A New Class of Vector Padé Approximants in the Asymptotic Numerical Method: Application in Nonlinear 2D Elasticity

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1 Word Journ of echncs, 204, 4, Pubshed Onne Februry 204 ( A New Css of Vector Pdé Approxnts n the Asyptotc Nuerc ethod: Appcton n Nonner 2D Estcty Abdeh Hdou, Rchd Hh, Bouzz Brt, Noureddne ouns, Noureddne D Lbortore d Ingénere et térux LIA, Fcuté des cences Ben, Unversté Hssn II ohed - Csbnc, Csbnc, roc E: bdeh.hdou@unvh2.c., hh_rchd@hot.co, b.brt@g.co, no_touns@ve.fr, noureddne.d@unvh2. Receved Noveber 4, 203; revsed Deceber 6, 203; ccepted Jnury 2, 204 Copyrght 204 Abdeh Hdou et. hs s n open ccess rtce dstrbuted under the Cretve Coons Attrbuton Lcense, whch perts unrestrcted use, dstrbuton, nd reproducton n ny edu, provded the orgn wor s propery cted. In ccordnce of the Cretve Coons Attrbuton Lcense Copyrghts 204 re reserved for CIRP nd the owner of the nteectu property Abdeh Hdou et. A Copyrght 204 re gurded by w nd by CIRP s gurdn. ABRAC he Asyptotc Nuerc ethod (AN) s fy of gorths for pth foowng probes, where ech step s bsed on the coputton of truncted vector seres []. he Vector Pdé pproxnts were ntroduced n the AN to prove the don of vdty of vector seres nd to reduce the nuber of steps needed to obtn the entre souton pth [,2]. In ths pper nd n the frewor of the AN, we defne nd bud new type of Vector Pdé pproxnt fro truncted vector seres by extendng the defnton of the Pdé pproxnt of scr seres wthout ny orthonorzton procedure. By ths wy, we defne new css of Vector Pdé pproxnts whch cn be used to extend the don of vdty n the AN gorths. here s connecton between ths type of Vector Pdé pproxnt nd Vector Pdé type pproxnt ntroduced n [3, 4]. We show so tht the Vector Pdé pproxnts ntroduced n the prevous wors [,2], re spec cses of ths css. Appctons n 2D nonner estcty re presented. KEYWORD Vector Pdé Approxnts; Asyptotc Nuerc ethod; Nonner Estcty. Introducton ny engneerng probes cn be reduced to sovng nonner probes dependng on contro preter λ. hese probes re wrtten n gener for: R U, λ = 0 () where { } ({ } ) n U s the unnown vector of, R s n vector functon wth vues n ssued to be suffcenty regur wth respect to ts rguents { U } nd λ. he Asyptotc Nuerc ethod (AN) [,2] s fy of gorths for pth foowng probes. he prncpe s spy to expnd the unnown ({ U}, λ ) of the nonner probe () n power seres wth respect to pth preter : N { V }( ) = { V }, 0, 0 = x (2) where { V }( ) { U } { U 0 } { U} = λ, { V }( ) { U }, = λ = s nown nd regur souton cor λ λ0 respondng to = 0 nd N s the truncted order of the seres. he nterv of vdty 0, x s deduced fro the coputton of the truncted vector seres (2). o, the step engths re coputed posteror by the foowng estton of x, whch hve been proposed n []: x { U} { U } = ε N N where ε s gven toernce preter nd. nd- (3) WJ

