: - z~ - (i :::: 0,,2,..,m)

28 hwever, if is dd ( = 2m + ) the we have (2.l0) r zi - \ z! l -i = - z!. (i=o,l,,m) (i = O,,,m), PROOF We will ly verify the ctets f the therem i case is eve; the prf i case is dd may be give i a similar way. First f all we remark that there is a uique sluti because the matrix f (2.8) is diagally dmiat. Assumig = 2m, it is sufficiet t shw that the exhibited umbers z! f (2.9) satisfy the first m+l equatis f (2.8). f. this is f.rue, the the symmetry relatis zim-i = - zi (i = O,l,,m) will guaratee that the remaiig equatis f (2.8) are als satisfied. Takig it accut a = we have 2z' + z' m -m a - a =-- ~3 m- -m+l v. Ct. - a ~--~--= 24m3 am + Ct.-m case i =,2,,m-J we get = 3 {2(Ct. m - a- m ) - (2 - ~am + (2 + l3)a-m} =_ 24m3 am + a 8m3 -m+i Ct. ) m-i-l -m+i+l} + a - a ad the expressi betwee brackets vaishes because a a- = 0. The (m+l)-th equati f (2.8) is als satisfied because z' = ad m Z~_l = - z~+l' This prves therem 4. umbers z! the sequel we will eed sme ifrmati abut the magitude f the. (i = O,l,,m). This we state i the frm f a lemma. LEMMA 2 Let the umbers z! (i = O,,,m) be give by (2.9), respectively. (2.0). The the fllwig assertis are true: (i) whe is eve, the 3 Z is icreasig with ad

29 (2. ) < 3 ---z < ' 3 (ii) whe S dd, the 3 z' is decreasig with ad a (2. 2) (2. 3) PROOF All three statemets ca be verified by elemetary calculatis based up therem 4. We mit the details. REMAl\.K view f (2.4) ad (2.7) the first derivative f the fucti fl(x,t)dt at the pit x = is equal t 24 z = - 24 zo As we remarked 0 page 23 this suffices fr the calculati f the cstat C appearig i (2.2). Usig (2.5), tgether with (2.9) ad (2.0) we btai (2. 4) - (ex - ex ) Mrever, usig (2.) ad (2.2), it is a csequece f (2.4) that (2. 5) 2.3. Therem 4, tgether with frmula (2.7), ca als be used t determie the shape f the fuctis fl(x,t)dt [O,J fr psitive iteger values a f. view f (2.2) it is ur purpse t give a estimate f J JLex,t)dtldX which hlds fr all psitive iteger values f ~ 2 ad 0 which is best pssible. Therefre we are particularly iterested where the fuctis jl(x,t)dt chage sig [O,J. Befre we g it this, we first a wat t remark that the fuctis uder csiderati are symmetric the uit iterval with respect t x = ~. tis z = - z.' This fllws frm (2.7) ad the rela- (i = O,l,,m). As fr the sig chages f jl(x,t)dt,

30 we te frm (2.7) that every subiterval we deal with a plymial f degree fur which vaishes at bth ed pits. Nw let be arbitrary but fixed ad csider the expressi (2. 6) the iterval [xi,x i + ], which ccurs i (2.7). Due t symmetry we may restrict urselves t 0 sis m-. f i is eve, the accrdig t therem 4 we have zi > 0 ad zi+ < 0 ad thus is JL(X,t)dt psitive defiite (X.,x. )' Hwever, if i is dd, the we btai z! < 0, z!+l > O. S the ~ ~+.. parabla i (2.6) is pulled dw, but due t the fact that z! <--- fr. 53 i = l,~".. m (this is a csequece f lemma 2), the fucti remais psitive whe x =!(x i + x i + ). Therefre the fucti simple zers jl(x,t)dt has tw the pe iterval (xi,x i + ), assumig i is dd. case is eve, fially, there is a duble zer i x =! because z' = 0; mrever, the furth zer will lie the iside f [x l'x ] whe m is eve ad stay m- m utside whe m is dd. These bservatis, tgether with the fact that the cefficiet f the leadig term f the plymial each subiterval is ~4 cmpletely determie the shape f the fuctis fl(x,t)dt [0,]. By way f illustrati we exhibit the graphs f these fuctis fr = 2,3,4,5. The values which are give fr the zers m the iside f a subiterval may be verified by simple calculatis based (2.7) ad therem 4. ' = 2! S 30 \ \ ~ + 30 S

