Numerical Algorithm for Analysis of n-ary Subdivision Schemes

Transcription

1 Avbe t htt://vueu/ A A Mth ISSN: Vo 8 Issue Decebe 64 6 Actos Ae Mthetcs: A Iteto Jou AAM Nuec Agoth fo Ayss of -y Subvso Schees Ghu Mustf Muh Zh Detet of Mthetcs The Is Uvesty of Bhwu ghuustf@ubeu; h_8@yhooco Receve: Febuy 6 ; Accete: August Abstct The yss fo cotuty of t cuves geete by -ot -y subvso schees s esete fo The yss s bse o the stuy of coesog ffeeces ve ffeece schees A uec goth s touce whch coutes the cotuty hghe oe ve ffeeces of schees effcet wy It s so fee fo oyo fctoto vso ue the we-ow Luet oyo goth fo yss of schees whch ees o oyo gebc oetos It oy ees o the thetc oetos Keywos: Subvso schee; ve ffeece; cotuty; yss; Luet oyo; uec goth AMS-MSC No: 65D7 65D7 65D5 Itoucto Coute e geoetc esg s the bch of coutto geoety whch es wth the goths fo esgg sooth cuves sufces voues Thee s vey cose etosh betwee coute e geoetc esg geoetc oeg The ost coo thg coute e esg s the costucto eesetto of fee fo cuves sufces by the set of ots usg oyos Subvso efes cuve o sufce fo t coto esh by ecusve efeet Thus 64

2 AAM: Ite J Vo 8 Issue Decebe 65 subvso schees e wey use coute ghcs coute e geoetc esg fo geetg sooth cuves sufces fo scete set of t ots s they ove effcet febe wy fo ths uose The cotuty of t cuve geete by subvso schee s vey ott So evey schee whe t s costucte ust be ye e wht s the oe of cotuty of the t cuve geete by ths costucte schee Dy esete the techue fo yss of by schees by the fos of Luet oyos Lte o ths etho ws etee fo tey utey schees [Hss Dogso Mustf Kh 9] By gebc oetos o such oyo suffcet cotos fo covegece of the subvso schee fo the soothess of the t cuve geete by the subvso schee c be chece the utotcy Gve the Luet oyo of -y subvso schee S etee fo of Luet Poyo Agoth LPA fo hghe ty schees c be estte s: Ste-: If ethe the schee oes ot covege Z Z Z Sto! Ste-: Coute Ste-: [] Set Ste-4: Fo L M : Coute b If L N L [ L] L L N S s coveget Sto! [ L c If N coute ] L L L Ste-5: S s ot cotctve fte M tetos Sto! Aothe but vey o etho fo yss of schees s Dve Dffeece Agoth DDA whch ws touce fo 4-ot by schee by Dy et 987 the ts geee veso fo -ot by schees by Dy et 99 Cuety DDA s ot cooy use fo yss of the schees LPA voves oyo fctoto vso Theefoe the yss of -ot schees wth hghe ty [L 9 Mustf Reh Zheg et 9] by LPA the ee to he hghe oe oyos the fctoto vso hs otvte us to touce goth fo yss whch shou be fee fo oyo gebc oetos Accog to Dy et 99 If the th oe ve ffeece of the og by schee s C -cotuous the the og by schee w be C -cotuous Moeove ccog to Sb Hghe oe ve ffeeces e ust ve ffeeces of ve

3 66 Ghu Mustf Muh Zh ffeeces I ths tce we hve geee these es to uec goths fo ve ffeeces the yss of the -ot -y subvso schees Cotbutos: The cotbutos of the e e Nuec goth fo ve ffeeces of -ot -y schees Nuec goth fo cotuty of -ot -y subvso schees I ths goth we hve ece Ste- Ste- of LPA by se thetc oetos A eostto by uec ees tht Poose Nuec Agoth PNA LPA gve the se esuts The est of the e s oge s foows: I Secto we scuss -y schees the ve ffeeces covegece We so eset uec goth fo ve ffeeces ths secto Secto s ecte to the soothess yss of the schees We eset uec goth fo cotuty of the schees uec ees coso of PNA LPA ths secto -y ts Dve Dffeece Schees I ths secto we eset -y schee coute the ffeece betwee ts two cosecutve coto oygos t ffeet subvso eves Covegece of the -y schee s so ove ths secto Its fst oe ve ffeece schee s esete t the e of ths secto -y Uvte Schees Let Z eote seuece of ots R N N -y subvso ocess efe by Aset s whee s o-egtve tege A whee coeffcets { } e ce subvso s If s the Luet oyo of the bove schee the etes coeffcets { } e ete s } { } The ecessy coto fo ufo covegece of schee s {

4 AAM: Ite J Vo 8 Issue Decebe 67 M Dffeeces betwee Coto Poygos I ths secto we coute the ffeece betwee the st eve coto oygo { } th eve coto oygo Le -y subvso schee Gve t coto oygo Z et the vues be efe ecusvey by subvso ocess togethe wth the whee 4 5 whee s the ty of the schee s eo thoughout the e Poof: Usg we get tht es AB whee A