2 A. HADAOUI E AL. 45 ctes stndrd nor. he step engths depend on the defnton of the pth preter nd we ust dd n uxry equton to defne ths preter []. By usng the evuton of the seres t = x, we obtn new strtng pont nd defne, n ths wy, the AN contnuton procedure. hs contnuton ethod hs been proved to be n effcent ethod to copute the souton of nonner prt dfferent equtons [,2]. he Vector Pdé pproxnts were ntroduced n the AN to prove the don of vdty x of vector seres (poyno) representton [2]. In order to extend the don of vdty of the representton (2) nd to reduce the nuber of steps needed to obtn the entre souton pth, n [2], rton pproxton, ced Pdé pproxnt [5-8], hs been used. In [2], the representton (2) hs been rewrtten n n orthonor bss but up fro the bss ( ) U generted by the AN nd strtegy to use Vector Pdé pproxnts hs been pped. hs hs been used n vrous feds [,9]. But ths strtegy hd the dsdvntge to generte gret nuber of poes nsde the don of vdty. An terntve, presented n [], s to use Vector Pdé pproxnts wth coon denontor, ced sutneous Pdé pproxnts [7,8]. he orthonorzton cn be done ccordng to the procedure of Gr-chdt or odfed Gr-chdt or tertve Gr-chdt [], or s t w be presented n ths pper for the frst te by usng the Househoder ethod. ny ppctons n structur echncs (for nstnce nonner estcty nd contct), [,2] hve estbshed tht Vector Pdé pproxnts wth coon denontor cn reduce the nuber of poes nd pert to obtn ore regur soutons. By usng ths rton representton n contnuton procedure, the nuber of steps to obtn the entre souton pth hs been reduced [0]. he Vector Pdé pproxnts hve so been consdered to cceerte the convergence of hgh order tertve gorths for ner or nonner [] probes. he of ths pper s to dscuss soe technques to defne new Vector Pdé pproxnts n the frewor of the AN nd to show tht ther utzton cn prove cery the cssc Vector Pdé representton. In the second prt, we propose new type of Vector Pdé pproxnt whch cn be drecty defned fro the vector seres () by extendng the defnton of the Pdé pproxnt of scr seres [5,8] nd wthout ny orthonorzton procedure. By ths wy, we show tht fy of Vector Pdé pproxnts s possbe. here s connecton between ths type of Vector Pdé pproxnt nd Vector Pdé type ntroduced n [3,4]. We show so tht the Vector Pdé pproxnt ntroduced n the prevous wors [,2] re spec cses of ths css. A the pproxnts re pped on soe expes fro nonner two-denson estcty whch re presented nd nyzed n the thrd prt. Aong ths fy of Vector Pdé pproxnt, we show on nuerc expes, tht there re soe pproxnts whch ncrese the rnge of vdty 0, x nd thus reduce the nuber of steps necessry for the ccuton of soutons. o ustrte ths, three nuerc tests n two-denson nonner estcty re consdered: trcton of n estc pte, bendng of n estc pte nd bendng of n estc rch. hese structures re dscretzed by the conventon fnte eeent ethod usng C eeent []. 2. Defnton nd Constructon of New ype of Vector Pdé Approxnt In ths ecton, we w gve the defnton nd the constructon of new type of vector Pdé pproxnts. 2.. Defnton A Vector Pdé pproxnt of vector functon n+ n+ { V}( ) fro to s Vector frcton whose yor expnson t gven order, concdes wth the vector functons one. ore precsey, the Vector Pdé pproxnt { V[ L, ]}( ) s the vector rton frcton of the for: ( ) ( { }) L { V[ L, ]}( ) = 0 [ B] A, = = 0 (4) [ B0] = [ In+ ] where [ B ] trces re of denson ( n+ ) ( n+ ) n+ A vectors re n. hs vector rton { }( ) dts the se yor expnson thn the vector functon { V}( ) up to order L+ ( L, re ntegers). he of ths pper s to defne Vector Pdé pproxnt { V[ L, ]}( ) foowng the se des s n the scr cse: A + B V L, = 0 (5) nd { } frcton V[ L, ] { }( ) [ ]( ){ [ ]}( ) where { A}( ) nos of degree L nd [ B]( ) s vector whose coponents re poy s trx whose eeents re poynos of degree, L, whch re s { A }( L ) = { A }, [ B ]( ) 0 = 0 [ B = = ] (6) he vector poyno { A}( ), nd the trx poyno [ B]( ) re derved fro the vector functon { V}( ) fro the condton: L { A}( ) [ B]( ){ V}( ) ( + + ) + =Ο (7) In Appendx, we show tht the trces [ ] B, n (6) re soutons of the foowng ner syste: WJ

