Differentiation and Numerical Integral of the Cubic Spline Interpolation
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1 JOURNAL OF COMPUTER VOL. NO. OCTOBER 7 Deretto d Nuercl Itegrl o te Cuc ple Iterpolto g Go cool o Coputer cece d Tecolog Jgsu Uverst o cece d Tecolog Zejg C El: go_sg@otl.co Zue Zg d Cuge Co Ke Lortor o Itellget Iorto Processg Isttute o Coputg Tecolog Cese Acde o ceces Bejg 8 C El: zzjzz@s.co cgco@ct.c.c Astrct Bsed o lss o cuc sple terpolto te deretto oruls o te cuc sple terpolto o te tree oudr codtos re put up orwrd ts pper. At lst ts clculto etod s llustrted troug eple. Te uercl results sow tt te sple uercl derettos re qute eectve or esttg rst d ger dervtves o equll d uequll spced dt. Te oruls sed o cuc sple terpolto solvg uercl tegrl o dscrete ucto re deduced. Te degree o tegrl orul s =.Te oruls s g ccurc. At lst tese clculto etods re llustrted troug eples. Ide Ters cuc sple ucto uercl deretto uercl tegrl rst dervtve secod dervtve I. INTRODUCTION ples d prtculrl cuc sples re ver populr odels or terpolto. Hstorcll ``sple ws coo drtg tool lele rod tt ws used to elp drw soot curves coectg wdel spced pots. Te cuc sple curve ccoplses te se result or terpolto prole. Te sple tecolog s pplctos CAD CAM d coputer grpcs sstes. We descre cuc sples ts ote d dscuss ter use terpolto d curve ttg.. Te cuc sple terpolto s pecewse cotuous curve pssg troug ec o te vlues te tle. Tere s seprte cuc polol or ec tervl ec wt ts ow coecets. Te rst dervtve d te secod dervtve o cuc sple re cotuous. For te pproto o grdets ro dt vlues t vertces o uor grd P. loère[] copre two etods sed o cuc sple terpolto wt clsscl etod sed o te dereces. For uvrte cuc sples p. loère use te so-clled de Boors Not Kot propert d ew etod gvg prett good slopes. J.. Ber. J. Estrd d M. V. Herádez [] ve developed G -cotuous cuc A-sple scee sootg te polgo deed te le segets jog cosecutve dt pots suc tt te sple curve les copletel o te se sde o te oudr polgo s te dt. Te proposed A-sple scee provdes ecet etod or geertg soot roots pt tt vods corers or polgol ojects or gve pled pt or desgg soot curve o polgol pece o terl etc. Petrovc Dvor [] presets two orultos o cusl cuc sples wt equdstt kots. Bot re sed o cusl drect B-sple lter wt prllel or cscde pleetto. I eter pleetto te cusl prt o te pulse respose s relzed wt ecet te-pulse-respose IIR structure wle ol te tcusl prt s pproted wt te-order te-pulse-respose FIR lter. Foruls or uercl dervtves re portt developg lgorts or solvg oudr vlue proles or ordr deretl equtos d prtl deretl equtos. Nuercl deretto s tecque o uercl lss to produce estte o te dervtve o tetcl ucto or ucto suroute usg vlues ro te ucto d perps oter kowledge out te ucto. Nuercl deretto s te process o dg te uercl vlue o dervtve o gve ucto t gve pot. I geerl uercl deretto s ore dcult t uercl tegrto. Ts s ecuse wle uercl tegrto requres ol good cotut propertes o te ucto eg tegrted uercl deretto requres ore coplcted propertes suc s Lpsctz clsses. Te deretto oruls o te cuc sple terpolto o te tree oudr codtos re put up orwrd ts pper. Nuercl tegrto s cocered wt developg lgorts to pprote te tegrl o ucto. Te ost cool used lgorts re Newto-Cotes oruls Roergs etod Guss qudrture d to lesser etets Hertes oruls d cert dptve tecques. We got terpoltor qudrture oruls wt equdstce kots usg tree tpes o cuc sple or osclltg tegrl estted te error d uercl eple ws gve to llustrte te g ccurc o our etod s wel. ACADEMY PUBLIHER do:./jcp...7-
