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1 U.P.B. Sc. Bull., Seres C, Vol. 7, Iss., 009 ISS 5-3 SYBOLIC COPUTATIO TUIG ETHOD FOR THE EVALUATIO OF ALGORITHS I SALL SIGAL PARAETERS EXTRACTIO Crsta ZORIO, rcea BODEA, Ioa RUSU 3 Problea ateatcă asocată problee de etracţe a paraetrlor uu crcut se reduce la deterarea ulu global al ue fucţ obectv obţută pr etoda celor a c pătrate. Î cazul aalze de seal c a uu crcut (câd se utlzează u odel larucţa asocată crcutulu este o fucţe raţoală, ş î cosecţă ş fucţa obectv are aceeaş foră. Aceasta perte a se lua î cosderare rezolvarea ssteulu de ecuaţ forat cu dervatele parţale ale fucţe obectv petru deterarea, î fal, a ulu global, etoda care, spre deosebre de etoda pur uercă, u a ecestă valor de start ale paraetrlor de etras ş î plus garatează faptul ca rezultatul obţut corespude ulu global. Această abordare coduce la problea ateatcă a rezolvăr uu sste de ecuaţ forat cu fucţ raţoale, care poate f trasforat îtr-u sste poloal echvalet. Î lucrare se deteră odul î care uărul de valor ăsurate ale ue fucţ de seal c (asocată uu crcut lar, care se au î cosderare, flueţează gradul acestu sste ţal de ecuaţ poloale. Se arată faptul că tpul de calcul total, (care depde de gradul ssteulu ţal de ecuaţ poloale ş, de aseeea, de algortul de reducere a ssteulu poloal la u sste echvalet quas tragular, rezolvabl pr etode uerce poate f cotrolat pr ajustarea acestu uăr. Utlzâd această propretate/depedeţă, care perte geerarea de problee ateatce (sstee de ecuaţ poloale ţale de copletăţ dferte, petru aceeaş probleă de etracţe, se aalzează î cazul uu crcut partcular, posbltatea obţer uu rezultat îtr-u tp rezoabl, cu algort cluş î două sstee CAD petru ateatcă. Cocluzle detfcă oportutatea utlzăr fecărua dtre aceste struete ateatce, petru pleetarea uu progra de etracţe u eapărat bazat pe u sste CAD. The ath proble assocated to the proble of paraeter etracto of a crcut ca be reduced to the proble of fdg the global u of a error fucto obtaed wth the least squares ethod. I the case of the sall sgal aalyss of a crcut (whe a lear odel s cosdered, the crcut s assocated fucto s a ratoal fucto ad as a cosequece the error fucto s of the sae for too. Ths akes possble to take to accout solvg the resultg Eg., Dept. of IT & Coputers, Roaa atoal Televso Socety, Bucharest, Roaa, crsta.zoro@gal.co Prof., Dept. of Devces, Crcuts ad Electroc Apparatus, Uversty POLITEHICA of Bucharest, Roaa, rceabodea@yahoo.co 3 Prof., Dept. of Electroc Techology ad Relablty, Uversty POLITEHICA of Bucharest, Roaa, oelv@yahoo.co