3 46 A. HADAOUI E AL. L+ = = [ ]{ L+ } { L+ } [ ]{ } { } B V = V, L B V = V, L, + + nd the L vectors { } A, 0 L, gven n (6) re derved fro the foowng retonshps: { } { } [ ]{ } = 0 (8) A = C = B V (9) he syste (8) cn be wrtten n the foowng trx for (see Appendx ) V B = C (0) It y be noted tht f the ters of the seres n Equtons (6) re scr, we fnd excty the syste defnng the scr Pdé pproxnt n [8]. Note so tht n the cse where the trx [ B]( ) s of the for P( )[ I n + ], where P( ) s poyno of degree I + s the unt trx, we fnd the defnton of nd [ n ] Vector Pdé type pproxnt { }( ) ( ) A P ntroduced n [3,4]. he new defnton so ows fndng Vector Pdé pproxnts csscy used n the AN gorth []. Rec tht csscy n the AN gorth, the Pdé pproxnt ssocted wth the vector seres s constructed by repcng ech scr seres coponents by scr Pdé pproxnt, or by repcng the scr poynos, whch pper fter orthonorzton of the vector bss, by scr Pdé pproxnts wth the se denontor []. hese cses w be found n the foowng prgrph Constructon of oe Vector Pdé Approxnts he constructon of the new type of Vector Pdé pproxnts requres the souton of the trx syste (0) verfed by the trces [ B ]. hs syste ows, n gener, n nfnte nuber of soutons. Indeed, fy of soutons of (0) cn be obtned n the foowng gener for: = ( ) [ n+ ] ( ) B V V V C + I V V V V W where W s ny rectngur trx hvng ( n + ) rows nd ( n+ ) couns. We esy chec tht the product of the trx V by the foru of the trx souton B gves the trx C. Note tht n ths fy of soutons, the trx V V s nvertbe becuse the row vectors of V re ssued nery ndependent. Note tht n the scr cse, the trx V n the syste (0) s squre trx. herefore, the syste (0) hs unque souton f the trx V s nvertbe. Accordng to the defnton of the Vector Pdé pproxnt (4), the constructon of ths new type of Vector Pdé pproxnt usuy requres hgh coputton cost due to the fct tht for ech vue of, we need to ccute the nverse of the trx [ B]( ) defnng the Vector Pdé pproxnt n (4). However, there re stutons n whch we cn expcty ccute the nverse of the trx [ B]( ) for vues of. For expe, f we oo for trces, [ B ], n dgon for, [ B] = dg ( b, b2,, b n + ), where b, b2,, b n + re re, then we fnd tht the coponents of the Vector Pdé pproxnt { V[ L, ]}( ) re gven by the foowng foru (see Appendx 2 ): ( bv ) L = 0 = 0 b 0 =, () It corresponds to the Vector Pdé pproxnt tht woud be but fro the scr Pdé pproxnt correspondng to ech coponent of the vector { V } s t ws ponted n AN frewor []. Another choce of the for of the trces [ B ],, bsed nty on the orthonorston of vectors { V }, (see Appendx 2), eds to the foowng Vector Pdé pproxnts: { V[ L, ]}( ) { } ( ) ( ) = V + V L L = 0 = { } L+ (2) where the poyno n (2) depends on the coeffcents b, whch re ccuted fro the orthonor- V, : zton procedures of the vectors { } ( ),0 = + b+ + b In Appendx 2, we show tht the scrs b, b2,, b re rbtrry nd so we cn use the new expresson of the Vector Pdé pproxnt (2) gvng the coeffcents b ny vues. Note tht for L = 0, the Vector Pdé pproxnt reduces to: { [, ]}( ) { } ( ) ( ) { } (3) V L V = V = We thus fnd the Vector Pdé pproxnt (3) ntroduced n the wor of the AN gorths [,2] where the coeffcents b, b 2,, b re deterned fro Gr-chdt orthonorston of the vectors { V},{ V2},,{ V } of the seres (2). herefore, we constructed new fy of Vector Pdé pproxnts gven by Equton (2) or (3) wth- WJ