2 8 JOURNAL OF COMPUTER VOL. NO. OCTOBER II. CUBIC PLINE INTERPOLATION N uppose tt { } re N pots were. Te ucto s clled cuc sple tere et N cuc polols d sts te propertes: I. s cuc polol o ] II. III. IV. [ V.. Ec cuc polol s our ukow costts ece tere re N coecets to e detered. Te dt pots suppl N codtos d propertes II III d IV ec suppl N codtos. Hece N N N codtos re speced. Ts leves us two ddtol degrees o reedo. Te coce o tese two etr codtos deteres te tpe o te cuc sple oted. Oe o te ollowg sets o oudr codtos s stsed [][][][7]: Clped sple: Curvture-djusted cuc sple: Perodc sple:. For te clped oudr codtos we use. pec we ot N ler equtos volvg te coecets. were [ ] Ts s trdgol ler sste. Te re ot Crout Fctorzto lgort. Te result s te ollowg epresso or te cuc ucto o [ ]. were d. III. DIFFERENTIATION OF THE CUBIC PLINE INTERPOLATION A. Fte derece orule Te splest etod s to use te derece pprotos. A sple two-pot estto s to copute te slope o er sect le troug te pots d ++. Coosg sll uer represets sll cge d t c e eter postve or egtve. Te slope o ts le s Ts epresso s Newtos derece quotet. Te slope o ts sect le ders ro te slope o te tget le out tt s pprotel proportol to. As pproces zero te slope o te sect le pproces te slope o te tget le. Tereore te true dervtve o t s te lt o te vlue o te derece quotet s te sect les get closer d closer to eg tget le: l ce edtel susttutg or results dvso zero clcultg te dervtve drectl c e ututve. A sple tree-pot estto s to copute te slope o er sect le troug te pots -- d ++. Te slope o ts le s B. Deretlto o cuc sple I te odes re equll spced tt s we te oruls c e epressed s Fro Eq. ACADEMY PUBLIHER
3 ] [ ] [ ] [ ] [ Use to get usttuto o equto produces te soluto 7 C. Clped sple Te d re kow. Fro Eq. 8 usttutg to equto d 7 respectvel te solutos re Te soluto o deretto o te Clped ple s sow tle. TABLE I. THE OLUTION OF DIFFERENTIATION OF THE CLAMPED 8 9 D. Curvture-djusted cuc sple Te d re kow. usttutg to equto 7d we ot te ler sste Te soluto re 7 7 Te vlues re susttuted te soluto s Te soluto o deretto o te Curvturedjusted cuc sple s sow tle. TABLE II. T THE OLUTION OF DIFFERENTIATION OF THE CURVATURE-ADJUTED CUBIC PLINE E. Perodc sple We d usg te soluto s.usg we d. usttutg to Eq. d Eq.7we get Usg Eq. we ot te ler sste Te soluto re Te vlues re susttuted d 7 respectvel te soluto s Te soluto o deretto o te perodc sple s sow tle. JOURNAL OF COMPUTER VOL. NO. OCTOBER 9 ACADEMY PUBLIHER
4 JOURNAL OF COMPUTER VOL. NO. OCTOBER TABLE III. THE OLUTION OF DIFFERENTIATION OF THE PERIODIC PLINE IV. DIFFERENTIATION EXAMPLE Eple : Clped sple: Fd deretlto or te pots were d. It s strgtorwrd tsk to copute te soluto d. Te cuc sple s Te cuc sple s sow gure Fg. : Te cuc sple o Eple Eple : Curvture-djusted cuc sple: Fd deretlto or te pots - were d. Te soluto s d. Te cuc sple s Te cuc sple s sow gure Fg. : Te cuc sple o Eple Eple : Perodc sple: Fd deretlto or te pots. Te soluto s d. Te cuc sple s 9 Te cuc sple s sow gure Fg. : Te cuc sple o Eple V. NUMERICAL INTEGRAL OF THE CUBIC PLINE INTERPOLATION A. Qudrture rules sed o terpoltg uctos Tere re severl resos or crrg out uercl tegrto. Te tegrd e kow ol t cert pots suc s oted splg. oe eedded sstes d oter coputer pplctos eed uercl tegrto or ts reso. ACADEMY PUBLIHER
5 A lrge clss o qudrture rules c e derved costructg terpoltg uctos wc re es to tegrte. Tpcll tese terpoltg uctos re polols. Te splest etod o ts tpe s to let te terpoltg ucto e costt ucto polol o degree zero wc psses troug te pot +/ +/. Ts s clled te dpot rule or rectgle rule. d Te terpoltg ucto e e ucto polol o degree wc psses troug te pots d. Ts s clled te trpezodl rule. d I uercl lss psos rule s etod or uercl tegrto te uercl pproto o dete tegrls. peccll t s te ollowg pproto: ] [ d B. Clped sple For te clped oudr codtos we use. pec we ot ler equtos volvg te coecets. N 7 were ] [. Ts s trdgol ler sste. Te re ot Crout Fctorzto lgort. Te result s te ollowg epresso or te cuc ucto o. ] [ 8 were d. ] [ d d d I te d 9 were d. k k It wll suce to ppl orul 9 over tervl [] wt te ve test uctos d. For te rst our uctos orul 9 s ect. For d For d } ] [ { For d JOURNAL OF COMPUTER VOL. NO. OCTOBER ACADEMY PUBLIHER
