On the Trivariate Polynomial Interpolation

Transcription

1 WSEAS RANSACIONS o MAHEMAICS Sle Sf O the vte Polol Itepolto SULEYMAN SAFAK Dvso of Mthetcs Fclt of Eee Do Elül Uvest 56 ıtepe c İ URKEY. sle.sf@de.ed.t Abstct: hs ppe s coceed wth the fole fo copt the coeffcets of the tvte polol tepolto PI pss thoh dstct pots the sold ectl eo. he PI s folted s t eqto s Koece podct d Kht-Ro podct of the tces d the coeffcets of the PI e copted s the eeled vese of t. I ddto the closed fole of the coeffcets of the bvte d vte polol tepoltos e obted b the se of the vese of the Vdeode t. It s see tht the tvte polol tepolto c be vestted s the t eqto d the coeffcets of the PI c be copted dectl fo the solto of the t eqto. Also t s show tht the bvte polol tepolto PI s the specl cse of the PI whe. Necl eples e epeseted. Ke-wods: Polol tepolto tvte polol bvte polol t eqto. Itodcto he polol tepolto pls pott ole both thetcs d ppled sceces. he poble of tepolt s fl coo eee d scetfc pplctos [-4 ]. he polol tepolto s vestted s dffeet fos d loths whch podce the se polol. Also t s esl solvble ethe ecll o b s copte leb pce. he ost cool sed polol tepoltos e the Le d Newto s fos [4 8-4]. he polol tepolto two o sevel vbles s the ost cool vestted poble of tepolt b eseches [ ]. he eslts o the Hete tepolto of two vbles e ve b s the pots o dffeet ccles [ ]. I [] the thos sed the techqes todced [] fo cosde the oe eel stto of hoff tepolto. Polol tepolto of two vbles bsed o pots tht e locted o ltple ccles ws stded []. he vlble techqes fo the polol tepolto d soe cte fo qeess of tepolt polol wee eteded b eet the cet dectos d b v vtos o the fdetl fol []. Copttol spects of the tepolto sevel vbles e ve b De oo d Ro [5]. he cbte fol d hoff tepolto of the tepolto sevel vbles wee vestted [4 6]. Fll Gsc d Se ve sve of eslts o ltvte tepolto the lst twet-fve es [9]. I ths ppe the pobles of the tvte bvte d vte polol tepoltos e cosdeed. he cotbto of ths ppe s to peset soe fole fo copt the coeffcets of the tepolt polols. hs ppe ddesses the poble of fd the coeffcets of tepolt polol oe two d thee vbles. he coeffcets of the polol tepoltos pss thoh ve dstct pots two d thee vbles e folted s the t eqtos. he t eqtos e solved b s the vese of the Vdeode t. hee s le ot of ltete o the coptto of the vese of the Vdeode t d ts pplctos [6 7 9 ]. he vese of the Vdeode t s vestted b s vos ethods d loths. LU fctoto of the Vdeode t d the vese of the t s setc fctos E-ISSN: Isse 8 Vole Ast

