TR/9 February 980 End Condtons for Interpolatory Quntc Splnes by G. H. BEHFOROOZ* & N. PAPAMICHAEL *Present address: Dept of Matematcs Unversty of Tabrz Tabrz Iran.
W9609
A B S T R A C T Accurate end condtons are derved for quntc splne nterpolaton at equally spaced nots. Tese condtons are n terms of avalable functon values at te nots and lead to 0( 6 ) covergence unformly on te nterval of nterpolaton.
- -. Introducton Let Q be a quntc splne on [ab] wt equally spaced nots x a ; 0l... (. ) were (b-a)/. Ten Q C 4 [ab] and n eac of te ntervals [x - x ] ;... Q s a quntc polynomal. Te set of all suc quntc splnes forms a lnear space of dmenson wc we denote by Sp(). Gven te set of values y ; -0l... were y y ) y c n [ab] n 6 we consder te problem of constructng an nterpolatory Q Sp() suc tat Q ) y ; 0.... (.) Snce dm Sp() te nterpolaton condtons (.) are not suffcent to determne Q unquely and four addtonal lnearly ndependent condtons are always needed for ts purpose. Tese are usually taen to be end condtons.e. condtons mposed on Q or ts dervatves Q (J) ; 4 near te two end ponts a and b. As mgt be expected te coce of end condtons plays a crtcal role on te qualty of te splne approxmaton. It s well nown tat te best order of approxmaton wc can be aceved by an nterpolatory quntc splne Q s Q - y 0( 6 ) were. denotes te unform norm on [ab]. Suc order of convergence s obtaned f for example te end condtons Q ) y Q ( x ) y ; 0
- - are used. However tese condtons requre nowledge of te frst dervatve of y at te four ponts x.x - ; 0 and n an nterpolaton problem ts nformaton s not usually avalable. Te natural end condtons Q (r) (a) Q (r) (b) 0 ; r 4 do not requre any addtonal nformaton but te resultng natural quntc splne does not ave 0( 6 ) convergence unformly on [ab]. Te purpose of te present paper s to derve end condtons for quntc splne nterpolaton at equally spaced nots wc depend only on te gven functon values at te nots and lead to 0 ( 6 ) convergence unformly on [ab]. We derve a class of suc end condtons n Secton by generalzng te cubc splne results of Beforooz and Papamcael (979) to te case of quntc splne nterpolaton.. Prelmnary Results To smplfy te presentaton we use trougout te abbrevatons () () m Q ) M Q ) n Q ) and N (4) Q ); I 0.. (.) Te followng quntc splne denttes are needed for te analyss of Secton : m - 6m - 66m 6m m {-y - - 0y - 0y y } ;.... - (.) M - 6M - 66M 6M M 0 { y - y - -6y y y };... - (.)
- - n - 6n - 66n 6n n 60 {-y - y - -y y } ;... - (.4) N - 6N -l 66N 6N l N 0 4 { y - - 4y - 6y - 4y y } ;... - (.) N - 4N - N - 6 { M _ - M M } 0;... - (.6) 60 { m - m m } - {n - 4n n } 0 {y -y - } ;... - l (.7) 8 {m - m - } - {M - - 6M M } 0 {y _ -y y } ;.... - (.8) m 6 { M M - } 60 { 8N 7N - } { y - y- } ;.... (.9) 6 m - { M M } { 8N 7N } 60 { y l- y } ; 0... - l (.0) m - { n - 8n n } {y - y. }; 0 M - { N - 8N N } {y -l- y y l }; 0... - (. )... - (.)
