Research Article A New Derivation and Recursive Algorithm Based on Wronskian Matrix for Vandermonde Inverse Matrix

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1 Mathematcal Problems Egeerg Volume 05 Artcle ID pages Research Artcle A New Dervato ad Recursve Algorthm Based o Wroska Matr for Vadermode Iverse Matr Qu Zhou Xja Zhag ad Xogwe Lu School of Iformato Scece ad Egeerg Hua Iteratoal Ecoomcs Uversty Chagsha Hua 4005 Cha College of Scece Natoal Uversty of Defese Techology Chagsha Hua Cha Correspodece should be addressed to Xja Zhag; jz 0075@63com Receved 6 Jue 05; Accepted 7 September 05 Academc Edtor: Xka Che Copyrght 05 Qu Zhou et al Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese whch permts urestrcted use dstrbuto ad reproducto ay medum provded the orgal work s properly cted For a aalytcal epresso of Vadermode verse matr a ew dervato process based o Wroska matr ad Lagrage terpolato polyomal bass s preseted Recursve formula ad mplemetato cases for the drect formula of Vadermode verse matr are gve based o dervg the ufed formula of Wroska verse matr For the calculato of symbol-type Vadermode verse matr the drect formula ad recursve method are verfed to be more effcet tha Mathematca whch s good at symbolc computato by comparg the computg tme Mathematca The process ad steps of recursve algorthm are relatvely smple The dervato process ad dea both have very mportat values theory ad practce of Vadermode ad geeralzed Vadermode verse matr Itroducto Vadermode matr ad geeralzed Vadermode matr ad ther verses are always wdely cocered may research felds such as umercal aalyss data ad sgal processg ad cotrol theory [ 8 There are may effectve ways to calculate the Vadermode verse matr Fo eample Tou [ ad Res [ obtaed the matr formula from the coeffcets of polyomal terms Neagoe [3 deduced a aalytc formula of comple-type Vadermode verse matr based o symmetrc polyomals Esberg ad Fedele [4 preseted a geeral eplct formula for the elemets of verse matr ad two dfferet algorthms were deduced I ths paper based o results [ a ew dervato process of a aalytcal epresso of Vadermode verse matr s preseted based o Wroska matr ad Lagrage terpolato polyomal bass ad recursve formula ad mplemetato cases for the drect formula of Vadermode verse matr are preseted based o deducg the ufed formula of Wroska verse matr The rest of ths paper s arraged as follows Frstly we summarze the ma dea ad aalytcal formulas preseted [3 ad gve a ew dervato process for the formula Secto I Secto 3 a recursve formula of Vadermode verse matr s deduced I Secto 4 umercal smulatos are preseted wth some cases Mathematca ad cocluso ad prospect are gve by comparg ad aalyzg the umercal results the last secto A New Dervato of Vadermode Iverse Matr If ( ) are dfferet umbers that s to say they are ot equal to each other the we defe the -order matr V ( )(V j ) ( j ) ( ) ( ) where j as a Vadermode matr For the verse of ths matr Neagoe [3 obtaed the calculato formula () accordg to relatos betwee the determat ()

2 Mathematcal Problems Egeerg ofthematr()adthedetermatofthematrwhch was obtaed by addg a row whose elemets were j (j ) ad a colum whose elemets were ( 0 ) to()adbasedothecalculatoformulaof verse matr by usg the matr determat: For a group of learly depedet fuctos Φ{φ () φ ()φ ()} whereφ () ( ) has the ( )th dervatve Wroska matr s defed as W[φ () φ () φ () (w j ) (φ () j ) j; 0 (4) V [(V j ) T [(( ) +j [ σ (j) k ( j k ) kj+ ( k j ) ) j T () If there s aother group of fuctos Φ { φ () φ () φ ()} whch satsfy φ () φ (t) ( t) ( )! (5) the the verse matr W [φ () φ ()φ () of (4) ca be obtaed [9 as where are colum subscrpts ad j are row subscrpts ad σ 0 ( ) σ k ( ) j j jk j < <j k k σ k (j) σ k ( j j+ ) Here we wll use a ew method etrely dfferet from [3 to deduce formula () (3) W [φ () φ () [( ) j φ ( j) () (6) where jad If we set y () (/( )!)( ) ( ) the we ca verfy that the Lagrage terpolato polyomal bass l () ( w ( ) j ) () j j where w() ( ) satsfy (5); that s to say (7) y () l (t) ( t) ( )! (8) For Wroska matr W[y ()y () W[y () y () [ ( )! ( j) ( )! ( ) ( )! ( ) ( )! ( ) ( )! ( ) ( )! ( ) ( )! ( ) [ [ (9)