2 WSEAS RANSACIONS o MAHEMAICS Sle Sf wee vestted d the pplctos wee ve [6 7 9]. Eplct closed fo epesso fo the vese t d loths of eeled Vdeode t wee ve b s the eleet setc fctos [7]. I [8] the vese t of lowe d ppe tl fctos of Vdeode t s setc fctos ws vestted. I ths std the coeffcets of the tvte polol tepolto pss thoh dstct pots the sold ectl eo e vestted s the t eqto s Koece podct d Kht-Ro podct of the tces. It s show tht the tvte polol tepolto c be folted s t eqto d the coeffcets of the tvte polol tepolto c be copted dectl fo the t eqto. I ddto t s show tht the bvte polol tepolto s the specl cse of the PI whe. Fll the closed fole of the coeffcets of the bvte d vte polol tepoltos whch e the specl cses of PI e obted s the vese of the Vdeode t. Polol Itepoltos I ths secto we ve soe bsc deftos ssocted wth vte bvte d tvte polol tepoltos d the Vdeode t. Fthe detls c be fod else [ 4- -4]. he splest d best ow w to costct th -deee polol ppoto p to cotos fcto f the tevl [ b] R s b tepolto. If e dstct pots [ b] fo bt el vles the thee ests qe polol of deee t ost p sch tht p f fo [4 8 4]. he coeffcets st stsf the eqtos. We c wte ths sste s follows: V b V b d the t V s Vdeode t. Whe pots e dstct the Vdeode t s osl [4 ] d coseqetl the sste hs qe solto V b V s vese of t V. Note tht ths solto depeds o the vese of the Vdeode t. Let X } d Y }. { { Assed tht dt e ve fo the fcto of two vbles t the pots the ectl thee s qe sfce of the fo p 4 of deee t ost el tht psses thoh ech pot the X Y X Y s Ctes podct of the sets X d Y. he polol 4 whch s clled bvte polol tepolto stsfes fo p. hs t pot ˆ ˆ whch s ot the X Y c be ˆ ˆ p ˆ ˆ [ 5-7 ]. estted b E-ISSN: Isse 8 Vole Ast

3 WSEAS RANSACIONS o MAHEMAICS Sle Sf Also we ow cosde the tvte polol tepolto. Gve h to ppote ove sold ectl eo tht s dded b o R. Assed tht h dt e ve fo the fcto of thee vbles t the dstct pots the sold ectl eo thee s qe hpe sfce 4 o R of the fo p 5 of deee t ost el tht psses thoh ech pot the sold ectl eo. We s tht 5 s tvte polol tepolto whch stsfes p h fo ll d. It s cle tht h t pot ˆ ˆ ˆ whch s ot sold ectl eo o R c h ˆ ˆ ˆ p ˆ ˆ ˆ [9 ]. be estted b A A F F A [ ] F [ ] I the et secto we fst folte t eqto to clclte the coeffcets of the tvte polol tepolto defed 5 s Koece podct d Kht-Ro podct d the vestte the solto of ths t eqto. he vte Polol Itepolto Let V V d A A A A ] [ [ F F ] F F V 6 7 [ ] d F s t d A s t. Note tht d h e the eleets of the tces A d F defed 7 espectvel d. Us 6 d 7 we c folte the coeffcets of the tvte polol tepolto 5 s follows: V A I I I ] V I F 8 [ I s the dett t d s the Koece podct. Also we c epess 8 whch s the t eqto of 5 s follows: V I V I A V F 9 E-ISSN: Isse 8 Vole Ast

4 WSEAS RANSACIONS o MAHEMAICS Sle Sf s the Kht-Ro podct d s the vecto whose etes e. Cooll Let I V be the t defed 9. he ts vese s V I V I V V I. Poof. Obsev the of the t defed s d s the eeled vese of the t of fll d Kht-Ro podct of the tces [5 5] s esl obted s V I [ I I I ] I I I V V I. hs the poof s copleted. We se the follow Le to solve the t eqto 9. Le. A ecess d sffcet codto fo the eqto XC D to hve solto s tht DC C D ths cse the eel solto s X DC Y YCC Y s bt t d s the eeled vese of the t [ ]. heoe he eqto 9 hs qe solto s follows: V I V I A V F o eqvletl F V I V I V V I V A Poof. Sce I V I V I V I V V F V I F V I I V I I FI F the eqto 9 hs solto. Also s Le we obt the eel solto s A A p A h V I V I A V F h p A W V V I. V I V I V I W V I I V I W I W V W WI W s bt t d Ap d A h e the ptcl d hooeos soltos espectvel. If A p A the 9 hs qe solto. hs the poof s copleted. heoe Let A d F be sbtces defed 7. he the solto of the eqto 9 s V [ V V I ] A V F o A V [ V V e ] V F 4 e [ ] th et fo d. Poof. o pove the theoe we c se heoe d the tces defed 7. Ptt 7 d s Koece podct d Kht-Ro E-ISSN: Isse 8 Vole Ast