- 4 - M {m - m - - m - m } {y - 6y -l - 4y 6y l y } ;... - (.) N { M - 8M M } 0 4 { y - - y y. } ;... K-. (.4) Te relatons (.) (4) can be derved from te results of Albasny and Hosns (97) Fyfe (97) Alberg Nlson and Wals (966) and Saa (970). Te most convenent way for constructng Q s probably troug ts B-splne representaton were as usual. Q ( x ) c ( x x ) l ( t x ) ( t 0 x ) t t < x x. However t s mportant to observe tat te unque exstence of Q can be establsed by sowng tat any of te four ( ) () lnear systems obtaned by usng eter of te relatons (.) (.) (.4) or (.) togeter wt te four equatons derved from te end condtons of Q > s non-sngular. For example f te lnear system correspondng to (.) as a unque soluton M ; 0 l... ten (.4) and (.6) gve te parameters N. ; 0l... and Q can be constructed n any nterval [x - x ] by ntegratng Q (4) ) _ { N _ x) N -x _ ) } four tmes wt respect to x and settng (.) Q ) y and Q () ) M ;.
- - for te determnaton of te four constants of ntegraton. Smlarly f te lnear system correspondng to (.) as a unque soluton m ; 0... ten (.) and (.8) gve te parameters M ; 0... and Q can be constructed n any nterval [x - x ] by use of te two pont quntc Hermte nterpolaton formula. Smlar arguments establs te unque exstence of Q n te two cases were te lnear systems correspondng to (. 4) and (.) are non-sngular. Te followng two lemmas are also needed for te dervaton of te results gven n Secton : Lemma.. If y C 6 [ab] ten for x [x - x ] ;... (r) Q ) (r) y ) l A r - r max 0 m y 6 0( r ); r0..6 (.6) were te A r are constants ndependent of. Lemma.. Let λ m - y. If y C 7 [ a b] ten were λ - 6λ - 66λ 6λ λ β ; - (.7) β 6 y (7) ; -...-. (.8) Lemma. can be establsed from te representaton of Q by means of te two pont quntc Hermte nterpolaton formula by usng a trval generalzaton of a result due to Hall (968: p.4). Lemma. follows easly from (.) by Taylor seres expanson about te pont x. End Condtons We let Q be a quntc splne nterpolatng te values y. y ) ; 0 l... at te equally spaced nots (.) and assume tat 6. As before we use te abbrevatons (.).
- 6 - We consder end condtons of te form () 0.. ; y a 60 m m m m y a 60 m m m m l l γ β α γ β α and see to determne te scalars αβγ and a ; 0... so tat Q exsts unquely and Q (r) _ y (r) 0 ( 6-r ) ; r 0.... (.) For ts we let λ -m -y ; 0... and assume tat y C 7 [ab]. Ten te equatons (.) and (.) gve (.4) β λ αλ βλ γλ β λ αλ βλ 4 γλ... ; β λ λ 6 λ 66 λ 6 λ β 4 γλ βλ αλ λ 0 β γλ βλ αλ 0 λ κ κ κ κ κ κ κ κ κ κ κ ι were ) 4. ( 0 ; y a 60 ) y ( y a 60 ) y ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) γ βγ αγ γ β γ βγ αγ γ β 6 y (7) ;... -. (.) ß Te matrx of coeffcents n (.) s te matrx of te ( l) ( I) lnear system wc determnes te parameters m of te quntc splne Q. We denote ts matrx by A let I be te set I {(α.β. γ) ; det A 0 } (.6)