5 68 Ghu Mustf Muh Zh B Futhe t es tht Sce by fo so Let eote the u o the whee Sy Ths es whee Now by 6 7 we get Ths coetes the oof 6 7

6 AAM: Ite J Vo 8 Issue Decebe 69 By eteg the techue gve Le of Dy et 99 we get the foowg e Le Gve t coto oygo Z et the vues be efe ecusvey by subvso ocess Suose s the ecewse e teot to the vues The the ffeece betwee s * * whee s soe e ube Theoe 8 Gve Z et the vues be efe ecusvey by subvso ocess togethe wth the ecessy coto be the ecewse e teot o to the vues The fo whee s efe by 4 thee ests C whch es schee s C -cotuous Poof: Cose the ecewse e teot o to the vues et eote the ufo o o C We w show tht { } efes Cuchy seuece o C Sce the ffeece betwee s tte t ot o the st esh the by 8 we get * * whee s soe e ube Utg we get * whee s efe by 4 If the t foows tht C { } efes Cuchy seuece o C

7 6 Ghu Mustf Muh Zh Ths coetes the oof Re Hee we ote tht s suffcet coto fo s efe by 4 5 Fst oe ve ffeece schee Le C -cotuty of the -y schee whee The fst oe ve ffeece ocess of schee efe by s efe s 9 Poof: If gve by s the ecewse e teot to the vues the the fst oe ve ffeece s By ecg by by bove euto we get Usg we get Sfyg we get { }

8 AAM: Ite J Vo 8 Issue Decebe 6 We wt to e euto of the fo Ths es Sfyg we get { } Cog the coeffcets of s ' sovg suteousy we get Substtutg we get { } Ths es By ecg by by we get Usg s oceue s bove we get

9 6 Ghu Mustf Muh Zh Ths coetes the oof Note: Wthout oss of geety we c vy ste of the eutos eesetg : 5 : 9 s the ety fo s eo Ths s ust to vo et coutto goth 4 Nuec Agoth fo Dve Dffeeces Sce hghe oe ve ffeeces e ust ve ffeeces of ve ffeeces by Sb utg 9 ecusvey we get uec goth fo coutg hghe oe ve ffeeces of -ot -y subvso schees fo Ths goth s fst effcet becuse se gebc oetos Stes - e efoe o the ght h ses of eutos esuts ssge to the eft h se of the eutos wthout usg et coute eoy to sve ewy coute vues Iut: Ete the s of the schee to to whee st fo the coety e ube of ots vove to set ew ot the coto oygo ty of the schee esectvey Ste : Coute Fget of 5 9 whee fo Ste : Coute Fget of 9 Goto Ste- fo et hghe oe ve ffeece othewse et Outut: s th oe ve ffeece c be obte by cycg s-tes Stes -

10 AAM: Ite J Vo 8 Issue Decebe 6 Nuec Agoth fo Cotuty of Schee Hee we fst sue the bove esuts the eset the uec goth fo cotuty of -ot -y schee The ecessy suffcet cotos fo C -cotuty e gve Theoe e < esectvey Fo hghe oe cotutes we ee hghe oe ve ffeeces whch c be coute by usg the bove oose uec goth By Dy et 99 gve schee w be C -cotuous f ts th ve ffeece schee s C -cotuous These esuts e to estbsh foowg uec goth fo coutg the oe of cotuty of the -ot -y subvso schee: Iut: Ete the s of the schee to to whee st fo the coety ty of the schee esectvey Ste : Utto of : If fo to the goto Ste othewse et Ste : Fget of 5 9 whch s coo both eutos: Do couttos o the ght h ses of the foowg eutos ssg the esuts to the eft h ses of eutos: fo whee Ste : Fget of 5 suffcet coto: By usg ute vues by Ste coute fo If the goto Ste 4 othewse et Ste 4: Fget of 9: By usg ute vues by Ste ght h ses of the foowg eutos ssgg the esuts to the eft h ses of the eutos coute Go to Ste Outut: The s-tes successfu coeto of Stes -4 e og schee s cotuous C s -

11 64 Ghu Mustf Muh Zh The vty of bove goth hs bee chece by coutg the cotuty of soe weow schees The foowg esuts obte by the oose uec goth coce wth the esuts obte by the geee Luet oyo goth Cooy Gve Z et the vues be efe ecusvey by foowg 4-ot teotg by subvso schee touce by Dy et 987 the schee s C -cotuous ove the etc tev 8 Poof: Hee 4 the s of the schee s: st Rou: Ste : Cey Ste : Ste :

12 AAM: Ite J Vo 8 Issue Decebe 65 Sce fo 4 theefoe schee s C Ste 4: Rou: Ste : Cey Ste : Ste : Sce fo schee s C whe schee s 8 theefoe fst oe ve ffeece C -cotuous Ste 4:

13 66 Ghu Mustf Muh Zh Rou: Ste-: Cey et The schee s C -cotuous fo 8 Cooy The schee s C -cotuous by tetg och LPA Poof: The Luet oyo fo the s of the 4-ot by schee c be wtte s Now S As we ow S s coveget C -cotuty ff S fo soe Z L So fo S L f 4 Now by coutg we get Ths es to 4 S 4