5 WSEAS RANSACIONS o MAHEMAICS Sle Sf podct the eqto s obted. the dett t s we et the eqto 4. We show tht whethe the bvte polol tepolto 4 s the specl cse of the tvte polol tepolto 5 o ot. Let s sse tht 9 d fo 7. d s the eqtos 7 d 9 we obt V AV Z 5 A Z Sce Vdeode tces V d V e osl [7 ] the sste of t eqto 5 hs qe solto [ ]. So thee s oe d ol oe solto set of coeffcets fo polol 4. It s well ow tht the eqto 5 s solvble ethe ecll o s copte leb pces. I ths std the coeffcets of the polol tepolto e to be copted dectl b eet specl fole whch c be ppled esl to the polol tepoltos stsf the ve dstct pots. We ow cosde the Vdeode t V. We c ewte the closed fo of the vese of V s V [ v v v ] the cols vectos 9 v v v e A d Z e the tces. I ths cse t s cle tht I V d V I V. he eqto 5 c be defed s the t eqto whch s sed to fd the coeffcets of the PI 4 stsf p fo ll the ectl. I ddto we c stte the PI 4 s t eqto the follow fo: p A 6 v A s defed bove. hs eqto c be esl obted fo p A I 7 whch s the t eqto of 5 whe. Note tht the solto of the t eqto 5 A V Z V 8 c be fod fo heoe. o heoe. d the PI c be vestted s specl cse of the PI. v E-ISSN: Isse 8 Vole Ast

6 . v All the fole fo the etes of the tces L d U be L d U the tl tces the LU fctoto of V c be fod b eplc vles of the eleet setc fctos [9]. he veses of tces L d U e folted closed fo s U d L < < <... d l l l.... he vese of the Vdeode t defed s 9 c be sed fo copt the coeffcets of the bvte polol tepolto. If 9 s ppled to the veses of the Vdeode tces V d V defed s we obt the closed fole of the coeffcets of the PI. hs leds to the follow eslts. Cooll Let be the podct of th ow of V wth th col of the t Z. he s ' e.... d... Poof. o pove the cooll we c ppl to the t V defed 6. Mltpl th ow of V wth th col of the t Z we obt s [ ] Z V. hs the poof s copleted. WSEAS RANSACIONS o MAHEMAICS Sle Sf E-ISSN: Isse 8 Vole Ast

7 Cooll Let be the coeffcets of the polol tepolto 4 stsf dstct pots the ectl. he the coeffcets of the PI e... fo.. d.... Poof Us the coeffcets e esl copted fo the podct of th ow of V Z wth th col of V. Fo eple fo d the coeffcets of the polol tepolto two vbles e copted esl fo the bove fole s follows: d. Eple : Asse the follow vles fo fcto two vbles of twelve dstct pots: 7 7. he coeffcets of the polol tepolto two vbles e esl obted b wt d bove fole. Us these coeffcets the polol tepolto two vbles stsf twelve pots s wtte s P. Fll t d fo 5 we obt s follows: V b V ] [ d ] [ b. It s cle tht hs the se solto wth d coseqetl t s see tht the vte polol tepolto s the specl cse of the PI. Now we c obt the coeffcets of the vte polol defed o fo the closed fo of the vese of V. he follow eslt s coceed wth the coeffcets of. WSEAS RANSACIONS o MAHEMAICS Sle Sf E-ISSN: Isse 8 Vole Ast

8 Cooll 4 Let fo.. be the coeffcets of the polol tepolto. he... Poof. Us d ppl the podct of the t V d the vecto b t s esl poved. Us Cooll 4 the follow eples e peseted. Eple he le eqto stsf d two dstct pots s copted s d the qdtc eqto pss thoh d thee dstct pots s wtte s. P Eple Cosde set { } 4 S of fo dstct pots. he coeffcets of thd deee polol tepolto pss thoh fo dstct pots e obted b wt s d d e clclted b s these fole d so the polol tepolto stsf fo dstct pots the set S s p. 4 Coclsos We cosde the tvte polol tepolto PI pss thoh dstct pots the sold ectl eo. We coclde tht the coeffcets of PI c be copted dectl fo the t eqto b the se of eeled veses Koece podct d Kht-Ro podct. WSEAS RANSACIONS o MAHEMAICS Sle Sf E-ISSN: Isse 8 Vole Ast