2 6 Crsta Zoro, rcea Bodea, Ioa Rusu equato systeored wth the partal dervatves of the error fucto, order to fd, the ed, the global u of the error fucto. The ethod, ulke ay pure uerc ethod, o loger requres "start values for the paraeters beg etracted ad also guaratees that the fal result correspods to the global u. Ths approach leads to the atheatcal proble of fdg the solutos of a equato syste fored wth ratoal fuctos, whch ca be trasfored a equvalet polyoal syste. The paper hghlghts the depedecy of the degree of ths tal polyoal equato syste, wth the uber of easured values of a lear crcut sall sgal fucto whch s cosdered a partcular etracto proble. It s show that the total eecuto te (whch depeds o the degree of the tal polyoal syste ad also o the algorth for reducg the syste to a equvalet quas-tragular, uercally solvable for ca be tued by adjustg ths uber. Usg ths property/depedecy whch akes possble to geerate several atheatcal probles (tal polyoal equato systes havg dfferet copletesor the sae etracto proble, we aalyze, usg a partcular crcut, the possblty of gettg a soluto a reasoable aout of te, wth the algorths pleeted two dfferet ath-cad systes. The coclusos detfy whch of the atheatcal struets used, could be used to pleet a stadaloe progra for etracto, whch should ot ecessarly be based o a CAD syste. Keywords: sybolc aalyss, odfed odal aalyss, sall sgal aalyss, paraeter etracto, SPICE put forat. Itroducto Etracto of the paraeters of a crcut geerally leads to the atheatcal proble of fdg the global u of a error fucto whch s detered by the atheatcal odel of the crcut ad the easured values of soe crcut sgals. A classcal approach for solvg the paraeter etracto proble (fdg the put values of the error fucto, correspodg to the fucto s global u s by usg a uercal ethod for geeratg a coverget descedet sequece of error fucto values. Ths ethod s based o a algorth whch evaluates the fucto ad selects for the curret put values those that reduce the gap betwee calculated ad evaluated values, leadg to a u of the error fucto. The algorth stops whe all possble odfcatos of the paraeters represetg put values for the error fucto could ot further reduce the value of the error fucto. Ths approach s bascally a uerc refeet process ad although t always leads to soe u of the error fucto, t ca ot guaratee that the sequece of errors coverge to the global u of the fucto (ad ot to soe dfferet local u. The result of the covergece process depeds o a tal set of paraeter values (the start values represetg the terato startg pot. The approprate guess ethod for obtag these values s by

3 Sybolc coputato tug ethod for the evaluato of algorths sall sgal para. 65 usg the egeer s tuto based o practcal eperece ad/or by usg a alteratve splfed, easy to copute, odel of the crcut to estate soehow the (approate values of the paraeters to be etracted. However, ay user applyg ths ethod, could ot voke coclusve rgorous arguets whch ght prove that ay of the guessed startg pots wll lead through the process of refeet to the global u of the error fucto, ad ot to ay of ts local us. A dfferet approach, ot based o uercal pure algorths usg covergece of a sequece of error values, s a sybolc coputato ethod that could potetally fd all the us of the error fucto. I theory, ths s always possble by coputg all the partal dervatves of the error fucto ad solvg the resultg equato syste. Two al codtos are requred for ths approach: - a sybolc forula of the error fucto ust est (ot oly a procedure/algorth for uercal evaluato - the equato syste fored wth partal dervatves has to be solvable usg uercal ad/or sybolc coputato ethod. A eaple of paraeter etracto the case of a lear crcut, that use aple [] ath-cad syste to perfor sybolc calculato of all the us of a error fucto cobed wth uerc procedures for fdg real roots of polyoals of oe varable, was already preseted [].. atheatcal odel of the etracto proble The geeralzato of the followg eaple wll help us to elate the task of gvg foral deftos for all the eleets of the atheatcal odel assocated wth the paraeter etracto proble. Assue a gve crcut havg a well detered schea for whch soe of the values of the crcut eleets (dscrete devces are ukow ad have to be detered by drect easureets. Also assue that we dspose of the real crcut whch ca be subject of dfferet easureets at ts put ad output gates. 3. Eaple of a crcut ad easureets The crcut of Fg. whch has already bee subject of sybolc paraeter etracto, [], (whe the resultat ethod for varable elato was used wll be used aga, the et sectos for geeratg a eaple of polyoal equato syste to be solved wth dfferet algorths. Eaple:

4 66 Crsta Zoro, rcea Bodea, Ioa Rusu r μ r b rb cμ rc o c π r b r b 5 0 r b r b rb V r π ro V 3 c v o 0 r b 0 rb re c g v v r v 8 Fg. - Crcut Eaple The Gacoletto equvalet crcut of a bpolar trasstor Solvg the paraeter etracto proble for the crcut of Fg. whe kowg the values of a subset of ts eleets (see Table eas fdg the values of the ukow copoets, r b ad g, by coparg calculated ad easured values of soe sall sgal put/trasfer/output fuctos of the crcut. Crcut s Iput/Trasfer/Output fucto Sce the crcut s schea s well detered ay sall sgal put/trasfer/output fucto ca be coputed as a sybolcal epresso whch cotas the schea eleets. Cosdergor stace, the put pedace z z (g, r b, r π, r μ, r e, r o, r c, c π, c μ,f of the crcut (whch was coputed wth the odal ethod [5] ad usg a specalzed scrpt [3] leads to (: z v Vo 0 R( rb,rπ,rμ,re,ro,g,c z ( rb,rπ,rμ,re,ro,g,cπ,cμ S( r,r,r,r,r,g,c b π μ e o π π,c μ,c μ wth: R (r b, g -r b r o r e -r b r π r c -r b r µ r e -r b r π r o -r b r µ r o -r o r e r µ -r b r µ r c -r o r e r c -r π r o r µ -r π r e r µ - jπf Cπ r π r o r e r c -jπf Cπ r π r e r µ r c -r b r π r e -r π r e r c -g r π r o r e r c -g r π r o r e r µ -r b r π g r o r e - r b r π r c g r o +π f Cπr π r o r ecµ r µ r c +π f Cπr b r π r µ r ocµ r c -r π r o r c +π f Cµ r b r π r µcπ r o r e -r b r o r c -r π r µ r c -r e r µ r c ( +j(-πf Cπ r π r o r e r µ -πf r o r e Cµ r µ r c -πf g r π r o r ecµ r µ r c -πfr π r e Cµ r µ r c -πf r π r o Cµr µ r c -πf Cπ r b r π r µ r e -πf Cπ r b r π r o r c -πf Cπ r b r π r µ r c -πf Cπ r b r π r µ r o -πf r b r µ r o Cµr c -πf r b r π Cπ r o r e -πf Cµ r b r µ r o r e -πf Cµ r b r π r µ g r o r e -πf Cµ r b r π r µ r e - πf Cµ r b r π r µ r c -πf Cµ r b r π r µ r o -πfr b r π r c g r µ r ocµ S (r b, g -r µ r c -r µ r o -r π r e -r π r o -r e r µ -r π r c -r o r c -r o r e -g r π r o r e -r π r c g r o +π f Cπr π r µ r ocµ r c +π f Cµr π r µcπ r o r e

5 Sybolc coputato tug ethod for the evaluato of algorths sall sgal para. 67 +j(-πf r π r ocµ r µ -πf r π r c g r o Cµ r µ -πf r e r µ Cπ r π -πf Cπ r π r µ r c -πfr o r e Cµ r µ - πf Cπ r π r µ r o -πf g r π r o r e Cµ r µ -πf Cπ r π r o r c -πf Cµ r µ r o r c -πf Cπ r π r o r e -πf r π r e Cµr µ -πf r π r c Cµ r µ The values for the kow eleets are gve Table (see the Ital otato row: Table Ital otato g r b r π r μ r e r o r c c π c μ Ital otato - uerc dees /r r r 3 r r 5 r 6 r 7 c c otato usg vector eleets Values Uts Ω Ω F The kow values for r π,, r μ, r e, r o, r c, c π, c μ, ca be substtuted the sybolc forula, resultg a epresso ( whch depeds oly o the ukow values r b ad g, to be etracted ad the frequecy f. 3 wth: R( rb,rπ A( rb,rπ + jb( rb,rπ z ( rb,rπ S( r,r, C( r,r + jd( r,r b π R (r b, g A (r b, g + jb (r b, g g r b g r b f r b f j( r b f g r b f g f f S (r b, g C (r b, g + jd (r b, g g f j ( g f f R ad S are polyoals havg cople (pure real or pure agary coeffcets ad A, B, C, D, are polyoals havg real coeffcets (to be approated by ratoal uber. b π b π (. Error fucto forula Wthout loosg geeralty, usg the eaple of the partcular crcut of Fg. ad cludg also the data of Table, a etracto odel ca defed. I order to llustrate the result s coplety ad sze, the huge partcular sybolc