- 7 - and for any (α ß ϒ) I we assume tat tere exsts a number M ndependent of suc tat A - M. Ten for any (α ß ϒ) I Q exsts unquely and snce max o max o λ A β Lemma. sows tat Q (r) - y (r) 0( 6-r ) ; r 0... only f ß 0 ( ); 0 -. Teorem.. Let Q be an nterpolatory quntc splne wc agrees wt y C 7 [ab] at te equally spaced nots (.) and satsfes end condtons of te form (. ) were (α ß ϒ). I. Ten Q (r) - y (r) 0( 6-r ) ; 0 only f n (. ) Proof. a a a a a a 0 7 α β γ 00 6 α 0 β γ 00 0 α 0 β 60 γ 00 60 α 60 β 0 γ 4 7 0 α β 0 γ α β γ By Taylor seres expanson we fnd tat (.7) β 0 ( ); 0 - only f te scalars α β γ and a ; 0... satsfy te relatons (.7). we fnd tat β β More specfcally wen te relatons (.7) old (6) (0 α β γ) y F 6 60 (6) (0 α β γ) y F 60 0 6 (.8)
- 8 - were y (7) F {6880 40( α γ ) 0400 a a 8( a o a 4 ) 87 a }; 0 l -. (.9) Defnton.. A quntc splne Q wc nterpolates te values y ; 0..... at te equally spaced nots (.) and satsfes end condtons of te form (.) wt te a.; 0... gven by (.7) wll be called an E(α ß ϒ) quntc splne. For any (α ß ϒ) I an E(α ß ϒ) quntc splne exsts unquely and by Teorem. f y y ) were y C 7 [ab] ten (r) Q _ y (r) 0( 6-r ) ; r 0 0.... In general for any (α ß ϒ) I ß 0( ); - 0 - and tus m - y 0( ); - 0.... However f te values α ß ϒ are suc tat 0 - α ß ϒ 0 (.0) ten from (.8) ß - 0( 6 ); - 0 -. For ts reason te class of E(α ß ϒ) quntc splnes wose parameters satsfy (.0) s "best" n te sense tat for any member of ts class m - y 0( 6 ); 0.... (.l)
- 9 - Corollary.. Te end condtons of an E(α ß ϒ) quntc splne can be wrtten as m m αm αm βm βm γm γm p ) ap ) βp ) γp p ) ap ) βp ) γp ); ); 0 (.) were p. denotes te quntc polynomal nterpolatng te values y.y l.... y at te ponts x x.... x. Proof. p p p p p p Te proof follows from (.)(7) and te results ) { 7y 00y 00y 00y 7y 4 y } 60 ) { y 6y 0y 60y 0y 4 y } 60 ) {y 0y 0y 60y y 4 y } 60 ) { y y 60y 0y 0y 4 y } 60 4 ) {y 0y 60y 0y 6y 4 y } 60 ) { y 7y 00y 00y 00y 4 7y 60 (.) }. Corollary. sows tat te end condtons of te E(000) quntc splne are m m p ); 0 p );. (.4) Clearly (000) I and terefore an E(000) quntc splne exsts unquely. In fact t s easy to see tat n ts case te (-4) (-4) matrx A of (.) s suc tat A.
- 0 - Te corollary also sows tan an E ( ß ϒ) quntc splne can be nterpreted as an nterpolatory quntc splne wt end condtons m m p p ( ) ( ) 4 ( ( x x ); );. (. ) Smlarly te E (α ß ϒ) and E(α ß ) quntc splnes can e nterpreted as nterpolatory quntc splnes wt end condtons respectvely m p ); m p ); and m p ); 4 m p ); 4. (.6) (.7) Te unque exstence of eac of te E( ß ϒ E( α ϒ) and E(αß ) quntc splnes can be establsed easly by consderng te lnear systems for te m 's derved from te consstency relaton (.) and te end condtons (.)- (.7). However te exstence of te E(α ϒ) ) and E(αβ ) splnes can only be establsed under te assumptons 7 and 9 respectvely. Te corollares stated below establs varous alternatve representatons for te end condtons of an E(α ßϒ) Quntc splne. Tey are establsed by usng te quntc splne denttes lsted n Secton and te expressons for te dervatves p () ); 4 of te nterpolatng polynomal p ). Altoug te algebra nvolved n te dervaton of tese results s very laborous te proofs are oterwse elementary and for ts reason te detals are omtted.