4 A. HADAOUI E AL. 47 out ny condton on the coeffcents b, b2,, b. In the foowng nuerc ppctons, we deonstrte tht there re choces for these coeffcents for whch the rnge of vdty s rger thn n the cses conventony used Contnuton Procedure he representtons (2) or (3) pert to copute ony prt of the souton pth of the nonner probe (). o obtn the entre souton pth, Cochen [] proposed contnuton procedure for the vector seres representton (2) bsed on the crteron (3) whch gves n evuton of the don of vdty of the poyno representton. Once the deternton of the don of vdty s done, by the coputton of the rdus of vdty x for fxed toernce εε, the vector seres representton (2) cn be pped n contnuton procedure to obtn the entre souton pth step by step. o ntroduce the vector Pdé representton n contnuton gorth, Ehge et. [0] proposed nother crteron defned by: P P ( ) ( ) P ( x ) V V x 2 x V V = ε P whch gves n evuton of the rdus of vdty x of the rton representton for fxed toernce ε pd, by usng dchotoy process. We sh use the se crteron to ntroduce the proposed Vector Pdé representtons (3) n contnuton process. 3. Nuerc Appctons he nuerc robustness of the pproxte soutons obtned by the vector seres representton (2) nd by the new fy of Vector Pdé representton (2) s dscussed on the bss of tests entng fro pne stress two-denson nonner estcty nyss. he studed structure s dscretzed usng cssc C fnte eeent [] nd s subected to odng proporton to contro preter λ. We see the souton of ths probe by representng the contro preter λ s functon of dspceent. he quty of AN steps s evuted fro od-defecton curves nd resdu defecton curves nd the n crteron s the step engths. ore precsey, we pot the od-dspceent curves wth three AN steps. hree ccutons re crred out: the frst ccuton w be de usng AN contnuton wth seres representton (2), the second ccuton w be de usng AN contnuton wth cssc Vector Pdé representton (2), the coeffcents b re derved fro n orthonorzton procedure, here by the ethod of Househoder, pd the thrd ccuton w be de usng AN contnuton wth the proposed Vector Pdé representton (2) but ths te the coeffcents b re rbtrry. he perfornce of the three ccutons re copred n ters of the step engths of the three AN contnutons, the quty of the soutons s gven by the resdu curves. 3.. Bendng of Pte he frst nuerc expe concerns the bendng of pte; see Fgure, the pte hs ength of 00 nd wdth of 0. he pte s cped on the eft sde nd subected to bendng force proporton to contro preter λ on the other end. ter chrcterstcs re: Young s oduus E = 0,000 P, Posson s rto υ = 0.3. he pte ws dscretzed usng 4 nodes ong the ength nd sx nodes ong the wdth; tot of 400 eeents nd 492 degrees of freedo s used. In Fgure, we represent the response curves obtned by the AN contnuton gvng the odng preter