6 68 Crsta Zoro, rcea Bodea, Ioa Rusu forula of ths crcut s Error fucto was already coputed, [], ad wll ot be reproduced here aga. The crcut s fucto - geeral case Table also provdes a ethod of reag the crcut s eleets, whch ca be easly geeralzed to ay crcut. The eleets havg ukow values, whch ust be detered by etracto are the frst colus of the table so that, whe usg the, otato, substtutg sybolc aes wth uercal values should always replace the last k,, k>, sybols wth ther correspodg values havg the effect of reducg. The geeral, otato, wll be used etor a put/trasfer/output fucto assocated to ay crcut so that the left part of the epressos ( or ( ca be rewrtte as: ψ ψ ( g,rb,rπ,rμ,re,ro,c,c, f ψ (, (3 wth 8 whe rewrtg ( or whe rewrtg (. Whe dealg wth lear crcuts, epresso ( or ( ca be regarded as ratoal fuctos, as a partcular for of a geeral epresso (3. Also, the lear case, the uerc values replacg sybolc eleets, are ebedded the coeffcets of the resultg ratoal fuctos (see 0. The followg geeral deftos for dstace ad global error fucto do ot ecessarly restrct ψψ(,,, the fucto assocated to a lear crcut, to be a ratoal fucto. Dstace forula Sce we dspose of easureets results for the values of the crcut s assocated fucto, ad sce we ca also copute such values by evaluatg ts forula for dfferet values of the paraeters (the schea eleets wth udetered values, t ake sese to defe a dstace dd (ψ Cal, ψ, betwee easured values, ψ, ad calculated values ψ Cal (: d d ( ψ Cal, ψ {Re [ ψ (, ] Reψ } + {Iψ (, ] Iψ } ote that each dstace d d (ψ Cal, ψ depeds of the frequecy values f chose for each easureet of ψ ( the good cople value for ψ. Error fucto forula (global dstace The geeral ethod for paraeter etracto uses a error fucto that estates a global dstace (5, (6: 0 d[ (,, f ] E(, d, ψ (5 0 (

7 Sybolc coputato tug ethod for the evaluato of algorths sall sgal para. 69 E (, {[Re ψ (, Re( ψ ] + (6 + [I ψ (, I( ψ ] } The values o,..., o whch put the error fucto ts global u, represet the values whch gve a optal ft of the crcut s calculated values (usg ψ s forula wth the easured values, ad represet etracted paraeters. Ths approach leads to the atheatcal proble of fdg the global u of a fucto, havg a ore or less cople forula. The case of a lear crcut I the case of a sall sgal lear crcut the put/trasfer/output fucto ψψ(,,, whch represets the behavor of the crcut, has of the for of a ratoal fucto, [5]: R(, A(, + jb(, ψ (, S(, C(, + jd(, ψ (7 where R ad S are polyoals wth real or pure agary coeffcets, ad A, B, C, D are polyoals wth real coeffcets such as: R(, A(, + jb(, C(, + jd(, ad S(, Ths was the case of epressos ( ad ( for z z (g, r π,, r μ, r e, r o, r c, c π, c μ,f. As a cosequece the error fucto wll be a ratoal fucto wth real coeffcets, (9: E (, R (, R (, Re Re( ψ + I I( ψ (9 S (, S (, (8 whch ca be rewrtte as (0 takg to accout the otatos (8: E (, A( A( +,, C (, B ( [ C (, ] + [ D (, ], D (, r D (, B (, C (, [ C (, ] + [ D (, ] + (0