6 For d For d Tereore te degree o orul 9 s. C. Curvture-djusted cuc sple We re kow we ot ler equtos volvg te coecets. N Were ] [. Fro equtos we c get te vlue d re susttuted to equtos 9 te ot te tegrl. D. Perodc sple Te codto o perodc sple re d. Te uercl tegrl orul s d It s copoud trpezod orul. VI NUMERICAL INTEGRAL EXAMPLE Eple : Fd tegrl or te pots d were d. 9 Usg equto 9 te uercl tegrl d Te cuc sple s Te cuc sple s sow gure. JOURNAL OF COMPUTER VOL. NO. OCTOBER ACADEMY PUBLIHER
7 JOURNAL OF COMPUTER VOL. NO. OCTOBER - - Fg. : Te cuc sple o Eple Eple : Fd te tegrl or te pots 9 d 8 were d. Usg equto we ot Te soluto s. d Te cuc sple s Te cuc sple s sow gure. 8 Fg. : Te cuc sple o Eple Eple : Fd te tegrl or te pots d. Usg equto te uercl tegrl d Te cuc sple s 9 Te cuc sple s sow gure Fg. : Te cuc sple o Eple VII. CONCLUION Cuc sples re populr ecuse te re es to pleet d produce curve tt ppers to e seless. As we ve see strgt polol terpolto o evel spced dt teds to uld dstortos er te edges o te tle. Cuc sples vod ts prole ut te re ol pecewse cotuous eg tt sucetl g dervtve trd s dscotuous. o te pplcto s sestve to te sootess o dervtves ger t secod cuc sples ot e te est coce. Te oruls solvg uercl deretl d tegrl o dscrete ucto re deduced ts pper. B uercl sulto te prctclt d eectveess re vered. Te tree-pot sple uercl deretto oruls re gve ts pper. lrl te oter pots sple uercl deretto oruls c lso coputed. ACKNOWLEDGMENT Ts work ws prtll supported Ntol Bsc Reserc Progr o Jgsu Provce Uverst 8KJB d te Ntol Nturl cece Foudto o C uder Grt No.779. REFERENCE [] P. loère. Grdet pproto o uor eses te dereces d cuc sple terpolto. Lecture Notes Coputer cece cludg suseres Lecture ACADEMY PUBLIHER
8 JOURNAL OF COMPUTER VOL. NO. OCTOBER Notes Artcl Itellgece d Lecture Notes Boortcs v LNC p - 9. [] J.. Ber. J. Estrd M. V. Herádez. Costred terpolto wt plct ple cuc -sples. Lecture Notes Coputer cece cludg suseres Lecture Notes Artcl Itellgece d Lecture Notes Boortcs v 97 LNC p [] Petrovc Dvor. Cusl cuc sples: Forultos terpolto propertes d pleettos. IEEE Trsctos o gl Processg v p - 8. [] R. L. Burde J.D. Fres. Nuercl Alss. Hger Educto Press & Toso Lerg Ic. pp [] J. H. Mtews K.D. Fk. Nuercl Metods Usg MATLAB. Pulsg House o Electrocs Idustr pp.8-9. [] M. I.. Cuc sple terpolto predctors over plctl deed curves. Jourl o Coputtol d Appled Mtetcs 7 pp.8-9. [7] T. L. Ts I. Y. Ce. Ivestgto o eect o edpot costrt o te-le cuc sple terpolto. Jourl o Meccs v p - Jue 9. g Go ws or 97 d receved s M.. degree 99 d P.D degree. He ow works scool o coputer scece d tecolog Jgsu Uverst o cece d Tecolog. He s proessor d He s egge l sstes egeerg d sot coputg. Zue Zg ws or 9 d receved s M.. degree tetcs 99 ro te Deprtet o Mtetcs Ygzou Tecg College d te P.D. degree tetcs 99 ro te Isttute o otwre te Cese Acde o ceces. Hs curret reserc res re recurso teor kowledge represetto d kowledge resog. Cuge Co ws or 9 d receved s M.. degree 989 d P.D. degree 99 ot tetcs ro te Isttute o Mtetcs te Cese Acde o ceces. Now e s proessor o te Isttute o Coputg Tecolog te Cese Acde o ceces. Hs reserc re s lrge scle kowledge processg. ACADEMY PUBLIHER
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