3 Mathematcal Problems Egeerg 3 From (6) the verse matr of W[y ()y () s W [y () y () [( ) j l ( j) () ( ) l ( ) () ( ) l ( ) () l () ( ) l ( ) () ( ) l ( ) () l () [ [( ) l ( ) () ( ) l ( ) () l () (0) where are row subscrpts ad jare colum subscrpts the above two matrces Deote by D the dagoal matr composed of {( ) ( )!} ;thats ( ) ( )! ( ) ( )! 0 D [ d () [ 0 0 It s easly proved that D W[y () y () [( j ) ( ) ( ) ( ) ( ) ( ) ( ) [ [ Deoted by () A ( )[( j ) (3) usg verse matr formula of matr multplcato we have (D W[y () y ()) W [y () y () D A ( ) From (0) ad (4) we ca get A ( )[ ( ) j ( j)! ( ) j l ( j) [ l( j) () ( j)! () (4) (5) Let 0;the()chagesto A (0 ) (6) [ [ Its verse matr s A (0 ) [ l( j) (0) ( j)! (7) For the values of σ k ( ) (k ) () we specfy them by the followg case For eample takg 3 from (3) we ca get Thewehave l ( j) σ 0 ( 3 ) σ ( 3 ) σ ( 3 ) σ 3 ( 3 ) 3 (8) (0) ( ) +j ( j)!σ j () k ( k ) k+ ( k ) (9) where are row subscrpts ad jare colum subscrpts ad from (7) ad (9) we ca get A (0 ) [( ) +j σ j () k ( k ) k+ ( k ) (0) Thus accordg to the correspodg relatos of elemets betwee A (0 ) ad V ( ) ad A (0 ) ad V ( ) we ca obta (); that s V ( ) [ ( ) +j σ j () k ( k ) k+ ( k ) () where are row subscrpts ad jare colum subscrpts

4 4 Mathematcal Problems Egeerg 3 Recursve Algorthm of Vadermode Iverse Matr By the above method of fucto matr we deote V ( ) as V ( )[( j ) m ( ) ( ) ( m ) ( ) (( ) ( ) ( m ) ) () Accordg to the property of Vadermode determat we have det V ( ) [( j ) ( ) <j ( j ) <j (3) For the coveece of descrpto we troduce the followg otatos A ( j ) deotes the algebrac complemet of the elemet of V ( ) whose row subscrpt s ad colum subscrpt s j A (k) ( j ) deotes kth dervatve o of A ( j ) ad B j () C j k kj <k kj It s easly obtaed that ( k ) ( k ) j (4) A () ( j ) ( ) A ( j ) (5) B j () ( )B j () B () ( j )B j () From (5) ad (6) we ca get A ( j ) ( )! A( ) ( j ) A (j)( ) j+ C j B j () (j ) () j A ( j ) ( )j+ ( )! C jb ( ) j () j (6) (7) Calculatg kth dervatve o at both sdes of (6) we ca obta [B j () (k) ( )B (k) j () kb(k ) j () [B () (k) ( j )B (k) j () kb(k ) j () They ca lead to A ( j ) ( )j+ C j ( )! ( ) B ( ) j () A ( ) ( )+ C ( )! ( ) B ( ) () It s easy to kow that C j C det V (t) k kj j [( )B ( ) j () j [( )B ( ) () ( k ) C j ( k ) C k ( k ) det V () k ( j ) Deoted by V ( )[Ṽ (j) the (8) (9) (30) Ṽ ( j ) A (j ) det V () (3) From (9) (3) we ca get Ṽ ( j ) [( )Ṽ ( j ) Ṽ ( j ) j Ṽ ( j ) p [( )Ṽ ( j ) Ṽ ( j ) j (3) Ad set Ṽ ( j ) 0 whe 0 0 j 0 >ad j> Let 0 (3); the recurso formula of elemets of Vadermode verse matr V (0 ) (Ṽ ( j 0)) cabeobtaed For smplcty deote Ṽ ( j 0) Ṽ ( j) Because 0 all elemets do ot cota so we deote -order Vadermode matr composed of as V ( )(V ( j)) (33) [( ) + Ṽ ( j) Thus each elemet of Vadermode verse matr ca be determed by the followg recursve formula:

5 Mathematcal Problems Egeerg 5 { [ Ṽ ( j) Ṽ ( j) Ṽ ( j ) { { p [ Ṽ ( j) Ṽ ( j ) j j (34) where p k ( k )/ k ( k ) ad set Ṽ ( j) 0 whe 0 0 j 0 >adj> Whe k ( k )ad k ( k ) Whe k ( k ) 4 Applcato Case ad Smulato We wll take the 3-order Vadermode verse matr as a eample to llustrate the recursve steps of (34) detal Let From (34) we have V 3 ( 3 )( 3 ) (35) 3 V 3 ( 3 )(V 3 ( j)) 3 3 [( ) + Ṽ 3 ( j) 3 3 (36) Takg Ṽ ( ) ad p as the tal values to calculate each elemet of [Ṽ 3 ( j) 3 3 thewecaget Ṽ 3 ( ) Sce Ṽ ( 0) 0 (j 0 0)the 3 [ 3 Ṽ ( ) Ṽ ( 0) (37) Ṽ 3 ( ) 3 3 Ṽ ( ) (38) As Ṽ 3 ( ) so from (36) ths leads to V 3 ( ) ( ) + Ṽ 3 ( ) Meawhle we ca also obta V ( ) ( ) + Ṽ ( ) For Ṽ 3 ( 3)theres Ṽ 3 ( 3) 3 ( )( 3 ) (4) (43) 3 [ 3 Ṽ ( 3) Ṽ ( ) (44) As j3> Ṽ ( 3) soṽ ( 3) 0 ad from (35) we ca obta Ṽ 3 ( 3) Ṽ 3 ( ) p 3 [ Ṽ ( ) Ṽ ( ) (45) As j> Ṽ ( ) soṽ ( ) 0 ad Ṽ ( ) ad p k 0 ( k ) k ( k ) (46) From (45) ad (36) ;thewecagetv 3 ( 3);thats From (34) aga we ca get Ṽ ( ) Sce Ṽ ( 0) 0 (j 0 0)the [ Ṽ ( ) Ṽ ( 0) (39) V 3 ( 3) ( ) + Ṽ 3 ( 3) ( ) + 3 ( )( 3 ) (47) Ṽ ( ) Substtutg Ṽ ( ) to (40) we have Ṽ ( ) (40) Ṽ 3 ( ) 3 Ṽ 3 ( ) ( )( 3 ) (4) Meawhle we ca also get V ( ) ( ) + Ṽ ( ) ( ) + [ p Ṽ ( ) p (48) The other elemets of Ṽ 3 ( j) j 3cabeobtaed by a smlar process Fally we ca get

6 6 Mathematcal Problems Egeerg Table : The total tme spet o symbolc Vadermode verse matr MMA Equato () Equato (34) Table : The total tme spet o umercal Vadermode verse matr MMA Equato () Equato (34) (Ṽ 3 ( j)) 3 3 ( 3 ( )( 3 ) + 3 ( )( 3 ) 3 ( )( 3 ) + 3 ( )( 3 ) (( 3 )( 3 ) + ( 3 )( 3 ) ( )( 3 ) ( )( 3 )) ( 3 )( 3 )) (49) Meawhle we ca get So the 3-order Vadermode verse matr of (35) s (Ṽ ( j)) ( ) (50) V ( 3 )(( ) + Ṽ 3 ( j 0)) ( )( 3 ) 3 ( ( )( 3 ) + 3 ( )( 3 ) + 3 ( )( 3 ) ( )( 3 ) ( )( 3 )) (5) ( ( 3 )( 3 ) + ( 3 )( 3 ) ( 3 )( 3 ) ) ad the -order Vadermode verse matr s (V ( j)) (( ) + V ( j)) ( ) (5) We choose dfferet order Vadermode matrces to compare the computg tme spet o calculatg ther verse matrces mathematcal software Mathematca 0 We dvde the calculatos to two types: symbolc verse matr ad umercal verse matr For the lmtato of computg tme we take oly the order as 30 ad the total tme of each calculato s the tme spet by dog 00 repeated calculatos; ts ut s secod Whe we calculatetheumercalvadermodeversematrwetake ()WeobtatheresultsasTablesad I Tables ad s the order of Vadermode verse matr MMA deotes the tme whch Mathematca speds o calculato by usg ts er fucto Iverse[ to calculate the verse matr Equatos () ad (34) deote the tmes whch Mathematca speds o calculatos by () ad (34) separately