λ s functon of the dspceent u t the node 246. We potted the od-dspceent curves usng three AN steps for the three ccutons nd for three choces of the truncton order: 0, 5 nd 20. In Fgure 2, for the three ccutons, we represent the resdu curve gvng the ogrth of the nor of the resdu og RUλ (, ) s functon of dspceent t node 246. he frst ccuton, AN contnuton wth seres representton (2), t orders 0, 5 nd 20, shows tht the step ength ncreses wth the truncton order. hree steps t order 20, ow obtnng the curve unt dspceent equ to 62 wth ccurcy of the order of 0 5. hs resut s cssc n the wors of AN gorth [], the ncrese of the order ncreses the step ength. he second ccuton, AN contnuton wth cssc Pdé representton, t orders 0, 5 nd 20, shows Fgure. Bendng of pte, od-dspceent curve, λ versus dspceent t node 246 for the three types of ccuton for truncton orders 0, 5 nd 20. WJ

5 48 A. HADAOUI E AL. Fgure 2. Bendng of pte, resdu curve, og RUλ (, ) s functon of the dspceent u t node 246 for the three types of ccuton for orders 0, 5 nd 20. tht the step engths re greter thn the frst ccuton usng seres representton. hree steps wth AN Pdé representton t order 0 ows obtnng the curve unt dspceent equ to 73 wth good quty s cn be seen on the resdu curve of Fgure 2. For ths second ccuton, the resuts obtned by usng the Househoder orthonorston ethod re copred wth those obtned by usng Gr-chdt orthonorzton n Fgure 3. In our nuerc experents, the Househoder orthonorzton ethod sees ore effectve thn Gr-chdt orthonorztons procedure. We w use n the foowng, the ethod of Househoder. he thrd ccuton, AN contnuton wth the proposed Vector Pdé representton (2) the coeffcents beng rbtrry, s perfored by usng the orders 0, 5 nd 20. We crred out the ccutons by sghty odfyng the vues of the coeffcents b, b 2,, b n +, ccuted by the ethod of Househoder for ech order. A the vues of the coeffcents were ncresed by vue equ to 0.. hs frst test ws very successfu. Indeed, t shows tht n rbtrry choce of the coeffcents b, b 2,, b n + cn gve good resuts s cn be seen n the response curve n Fgure nd the resdu curve n Fgure 2. hree steps wth the proposed AN Pdé representton t order 0 ows obtnng the curve unt dspceent equ to 78 wth good quty s cn be seen on the resdu curve of Fgure Bendng of n Estc Arch For the second nuerc experent, we chose the expe of the bendng of n estc rch, see Fgure 4, of rdus R = 2540, wdth of 5 nd n nge of 0: rd. he rch s cped t both ends nd s subected to bendng force proporton to contro preter λ t ts dde. ter chrcterstcs re: Young s oduus E = 0,000 P, Posson s rto υ = 0.3. he rch Fgure 3. Bendng of pte, od-dspceent curve, λ s functon of the dspceent u t node 246. Coprson of AN contnuton usng Vector Pdé representton for four processes of orthonorstons: Gr-chdt, odfed Gr-chdt, tertve Gr-chdt nd Househoder t order 29. Fgure 4. Bendng of n estc rch, od-dspceent curve, λ versus the dspceent u t node 52 for the three types of ccuton for truncton orders 0, 5 nd 20. ws dscretzed usng 4 nodes ong the rdus nd fve nodes ong the wdth; tot of 320 eeents nd 40 degrees of freedo s used. We represent n Fgure 4, the response curve obtned by AN gorths gvng the odng preter λ s functon of dspceent u t node 05. Resdu curves re gven n Fgure 5. WJ