14 AAM: Ite J Vo 8 Issue Decebe 67 Thus fo the ge 8 8 L we hve S As fo soothess yss C -cotuty cose S b wth b The S b s coveget ff S s cotctve Now b But fo the cse L S Theefoe to see the cotctvty we cose S 5 coto S gves the ge 54 Re 8 The Fo Cooy oe c see tht PNA s fee fo oyo oetos ee o oy thetc oetos whe LPA ees o oyo s we s o thetc oetos So t s obvous tht the coutto coety of PNA s ess th the coety of LPA Cooy Gve Z et the vues be efe ecusvey by the foowg 6-ot teotg tey subvso schee touce by Kh Mustf the schee s C -cotuous ove the tev Cooy 4 Gve Z et the vues be efe ecusvey by the foowg 4-ot otg utey subvso schee touce by Hss Dogso f f f f f f f f f f

15 68 Ghu Mustf Muh Zh f f f f f f f f f f the schee s C -cotuous ove the tev Cooy 5 Gve Z et the vues be efe ecusvey by the foowg 7-ot teotg tey subvso schee of L 9 f f f f f f f f f f 4 f f f f f f f f the schee 4 s Cooy 6 C -cotuous Gve Z et the vues be efe ecusvey by foowg 6-ot otg uy subvso schee of Mustf Reh f [88 f 7565 f 884 f 447 f 485 f 67 f ] 5 f5 [ 4769 f f 9 f 74 f 8555 f 558 f] 5 f [ f 965 f 475 f 475 f 965 f f ] 5 f [558f 8555 f 74 f 9 f f 4769 f ] 5 f [67 f 485 f 447 f 884 f 7565 f 88 f ] 54 whee 4 the schee 5 s C -cotuous 4 Coso Hee s coso betwee the oose uec goth PNA fo cotuty the Luet oyo goth LPA

16 AAM: Ite J Vo 8 Issue Decebe 69 We see tht Ste of LPA Ste- of PNA e se We so obseve tht Ste 4 & b of LPA Ste of PNA e se Ste Ste of LPA e ffeet fo Ste Ste-4 of PNA Hee we obseve tht oyo fctoto vso suto e vove Ste Ste of LPA but se thetc oetos such s subtcto utcto e vove Ste Ste 4 of PNA Obvousy fo hghe ty schees e tey utey etc the oyo fctoto vso suto vove Ste Ste of LPA eue oe couttos th se thetc oetos vove Ste Ste 4 of PNA Howeve fo few schees wth egtve ss wthout ete LPA gves sh bous fo cotuty th PNA I ths sec cse the PNA ees tetve of Ste- 4c of LPA to get sh bou We eve ths s oe uesto 4 Cocuso I ths e we hve esete uec goths fo coutg hghe oe ve ffeeces cotuty of -ot -y subvso schees fo Ou uec goth fo coutg ve ffeeces s etvey ew Poose uec goth fo cotuty s fee fo oyo gebc oetos uecy stbe fst effcet We hve eostte the vty of the uec goth by uec ees The esuts obte by ou uec goth coce wth those of the Luet oyo goth Acowegeet Ths wo s suote by the Igeous Ph D Schosh Schee of Hghe Eucto Cosso HEC Pst REFERENCES Aset N No-e subvso of uvte sgs scete sufces PhD thess E coe Poytechue Fe ' e' e e Luse Luse Swte Dy N Ayss of covegece soothess by the fos of Luet oyos A Ise E Qu M S Fote Es Tutos o Mutesouto Geoetc Moeg Sge 5-68 Chte Dy N Lev D J A Gegoy J A 987 A 4-ot teotoy subvso schee fo cuve esg Coute Ae Geoetc Desg Vo Dy N Gegoy J A Lev D 99 Ayss of ufo by subvso schees fo cuve esg Costuctve Aoto 7-47

17 6 Ghu Mustf Muh Zh Sb M Ayss esg of uvte subvso schees Geoety Coutg Sge 6 ISBN L J-A 9 O -y subvso fo cuve esg: III -ot +-ot teotoy schees Actos Ae Mthetcs: A Iteto Jou Vo 4 No Mustf G Reh N A The s of b + 4-ot -y subvso schee Coutg Vo 9-4 DOI 7/s Kh F Mustf G 8 Tey s-ot teotg subvso schee Lobchevs Jou of Mthetcs Vo 5-6 Hss M F Dogso N A Tey thee-ot uvte subvso schees TR No 5 Uvesty of Cbge Coute Lbotoy htt://wwwcccu/techreots/ucam-cl-tr-5f Mustf G Kh F 9 A ew 4-ot C utey otg subvso schee Abstct Ae Ayss Vo 9 Atce ID ges DOI:55/9/967 Zheg H Hu M Peg G 9 -y subvso geeg B-ses Seco Iteto Cofeece o Coute Eectc Egeeg DOI: 9/ICCEE94