8 70 Crsta Zoro, rcea Bodea, Ioa Rusu where: r Re(ψ, I(ψ ( otce that the case of a lear crcut, a sybolc forula of the error fucto ca always be detered ad the process of calculatg a fucto value wth the forula for kow put values eas evaluato of ths forula. 5. Aalyss of teredate sybolc epressos coplety whe coputg a polyoal syste fro a syste of ratoal fuctos Ital syste of polyoal equatos A straght forward ethod for deterg a global u of the error fucto ΕE(,...,, s to choose the sallest pot of etres of Ε, after deterg all the pot of etres. We wll assue that Ε has a fte set of pots of etres, as a cosequece of physcal terpretato of the sgfcace of global Error fucto Ε. Iterpretato of the algebra theory cocerg the deso of a affe varety [7] should lead to the sae cocluso. Sce we dspose of the sybolc epresso, (9, (0, of the Error fucto ΕE(,...,, a frst step ths drecto s calculatg the sybolc epressos of ts partal dervatves, ad costructg the syste : E, ( 0 ;... ( Ths s a syste of ratoal fuctos whch ca be trasfored to a syste of polyoal equatos ad solved wth oder algebra theory [7]. Solvg polyoal equato systes usg coo algebrac ethods, s always possble f cosderg fte coputg resources ad/or a fte coputg te [7], but whe takg to accout the possblty of obtag practcal results, the coplety of the tal equato syste ad of the solvg algorths ust be cosdered too. Ths troduces the ecessty of a ore precse defto for the oto of coplety of the put data of these algorths.e. the coplety of soe gve tal polyoal equato syste whch s to be solved. The quatfable propertes of ay specfc tal polyoal equato syste, used to easure ts coplety, s,, d, h, are varables o whch the forulae used to calculate the coplety of a solvg algorths depeds o, [9]: s s the uber of equatos, s the uber of varables ( our case s, d s total degree (assocated wth a gve ooal orderg [7] ad referred et as total degree, ad h (heght represetg the uber of bts eeded to store deoator/uerator of each ratoal ubers represetg polyoal coeffcets.

9 Sybolc coputato tug ethod for the evaluato of algorths sall sgal para. 7 Sce h affects uerc precso whch at least at ths stage, ca be cosdered, a refeet feature, ad ay aalyss cocerg s or, the uber of equatos/varables, s obvous ad straghtforward, the et paragraphs wll focus o the order of agtude of the total degree of the equato syste of ratoal fuctos ( ad splfyg t s sybolc forula. Sybolc trasforatos ad fluece of the applyg order The left part of each of the equatos of syste ( ca be regarded as a sybolc forula, (5 whch has to be trasfored a su of ratoal fuctos havg each the sae deoator, whch could be the elated, so that oly the uerators of each su/equato should for a polyoal equato syste. The set of solutos of ths polyoal syste wll always clude the solutos of the tal syste (. a coputatos for trasforg sybolc forulae to equvalet forulae The a sybolc coputatos (operators used to odfy sybolc forulaeor the above etoed purpose, to equvalet epressos are: Real ad Iagary part calculus/separato Addg/subtractg easured values to/fro ratoal epresso Calculatg squares each ter (3 Partal dervatve sybolc calculus Calculatg a coo deoator for all ters ad sug The result of ay of these sybolc trasforatos s always a ratoal fucto ad each possble trasforato have as put (see (5, (6 sybolc forulae of the type of a ratoal fucto or a su of ratoal fuctos cludg cople costats (easured values. The put selected operad of each a coputato depeds o the selected order for applyg the operads (lke eeplfed et for the dervato operator secto 0. The total effect of these operatos o the degree of teredate polyoals (preseted secto 0, after selectg a order of applyg these a trasforatos too wll gve the coplety (.e. degree of the fal polyoal syste. Splfyg sybolc coputatos The sybolc forulae (ratoal fuctos represetg partal results of the a trasforato process of sybolc forulae ay be the subject