7 Mathematcal Problems Egeerg 7 Remark Here we oly take Mathematca as the eecuto evromet maly because Mathematca has the superorty of symbolc computato compared wth other types of commoly used mathematcal software For eample whe we calculate the 6-order symbolc Vadermode verse matr by the er fuctos of Mathematca Maple ad MATLAB the computg tmes are ad 03 secods separately 5 Cocluso ad Prospect FromTablewecaseethatthetotaltmespetbyMathematca o calculatg the symbolc Vadermode verse matr by usg er fucto Iverse[ s far more tha that by () ad (34) Wth the crease of dmesos the tme grows dramatcally For the 0-order matr t almost speds early 7 hours From Table we ca see that the total tme spet by Mathematca (0 Table meas that the total tme s less tha 000 secods) o calculatg the umercal Vadermode verse matr by usg er fucto Iverse[ s far less tha that by () ad (34) For umercal verse matr Mathematca s er fucto has greater advatage tha () ad (34) ad the tme spet by () s less tha that by (34) Ths s because our algorthms are ot optmzed terms of umercal calculato We do these calculatos oly by a few fuctos wrtte by us Mathematca Therefore the total computg tmes are much more But for symbolc verse matr () ad (34) are better tha Mathematca er fucto I ths paper we use the dfferet methods from [ 5 to deduce () based o Wroska fucto matr ad propose a recursve algorthm for Vadermode verse matrtheprocesshasacertauversaltyadsomevalues for calculatg verse matrces of other specal matrces FromTablesadwecaseethattwomethodsths paper are more effcet tha Mathematca er fucto for symbolc Vadermode verse matr For umercal Vadermode verse matr how to optmze ad desg the effectve algorthms to carry out our methods wll be our future research [ G C Res A Matr Formulato for the Iverse Vadermode Matr 967 [3 V-ENeagoe IversooftheVaderModematr IEEE Sgal Processg Lettersvol3o4pp [4 A Esberg ad G Fedele O the verso of the Vadermode matr Appled Mathematcs ad Computatovol74 o pp [5 XFLuZXuQLuetal Afastalgorthmoftheverseof Vadermode-type matr Proceedgs of the 7th Iteratoal Coferece o Matr Theory ad Its Applcato vol7 pp Chegdu Cha 006 [6 Z Xu Fast Algorthm of Vadermode Matr ClassNorthwester Polytechc Uversty Press X a Cha 997 [7 L Zhao Z Xu ad Q Lu A effcet ad fast algorthm for the verse of geeralzed Vadermode matr Numercal Mathematcs: A Joural of Chese Uverstesvol33o4pp [8 PRMaeSGAdgaadMSathshKumar Performace evaluato of radom lear etwork codg usg a Vadermode matr Physcal Commucato vol 0 pp [9 X Zhag Theory ad Applcato of Reproducg Kerel Scece Press Bejg Cha 00 Coflct of Iterests The authors declare that there s o coflct of terests regardg the publcato of ths paper Ackowledgmets The authors thak the ukow revewers for ther careful readg ad helpful commets Ths work s supported by Scece Research Project of Hua Provce Educato Offce (4C0650) ad Research Project of Natoal Uversty of Defese Techology (JC-0-0) Refereces [ J T Tou Determato of the verse Vadermode matr IEEE Trasactos o Automatc Cotrol vol9o3p34 964

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