31 = 4 ~~ '" +~ /7 5/7 = i T3 90 ~ /5 '' tit' 4/5 \ 2 \ 2/5 3/5 \ \ \ :2f3 3 - :2f Usig the ifrmati prvided by the pictures, frmula (2.7) ad therem 4, we ca evaluate the itegrals J jl(x,t)dtdx i cases = 2,3,4. Ele 0 metary calculatis lead t the fllwig results: (2.) if = 2, the J f lex, t)dt dx = -- = (2.8) if = 3, the J fl(x,t)dtdx = (2.9) if J 9349 = 4, the L(x,t)dt J dx = -~-- = cases = 5,6, the itegrals uder csiderati ca be dealt with i the fllwig way. Fr ur purpse it is t ecessary t calculate them exactly; estimates f L(X,t)dt will be sufficiet. T this ed we use the represetati the iterval [x.,x. J as give i (2.). As a prelimiary ~ ~+

32 , result we eed the itegrals (2.20) (2.2) x. ~ A estimate f xi+ f (x x. ~ xi+l f (x x f fl(x,t)dtldx may w be derived frm (2.7) as fllws. 0 Takig tgether all ctributis t the itegral ver the subitervals ad usig (2.20), (2.2) ad therem 4, we btai (2.22) L(X,t)dtdX 0 ::;;;_l_{_l_+ CZ ' - 2z'l + 2z 2 ' - + (_l)m 2Zm'J} We will shw that fr a arbitrary but fixed psitive iteger 2 5 the right-had side f (2.22) is smaller tha --~ it is sufficiet t establish that Usig Z ::;;; ZO' - 2z ' + 2z ' - + (_l)m 2z' ::;;; l 2 m ad takig it accut (2.3) we btai S S (cf. (2.7». Accrdigly, ( 2 5). Z - 2zi + 2z Z -. + (_)m 2z' < _3_ + _2_ { + ~ + + } _ ~ <, m S3 S3 32 S3 42 This, tgether with (2.8) ad (2.9), shws that ( 2 5) (2.23) J fl(xtt)dtdx ::;;; ( = 2,3,4, ), ad the equality sig hlds if ad ly if = 2.

33 ,.t Nw we ca state ur defiite result. Takig it accut frmula (2.2), tgether with (2.5) ad (2.23), e has the fllwig therem, which is a imprvemet f Atkis's result as frmulated p. 2. THEOREM 5 Let f C 4 [0,lJ ad let the des x. (i = O,,,) be equally ~ spaced [O,J. f the weights w. are defied by (.7), the fr ~ = 2,3, (2.24) ff(x)dx w. i=o ~. ~ _- f"(o) + f"() + -- ~ with llf(4) = max f(4) (x). this iequality the cstats O~x~ ad 40 = = are best pssible. PROOF Oly the last asserti eeds t be verified. This will be de by chsig tw extremal fuctis f ad f2 havig the prperty that f ~4) = 0 ad fz(o) + f 2 () = O. fact, fl ca be set equal t x 3 [O,J. Applyig the umerical itegrati frmula (.5) we bviusly have ad it is a csequece f (2.5) that cl = whe = 3. S the 40 3 cstat! i (2.24) cat be replaced by ay smaller umber. t is als easy t shw that the cstat 3~0 i (2.24) is best pssible. Fr this purpse take = 2 ad csider the fucti f2 [O,J, defied by f 2 (x) = ~4 (x 4-2x 3 ). The fz(o) + f () 2 = 0 ad f~4)(x) =. Because f this ad the fact that fr = 2 the right-had side f (2.4) is -egative [O,]J (cf. p. 28), it fllws frm (.5) ad (2.7) that we have This prves ur asserti. ~ l.. w. f 2 (x.) = i=o

34 ., REMARK f fl(o) + fl() ~ 0, the the first tw terms i the right-had side f frmula (.5) behave like as +~; this is a csequece f 3 (2.5). Usig (2.7) ad therem 4 it ca als be shw that greater a rder tha - is pssible fr the last term i 4 (.5). We mit the details. Refereces [J Atkis J K.E., O the rder f cvergece f atural cubic splie iterplati. SAM J. Numer. Aal' J 5(), 89-0 (968). [2J Br, C. de, Best apprximati prperties f splie fuctis f dd degree. J. Math. Mech., 2(5), (J963). [3J Hlladay, J.e., A smthest curve apprximati. Math. Tables Aids Cmput.,, (957). [4J Meyers, L.F. ad A. Sard, Best apprximate itegrati frmulas. J. Math. Phys., 29, 8-23 (950). [5J Sard, A., Best apprximate itegrati frmulas; best apprximati frmulas. Amer. J. Math., 7, 80-9 (949). [6J Scheberg,.J' t Splie iterplati ad best quadrature frmulae. Bull. Amer. Math. Sc., 70, (964). [7J Scheberg,.J' t O best apprximatis f liear peratrs. dag. Math., 26(2), (964). [8J Seveld, P., Errrs i cubic splie iterplati. J. Egieerig Math., 3(2), 07-7 (969). [9J Walsh J J.L' J J.H. Ahlberg ad E.N. Nils, Best apprximati prperties f the splie fit. J. Math. Mech., (2), (962).