10 7 Crsta Zoro, rcea Bodea, Ioa Rusu splfyg operatos before akg further (a trasforatos by the rest of the above a trasforg operatos(3, order to keep uerators ad deoators of ratoal fuctos as sple as possble. Possble splfcato operatos are: o Elato of coo factors betwee uerator ad deoator of each ter o Factorzg uerators after dervato o Fdg a greatest coo ultple betwee all deoators for the splest coo deoator ( before sug o Choosg a set f,,f for factorzg [C (,...,, f ] + [D (,..., ] The splfcato operatos ( that ca be used after each a operato (3 are preseted secto 0. Ifluece of the order of sybolc coputatos A selecto for the order whch a operatos are appled ca be cosdered to be well defed after statg oe of the equvalet fors of the ath forula (5 (.e. lke forula (6 represetg the left ter of equatos (. The coputg power of oder ath-cad lke aple, [], or Sgular, [], akes possble to fd a fal forula of each dervatve E depedetly of the order of applyg the a operators (3. However, choosg the best order splfes sght o partal results ad ca evdece the fact that soe teredate sybolc forula of the error fucto could have spler teredate epresso. A approprate selecto of the order of applyg the a operators (3 perts a earler use of ay of the approprate splfyg operatos (, provdg saller teredate forulae represetg spler operads for the et a operator (3. As a eaple, take to accout coputg the sybolc dervates of ΕE(,...,, startg fro epresso (6 where we substtute ψψ (,...,, f wth ts ratoal epresso (7. Partal dervato ca be doe after applyg all other trasforatos (5: E (, R (, R (, Re Re( ψ + I I( ψ S (, S (, (5

11 Sybolc coputato tug ethod for the evaluato of algorths sall sgal para. 73 or, the dervato operator ca be dstrbuted the dervato operator to each ter of the su: E(, (6 R(, R(, Re Re( ψ + I I( ψ S(,, S,,... (... I both (5 ad (6 cases, the fucto to whch the dervato operator apples s a ratoal fucto ad the cosequece of dervato to degrees of the uerator ad deoator could be the sae ters of fal degree, as preseted et, secto 0, so t sees that the oet of usg does ot atter. O the other had applyg the dervato operator o each ter of the su, lke (6.e. before cosderg ultplyg the deoator ad uerator of each ter of the su (5 by soe polyoals ( order to geerate a coo deoator for each ter could lead to uerators of ters whch have a coo factor. Ths factor could be separated as a coo ultpler of each ter before dog the sybolc calculus of the su. As a cosequeceorula (6 gves a better order selecto for the dervato calculato tha (5. Effect of sybolc coputatos o total degree Degrees of the deoator ad uerator polyoals after each trasforato are always substatally bgger tha the oes of the ratoal fucto that represeted the (whole or partal put of the sybolc trasforato operator. Ths secto aalyzes the effect o the degree of teredate forulae of the a, (3 ad splfyg, (, operatos, ad also defes the selected order for the a operatos (whch ust be appled fro the sde to the outsde of relato (6. Real ad Iagary part calculus/separato Separato of the Real ad Iagary parts of the crcut fucto whch s cluded each ter, results (7: R (, A + jb ( A + jb ( C jd S (, C + jd ( C + jd ( C jd (7 A C B D A D B C j ( C ( ( ( + D C + D where A A(,...,, B B(,..., respectvely C C(,...,, D D(,..., are polyoals wth real coeffcets represetg the real

12 7 Crsta Zoro, rcea Bodea, Ioa Rusu ad agary parts of R R(,...,, the uerator respectvely S S(,..., the deoator of the crcut s fucto evaluated at f,(8: Re[ R(, ] A A(, A (,, I[ R(, ] B B(, B (,, (8 Re[ S(, ] C C(, C (,, I[ S(, ] D D(, D (, Relatos (7 ad (8 show that, before dervato, the fal degree of the two deoators of each ter the su (5 becoe the double of the au degree of the deoator s ad uerator s real ad agary part,(9: deg((c +( D a{ deg(c, deg[d } (9 For the uerator of the real ad agary parts, the ew degrees are, (0: deg(a C + B D a{ deg(a + deg(c, deg(b + deg(d } (0 ad respectvely ( for the agary part: deg(a D + B C a{ deg(a + deg(d, deg(b + deg(c } ( Addg/subtractg easured values to/fro ratoal epresso Icludg the easured values r Re(ψ ad respectvely I(ψ the ratoal epresso of each ter ( respectvely (3, does ot affect the degree of the (sae deoator of each ratoal fucto s ter represetg a real or agary part: R(, Re S(, AC Re( ψ B [ ] [ C ] + [ D ] D ( C + ( D r P ( C + ( D ( C + ( D AC B D r ( R(, I S(, A D B I( ψ ( C + ( D C [( C + ( D ] Q ( C + ( D ( C + ( D A D BC Relatos ( (3 show that the degree of the uerator each ter, correspodg to the real/agary parts, becoes the au of the uerator s ad deoator s degree of the prevous sybolc operato. ote: (3