- - Corollary.. Te end condtons of an E(α ß ϒ) quntc splne can be wrtten as A o Δ N 4 A Δ N A Δ N A Δ N 0 A o N A 4 N A N A 0 were N Q (4) ) N 0; (.8) A A A A o ( α β γγ) (69 7 α β γγ 0(7 α 7 β 9 γ9γ 0(0 α β γ) (.9) and A V are respectvely te forward and bacward dfference operators. Corollary. sows tat te fve E(α ß ϒ) quntc splnes lsted below ave partcularly smple representatons of te form Δ n N n N - 0. Splne End condtons E Δ Ν N - 0; 0 E() Δ N N - 0; E(9 9 ) Δ N N - 0; 0 (.0) E(7 9) Δ 4 Ν 4 Ν _ 0; 0 E(6) Δ N N - 0 ; 0 Te unque exstence of eac of tese fve E(α ß ϒ) quntc splnes can be establsed easly by consderng te lnear system
- - for te parameters N derved by usng te end condtons (.0) and te conastency relatons (.). However te exstence of te E() E(79) and E(6) splnes can be establsed only under te assumptons 7 7 and 8 respectvely. We note tat te parameters of eac of te E(99 ) E(79) and E( 66) quntc splnes satsfy (.0) and tus for eac of tese tree splnes max m - y 0 ( n ) (.) o were n 6 rater tan n. Corollary.. Te end condtons of an E(α ß ϒ) quntc splne can B B {m {m p () p )} 0 () )} 0; 0 (.) be wrtten as were M Q () ) B o ( 8α 7β 8γ-) B ( 77 α 8 β 67 γ 8) B ( 8 α 77 β 8 γ 67) B ( 7α 8β γ 8) and p denotes te quntc polynomal nterpolatng te values (.) y y.... y at te Ponts x x l.... x. Wen α 87/888 ß - 890/888 and ϒ 7/888 ten n (.) B B B 0 It follows from Corollary tat te end condtons of te E 87 888 890 888 7 888 can be wrtten as M M p p ( ) ( ) ( x ( x ); ); 0. (.4 )
- - Te unque exstence of ts splne follows at once by consderng te lnear system for te parameters M. derved from te consstency relatons (.) and te end condtons (.4). Corollary.4. Te end condtons of an E(α ß ϒ) quntc splne can be wrtten as C C {n {n p were n Q () ) C o α γ -8 C 8 α β C α 8 β C β 8 γ - () () p 6 γ - 6 6 γ - 6 )} 0 )} 0; 0 and p denotes te quntc polynomal nterpolatng te values (.) (.6) y y... y at te ponts x x... x. Wen α 74/78 ß - 040/78 and ϒ 9/78 ten n (.6) C C C 0. It follows from Corollary.4 tat te end condtons of te 74 040 9 E splne can be wrtten as 78 78 78 n n ( ) p ( ) p ( x ( x ); 0 );. Te unque exstence of ts splne follows at once by consderng te lnear system for te parameters n derved from te consstency relatons (.4) and te end condtons (.7). (.7 )
- 4 - Corollary.. Te end condtons of an E(α ß ϒ) quntc splne can be wrtten as D D {N {N p (4) p )} (4) 0 )} 0; 0 (.8) were NQ (4) ) D o (8 α β γ -9) D ( α 8 β 7 γ - 6) D (8 α β 6 γ -7) D ( α β 9 γ - ) and p denotes te quntc polynomal nterpolatng te values (.9) y y l... y at te ponts x x l.... x. Wen α 9/8 ß - 79/8 and ϒ 7/8 ten n (.9) D D D 0. It follows from Corollary 9 tat te end 9 79 7 condtons of te E can be wrtten as 8 8 8 N N p p (4) (4) ); 0 );. (.0) Te unque exstence of ts splne follows at once by consderng te lnear system for te N derved from te consstency relatons (.) and te end condtons (.0). Let d denote te ump dscontnuty of Q () at te not x. Ten from (.) d Q( )( X ) - Q ( ) ( X -) Δ N -l N ; -. (.)