6 A. HADAOUI E AL. 49 We represent n Fgure 6, the response curves obtned usng three ethods of AN contnuton t orders 0, 5 nd 20 used n the prevous cses. he resdu curves re gven n Fgure 7. hs test confrs the resuts obtned n the frst two nuerc tests. In prtcur, ths expe so shows tht n rbtrry choce of the coeffcents b, b 2,, b n + cn gve very good resuts s cn be seen on the response curve, Fgure 6, nd the resdu curve, Fgure 7. Fgure 5. Bendng of n estc rch, resdu curve, og RUλ (, ) s functon of the dspceent u for the three types of ccuton for orders 0, 5 nd 20. he resuts obtned by respectvey the frst ccuton, AN contnuton by usng seres representton (2) t orders 0, 5 nd 20, the second ccuton, AN contnuton by usng cssc vector Pdé pproxton t orders 0, 5 nd 20, the coeffcents b, b2,, b n + re derved fro the Househoder orthonorston ethod, nd the thrd ccuton, AN contnuton usng the proposed Vector Pdé representton (2) t orders 0, 5 nd 20, the vues of the coeffcents b, b 2,, b n + were ncresed by vue equ to 0. re reported on Fgures 4 nd 5. hree steps wth the proposed AN Pdé representton t order 20 ows obtnng the curve unt dspceent equ to 72 wth good quty s cn be seen on the resdu curve of Fgure 5. Whe three steps wth the cssc AN Pdé representton t order 20 ows obtnng the curve unt dspceent equ to 0 nd the three steps wth the AN wth the seres representton t order 20 ows obtnng the curve unt dspceent equ to 8, see Fgure 5. hs second nuerc test confrs tht n rbtrry choce of the coeffcents b, b 2,, b n + cn gve very good resuts s we cn see fro the response curve n Fgure 4 nd the resdu curve of Fgure rcton of n Estc Pte For the thrd nuerc experent, we consder the trcton of n estc pte; see Fgure 6, the pte hs ength of 00 nd wdth of 25. he pte s cped on the eft sde nd subected to tense force, on the other end, proporton to contro preter λ. ter chrcterstcs re: Young s oduus E = 0,000 P, Posson s rto υ = 0.3. he pte ws dscretzed usng 4 nodes ong the ength nd sx nodes ong the wdth; tot of 400 eeents nd 492 degrees of freedo s used. 4. Concuson In ths wor, we ntroduced new wy to bud drecty new type of Vector Pdé pproxnts fro truncted vector seres n the frewor of the syptotc nuerc ethod. We hve shown tht the vector Pdé pproxnts ntroduced n references [,2], re spec cse of ths css. he proposed Vector Pdé pproxnts cn be deterned wthout ny orthonorston procedure whch sves the te coputton for probes wth rge nuber of degrees of freedo. In Fgure 6. rcton of n estc pte, od-dspceent curve, λ versus dspceent u t node 246 for the three types of ccuton for truncton orders 0, 5 nd 20. Fgure 7. rcton of n estc pte, resdu curve, s functon of the dspceent u t node og RUλ (, ) 246 for the three types of ccuton for orders 0, 5 nd 20. WJ

7 50 A. HADAOUI E AL. fct, the orthonorzton procedure s te consung becuse of the very rge nuber of scr products to be evuted. It rens to expore dfferent choces of ths new css of Vector Pdé pproxnts. REFERENCE [] B. Cochen, N. D nd. Poter-Ferry, éthode Asyptotque Nuerque, Heres cence, Prs, [2] B. Cochen, N. D nd. Poter-Ferry, Asyptotc Nuerc ethod nd Pdé Approxnts for Nonner Estc tructures, Internton Journ for Nuerc ethods n Engneerng, Vo. 37, 994, pp [3] J. Vn Iseghe, Vector Pdé Approxnts, n