13 Sybolc coputato tug ethod for the evaluato of algorths sall sgal para. 75 Sce the chose frequecy values f ad respectvely the easured uerc values r, ad for the cople pedace, wll be ebedded the three dfferet polyoals represetg uerators or deoators of the resultg ratoal fuctos calculated wth the relatos ( ad (3, each of these polyoals wll respectvely have the sae ooals ad dfferet coeffcets for each. ote: A portat observato whch ca be ade at ths pot ad resultg fro the relatos ( ad (3, s that the f values, wll be ebedded each, the coeffcets uerc values of the uerator ad of the deoator polyoal of each ter, whle the easured values r respectve, wll be cluded oly the calculato of the uerc coeffcets of each real ( ad respectve agary (3 uerators polyoals, each ter. o Choosg a set f,,f for factorzg [C (,..., ] + [D (,..., ] ote: I a later stage of the sybolc calculato, after aalyzg the effect of ultplyg for each the ter s uerator ad deoator wth the sae polyoal, order to obta a coo deoator for all the ters before coputg the su, the property of depedece of the uerator s coeffcets of r respectve, could be used a sybolc splfyg coputato whch could fd a spler coo deoator. Calculatg the square each ter ad sug the two uerators of oe ter Each ter T of the left part of ay equato s, (: P Q ( P + ( Q T + ( ( C ( ( ( [( ( ] + D C + D C + D The deoator s coeffcets stll does ot deped o r ad, ad t s fal deoator (5 s four tes of the au of the tal real ad agary parts of the crcut s fucto deoator: deg([(c +( D ] a{ deg([c ], deg([d ] } a[ deg(c, deg(d ] (5 The uerator s degree creases wth a factor of the au of the forer two uerators correspodg to the real ad agary part of the crcut s fucto, because, (6: deg([p ] +[ Q ] a{ deg([p ], deg([q ] } a{ deg(p, deg(q } (6 ote:

14 76 Crsta Zoro, rcea Bodea, Ioa Rusu After coputg the square of each of the epressos of the resultg ratoal fuctos ( ad (3, ad sug each ter s uerators, all the et (a or splfyg sybolc operatos appled to each ter (before the teredate sybolc epresso allows perforg the su tself wll preserve the followg property of the ters: each ter s a ratoal fuctos wth the sae ooals but dfferet uerc coeffcets depedg oly of the f s values for the deoators ad of the f s values but also of the r s ad s for the uerators. Calculatg partal dervatves Takg to accout (7 ad (8 for partal dervato: R(, Re S ( Re( ψ, [( C + ( D ] P [( C + ( D ] P + P [ C + D ] [( C + ( D ] (7 R( I, I( ψ ( Q ( C + ( D S (, [ ] [( C + ( D ] ( Q + ( Q [( C + ( D ] [( C + ( D ] the calculus of the dervatve of each ter gves, (9: R(, R(, Re Re( ψ + I S(, S(, (8 I( ψ [( C + ( D ] P + P [( C + ( D ] + [( C + ( D ] Q + Q [( C + ( D ] {[ C(, ] + [ D(, ] } F (, [ G(, ] The degree of each resultg uerators s gve by (30: deg(f (,, a{ deg([p ], deg([q ] }+ deg([(p +( Q ] }- (30 where deg([(p +( Q ] could be replaced usg relato (5. I relato (30 subtractg s the cosequece of dervato ad the addto operato the cosequece of ultplyg deoator/uerator polyoals or dervatve of (9