- - E It follows at once from (. 0) tat te fft dervatves of te and E( ) quntc splnes are contnuous respectvely at te nots x ; - - and x ; J - -. Tese propertes are te specal cases G G G 4 0 and G G G 4 0 of te general result contaned n te followng corollary. Ts result s establsed easly by usng (.) and te result of Corollary.. Corollary.6. Let Q be an E(α ß ϒ) quntc splne and let d denote te ump dscontnuty of Q () at x x Ten G G d d G G d d G G d d G G 4 4 d d 0; 0; (.) were G G G G 4 6 76 09 6 α 9 α 49 α 76 α 6 β β 66 β 49 β 9 γγ γ 49 γ9 66 γ6 (.) A number of propertes of specal nterest wc emerge from te result of Corollary.6 are lsted below. Splne Property satsfed by te d ' s E d d d - d - 0 E( ) d d d - d - 0 E(9 9 ) d d d and d - d - d - E(7 9) Δ d d -. 0; (.4) E( 6 ) Δ d d - 0;
- 6-4. Numercal results In Table.. we present numercal results obtaned by tang y ) exp) x 0. 0 ; 0... 0 (4.) and constructng varous E(α ß ϒ) quntc splnes. Te splnes consdered are te E(000) quntc splne and te fve splnes of (..0). Te results lsted are values of te absolute error exp) - Q) (4.) computed at varous ponts between te nots. For comparson purposes we also lst results computed by constructng te natural quntc splne (N.Q.S.) wt nots (4.) nterpolatng te functon y) exp) at te nots. Te numercal results ndcate te serous damagng effect tat te natural end condtons Q (r) 0 ) - Q (r) 0 ) 0; r 4 (4.) ave upon te qualty of te approxmaton and demonstrate te consderable mprovement n accuracy obtaned by usng end condtons of te type consdered n te present paper nstead of (4.). Te results also sow tat as predcted by te teory te 'best' E(α ß ϒ) quntc splnes correspond to values α ß ϒ tat satsfy (.0).
TABLE Values of exp) - Q) X N.Q.S. E(000) E E( ) E(99) E(79) E(6 ) 0.0 0.9 0-0.7 0-9 0.X0-9 0.x0-8 - 0.70 0-0.84 0 0.7x0-0.0 0. 0-0.78 0-9 0. 0-0.8x0-8 0.84 0-0.x0-0.X0-0.07 0. x 0-0.7X0-9 0.6x0-0 0.4x0-9 0.x0 - - 0.94x0 0.69x0-0.09 0.X0-6 0.X0-9 0.4xl0 0 0.76 l0-0 0 l6-0.xl0-0.6xl0-0. 0.9 0-7 0.9X0-0 0.4x0-0.9X0-0.l0 l0-0; 0-0.X0-0.6 0.X0-8 0.40X0-0.xl0-0. l0-0.6xl0 0.6xl0-0.xl0-0.6 0.6X0-8 0.98X0-0.ll l0-0.x0-0.7 0-0.7X0-0.7X0-0.9 0.X0-0.4 0-8 0.xl0-9 0.49 l0-9 0. l0-0.0 l0-0.4x0-0.96 0.X0-0. x 0-8 0.6x0-9 0.xl0 8 0.7 0 - O.4 l0-0.94xl0-0.98 0.9 0-0. 0-8 0.xl0-9 0.8x0-8 0.x0-0 - 0.8x0 0.4 l0-0.99 0.77X0-0.9 0-9 0.4 l0-9 0.6xl0-8 0.0xl0-0 0.X0-0.l0 l0 -
- 8 - R E F E R E N C E S AHLBERG J.H. NILSON E.N. & WALSH J.L. 967. Te teory of aplnes and ter applcatons. London: Academc Press ALBASINY E.L. & HOSKINS W.D. 97. Te Numercal calculaton of odd degree polynomal splnes wt equ-spaced nots. J.Inst.Mats Apples 7 84-97. BEHFOROOZ G.H. & PAPAMICHAEL N.979. End condtons for cubc splne nterpolaton J. Inst.Mats Apples -66. FYFE D.J. 97. Lnear dependence relatons connectng equal nterval Nt degree splnes and ter dervatves. J.Inst.Mats Apples 7 98-406. HALL C.A. 968. On error bounds for splne nterpolaton. J.Approx.Teory 7 4-47. SAKAI M. 970. Splne nterpolaton and two-pont boundary value problems. Memors of Faculty of Scence Kyasu Unversty Seres A XXIV 7-4. XB 640 X