Nuerc thetcs nd Appctons, North Hond, Asterd, 985, pp [4] C. Brezns, Coprsons between Vector nd trx Pdé Approxnts, Journ of Nonner thetc Physcs, Vo. 0, upp. 2, 2003, pp [5] H. Pdé, ur Représentton Approchée D une Foncton pr des Frctons Rtonnees, Annes de Ecoe Nore upéreur, Vo. 9, 892, pp [6]. Vn-Dye, Coputed-Extended eres, Annu Revew n Fud echncs, Vo. 6, 984, pp [7] C. Brezns nd V. Iseghe, Pdé Approxnts, In: P.G. Cret nd J. L. Lons, Eds., Hndboo of Nuerc Anyss, Vo. 3, North-Hond, Asterd, 994. [8] G. A. Bcer Jr. nd P. Grves orrs, Pdé Approxnts, Encycoped of thetcs nd Its Appcton, Vo. 2, Cbrdge Unversty Press, Cbrdge, 996. [9] H. De Boer nd F. Vn Keuen, Pdé Approxnts Apped to Non-Lner Fnte Eeent outon trtegy, Counctons n Nuerc ethods n Engneerng, Vo. 3, No. 7, 997, pp ::AID-CN04>3.0.CO;2-V [0] A. E Hge-Hussen,. Poter-Ferry nd N. D, A Nuerc Contnuton ethod Bsed on Pdé Approxnts, Internton Journ of ods nd tructures, Vo. 37, No , 2000, pp [] J. L. Btoz nd G. Dhtt, odéston des tructures pr Eéent Fns, Edton Herès, Prs, Vo., 990. WJ

8 A. HADAOUI E AL. 5 Appendx : Equtons tsfed by the trces [ B ] nd Vectors { A } o deterne the equtons stsfed by the trces [ B ], 0, nd vectors { A } 0 L we strt fro the truncted seres of order L+ of the V vector functon { }( ) { V }( L L ) = + { V } + ( + + ) Ο = 0 Inectng (6) nd (4) nto (7) yeds: (4) ( ) L+ ( [ B ]) { } 0 V = = { } ( ) L L A Ο = 0 = + whch cn be wrtten s: where L { } = { } + ( ) L + L C A Ο + + = 0 = 0 (5) (6) { } [ ]{ } C = B f 0 r V r (7) r = { } [ ]{ } C = B V f r = 0 r r > (8) By dentfyng the ters correspondng to coeffcents L+ L+ 2 L+,,, (6) we get the frst L equtons verfed by the ters [BB ] : [ B0]{ VL+ } + [ B]{ VL} + [ B2]{ VL } + + [ BL+ ]{ V0} = 0 [ B0]{ VL+ 2} + [ B]{ VL+ } + [ B2]{ VL} + + [ BL+ 2]{ V0} = 0 B V + B V + B V + + B V = 0 [ ]{ } [ ]{ } [ ]{ } [ ]{ } nd tht the st L equtons re [ B0]{ V+ } [ B]{ V} [ B2]{ V } [ B ]{ V} [ B ]{ V } [ B ]{ V } [ B ]{ V } [ B ]{ V } [ B0]{ V+ L} + [ B]{ V+ L } + [ B2]{ V+ L2} + + [ B ]{ V } = 0 (9) = = 0 L (20) As [ B0] = [ I n + ], the trces [ B ], shoud verfy, s n the scr cse, the foowng syste of equtons: [ B]{ VL} + [ B2]{ VL } + + [ BL+ ]{ V0} = { VL+ } [ B ]{ V } + [ B ]{ V } + + [ B ]{ V } = { V } L+ 2 L L+ 2 0 L+ 2 [ B]{ V } + [ B2]{ V2} + + [ BL]{ VL} + + [ B ]{ V0} = { V} [ B]{ V} + [ B2]{ V } + + [ BL ]{ VL} + + [ B ]{ V} = { V+ } [ B ]{ V } + [ B ]{ V } + + [ B ]{ V } = { V } whch re wrtten n trx for where: [ B ]{ V } + [ B ]{ V } + + [ B ]{ V } = { V } + L 2 + L 2 L L+ V B = C (22) VL VL V0 0 0 VL+ VL V V0 0 0 V = V V V 2 0 V V V V V V + L + L2 L VL+ [ B ] V L+ 2 [ B2 ] B = nd C = V V + [ B ] VL+ 2) (23) (24) WJ

9 52 A. HADAOUI E AL. Appendx 2: Constructon of oe Vector Pdé Approxnt A2.: A Frst Vector Pdé Approxnt Used n the AN Agorth We w oo n ths prt to prtcur soutons [ B ],, of the syste (0) or (22) n the for of dgon trces: [ B] dg ( b, b2,, b n + ) = (25) where B trces. If, for ny, 0 L+, we denote by v, v 2,, v n + the coponents of the vector { V }, then B V re b re the dgon coponents of [ ] the coponents of the vector [ ]{ } vb, vb 2 2,, vn+ bn+. herefore f we repce n the syste (8) or (6), ech vector [ B]{ V },, L+ L+, by ts coponents, we deduce, for ech, n+, tht, 2 b b,, b