9 WSEAS RANSACIONS o MAHEMAICS Sle Sf It s see tht the tvte polol tepolto c be vestted s the t eqto d ts coeffcets c be copted dectl fo ths t eqto. I ddto t s show tht the bvte polol tepolto PI s epessed s the specl cse of the PI whe d the specl fole of the coeffcets of the PI e obted s the vese of the Vdeode t. Also the coeffcets of the polol p e copted s the closed fole. Refeeces [] M. oov d Y. X O Hete tepolto b polols of two vbles SIAM J. Ne. Al. Vol. 9 No. 5 pp [] M. oov d Y. X O polol tepolto of two vbles J. Appo. heo pp [] S. L. Cpbell d J C. D Mee. Geeled Iveses of Le sfotos Pt Pblsh Lodo 979. [4] S. D. Cote d C. de oo Eleet Necl Alss McGw Hll Jp 97. [5] C. de oo d A. Ro Copttol spects of Polol tepolto sevel vbles Mth. Copt. Vol. 58 No pp ] A. Esbe d G. Fedele O the veso of the Vdeode t Appl. Mth. Copt pp [7] M. E. A. El-Mw Eplct vese of eeled Vdeode t Appl. Mth. Copt. 46 pp [8] L. Fsett Necl Methods Aloths d Applctos Peso Edcto Ic. New Jese. [9] M. Gsc d. Se Polol tepolto sevel vbles Adv. Copt. Mth. pp [] C. F. Geld d G. Whetle Appled Necl Alss Seveth Itetol Edto Peso Edcto Ic. USA 4. [] C. S. Jo he ccte veso of Vdeode tces Copt. Mth. Appl pp [] L. W. Johso d R. D. Ress Necl Alss Addso Wesle Pblsh Cd 98. [] D. Kcd D. d W. Chee Necl Alss Mthetcs of Scetfc Copt oos/cole Pblsh Cop Pcfc Gove Clfo 99. [4] E. Kes. Advced Eee Mthetcs 7th Ed. Joh Wle &Sos New Yo 99. [5] S. L Mt eslts o the Kht-Ro d c-sh podcts Le Aleb Appl pp [6] J. J. Mte d J. M. Pe Fctotos of Cch-Vdeode tces Le Aleb Appl pp [7] H. Oç d G. M. Phllps Eplct fctoto of the Vdeode t Le Aleb Appl. 5 pp. -. [8] H. Oç d H. K. A Setc fctos d Vdeode t J. Copt. Appl. Mth. 7 4 pp [9] H. Oç L fctoto of the Vdeode t d ts pplctos Appl. Mth. Lettes 7 pp [] C. R. Ro d S. K. Mt Geeled Iveses of Mtces d ts Applctos Joh Wle New Yo 97. [] H. C. J hche d W. E. Mle Itepolto sevel vbles J. Soc. Ids. Appl. Mth.Vol. 8 No. 96 pp. -4. [] M. A. d M. Delp Copttol Coplet Ivesttos fo Hh Desol Model Repesetto Aloths Used Mltvte Itepolto Pobles th WSEAS Itetol Cofeece o Appled Mthetcs Decebe 9- Co Ept 7 pp. -9. [] L. R. e Ivese of the Vdeode t wth pplctos NASA echcl Note NASA-N-D pp. -4. [4] Y. X Polol tepolto sevel vbles cbte fole d dels Adv. Copt. Mth. pp [5] X. Zh Z. P. Y d C. G. Co Mt eqltes volv the Kht-Ro podct Achv Mthetc RNO os 8 pp [6] P. Zh O hoff tepolto b polols sevel vbles J. Copt. Appl. Mth pp.6-7. E-ISSN: Isse 8 Vole Ast