stsfy syste of the se for. If the syste hs souton, then by (9), the A, A, s gven by: coponent of the vector { } bv = 0 B = 0 b = A = (26) As the trx [ ] s dgon nd ts dgon eeents re gven by, n+, we concude fro (4) tht the coponent 0 of the Vector Pdé pproxnt { V[ L, ]}( ) s gven by the foru (). A2.2: A econd Vector Pdé Approxnt Used n the AN Wth the of budng Vector Pdé pproxnt such tht ts coponents re rton frctons wth the se denontor, we denote by { Y } vector of n+ of the for: { Y } { V * * L} σ { V L+ } = + (27) = where σ, re rbtrry scrs gven n nd { V * } { * } { * L, VL+,, VL+ } re vectors but fro n orthonorzton procedure of the vectors { V0},{ V},{ V2},,{ V L + } such s Y V = nd Y V = 0,0 L. (28) { L} { } Usng ths vector { } n the for [ ] { } Y (27), we oo for the trces [ B ], B = X Y where the vectors { X},, re deterned n order to stsfy the syste (8). By repcng, n the syste (8), the trx [ B ] by { X} Y nd ettng β = Y { V} we obtn by usng (28), the foowng equtons: β { } { }, L + X = VL + (29) = hese Equton (29) show tht the vectors { } re gven by: { X} = { VL+ } { } { L+ } βl+ { } X = V X,2 = Usng Equton (9), we deduce tht { } { },0 It s obvous tht f the trx X, (30) A = V L (3) [ ] = n+ + { } { } where { } { } = 0 = 0 B I X Y = I + X Y X = X n+ = 0 s nvertbe, then ts nverse s of the for I + n+ x{ X} Y, (32) where x s re nuber. If ths s the cse, xx stsfes ( { } )( { } ) I + x X Y I + X Y = I (33) n+ n+ n+ Whch s equvent to { } { } ( { }){ } I + X Y + x X Y + x Y X X Y = I (34) n+ n+ As resut, x = = + Y { X} + Y { X} By choosng { X0} { VL} b = Y{ X},0 ( ) = 0 = nd posng nd = + b+ + b,0 equty (4) s wrtten V L, { }[ ]( ) ( ) ( ) { } L = I { } n+ X Y V 0 = = { V} X Y Y V X (35) (36) ( ) { } L ( { }){ } L = 0 = 0 L L { V} ( ) { }. 0 X = = o express the Vector Pdé pproxnt { V}[ L, ]( ) s functon of the vectors V, V,, V, we see tht f we set L L+ L+ b b2 b b b 2 Ψ = b [ ] (37) WJ

10 A. HADAOUI E AL. 53 nd [ ] βl+ βl+ 2 βl+ βl β + L+ 2 Φ = βl+ then, tng nto ccount the equty (29), we obtn (38) [ Ψ][ Φ] = [ I ] (39) Hence [ Φ] = [ Ψ]. As Equton (29) s equvent to { VL+ },{ VL+ 2},, { VL+ } (40) = { X },{ X },, { X } [ Φ] t s concuded tht: 2 2 X = X X2 X { } { },{ },, { } { },{ },, { } [ Ψ] 2 = VL+ VL+ 2 VL+ = ( ){ V } L+. ( ) 2 = { VL+ },{ VL+ 2},, { VL+ } 2( ) = (4) Usng ths Equton (4) n the expresson of the Vector Pdé pproxnt { V}[ L, ]( ) (37), we fnd the Equton (2) whch generzes the foru of Vector Pdé pproxnt. A2.3: Generzton of (2) We show n ths secton tht n the defnton of the Vector Pdé pproxnt (37), the scr b, b2, + b cn be chosen rbtrry n. ore precsey, for b, b2, + b rbtrry but fxed, there re re nubers σ, σ2, + σ such tht the vector { Y } defned n (27) stsfes the equtes (36). Indeed, for =, we hve * { } { } b = Y X = V V σ L L+ σ = VL VL+ b, so tht equty (36) s stsfed. In gener, for ny,2, we hve: * herefore, t suffces to te { } b Y X V V = { } = L + σ L+ = { VL+ } ( Y { VL+ } ){ X} = L L L L = { + } σ + { + } = V V V V VL { VL+ } σ VL+ { VL+ } b = = L L L L = { + } σ σ + { + } = V V V V VL { VL+ } σ VL+ { VL+ } b = = Hence, f one chooses σ = VL VL VL VL = { + } σ + { + } VL { VL+ } σ VL+ { VL+ } b b = = then we hve the equty (2) for. Note so tht f L = 0, we fnd the Equton (3). WJ