15 Sybolc coputato tug ethod for the evaluato of algorths sall sgal para. 77 deoator/uerator polyoals wth dervatve of uerator/deoator or respectvely uerator/deoator. The degree of each deoator creases by a factor of ad aga, the resultg deoator stll does ot deped o r ad. ote: Sce fro the partal dervato relato (9, t appears that each equato, the dervato operator does ot appear at the deoator of ay -th ter, the sybolc epressos of the deoator of the -th ter do ot dffer at two equatos. At ths pot we ca cosder splfyg the forula of each E. o Factorzg uerators after dervato If the uerators of all ters of a su have, after dervato a coo factor for the whole su, ths could lead to leavg each ter a spler uerator forula. Calculatg a coo deoator for all ters ad sug ote: As a cosequece of the fact that the deoator of ay -th ter s the sae ay two equatos, k, of the syste (, the proble of fdg a coo deoator for the ters of ay equato s uque ad the assocated sybolc calculato does ot deped o the partcular equato. I order to effectvely calculate a sybolc forula for each su, oe ust ultply the deoator ad the uerator of each curret ter wth a polyoal that s the less coo ultple, LC, of all the deoators dvded by the deoator of the ratoal fucto represetg the curret ter (3: E(, [ G(, ] F (, [ LC(,, f G(, ] [ G(, ] [ LC(,,... G(, ] F (,..., F (, [ LC (,,... G (, ] [ LC (,,... ] (3 LC G(, [ (,,... ] wth G ( F (, [ LC(,,... ] [ C (, ] + [ D(, ],

16 78 Crsta Zoro, rcea Bodea, Ioa Rusu Ths operato creases the tal degree of each uerator F F (,..., wth the degree of the polyoal LC(,,... /G(,..., (3: deg(f (,..., LC(,,... /G(,..., (3 deg(f (,..., + deg(lc(,,... - deg(g(,..., I the worst case the fal coo deoator of all ters s the product of all deoators ad the polyoal LC(,,... /G(,..., s the product of all the deoators eceptg the curret deoator. The effect o total crease of the degree the worst case s gve by relato (36 secto 0. The et two splfyg operatos are preseted oly as theoretcal aspects sce the classc solvg algorths could have upractcal coputg te the geeral case. o Fdg a greatest coo ultple betwee all deoators for the splest coo deoator before sug A algorth for calculatg the less coo ultple of two ultvarate polyoals, (whch ca be aturally geeralzed for ore the two polyoals s based o coputg a Gröber bass for the tersecto of the two (prcpal deals geerated by the two polyoals. Ths s preseted [7] chapter paragraph 3 after Theore. The use of ths category of algorths wll requre a later aalyss that wll put balace the coputg te cosued for perforg the splfcatos o oe had ad, o the other had, the possble beefts coputg te whe avodg the worst case of usg the product of all the deoators stead of ther less coo ultple, the process of elatg uerators. o Choosg a set f,,f for factorzg [C (,..., ] + [D (,..., ] Sce the deoators of the ters cota ebedded ther coeffcets oly the arbtrary ( ters of coputato f values, a terestg questo s whether soe restrctos posed o these values ( beg the uber of easureets equal to the uber of ters could have as cosequece that the sybolc epresso of the polyoals represetg the deoator of each ter could have a for whch could lead to a less coo ultple havg a aal degree (.e. tha a case of arbtrary f s. 6. Tug the coplety of the polyoal syste by apulatg the uber of easureets The left sde of the polyoal equato whch results durg the last step of the process of obtag a polyoal equato syste: the elatg the

17 Sybolc coputato tug ethod for the evaluato of algorths sall sgal para. 79 coo deoator [LC(,,,,F ] (33 : P(,,... F (, LC(,, f G(, F (, [ LC (,,... ] wth [ G(, ] G ( of relato (3, s of the for 0 [ C (, ] + [ D(, ], (33 I the worst case, the less coo ultple s the product of all deoators, (3: P(,,...,, f G(, k k LC( (3 ad each equato (33 ca be rewrtte as (35: F (, G(, k k [ G(, ] F (, G(, k k k (35 F (...,, G (, k 0 k [ G(, ] wth G ( [ C (, ] + [ D(, ], Sce the uber of ters s equal to the uber of frequeces f, used for easurg r ad (of z r + j, ad the worst case each suppleetary frequecy f + (whch creases the uber of easureets to + would add to the coo deoator a ew factor of [ G ( ] [ ] [ ] ], + C(, + + D(, +, each addtoal ter/easureet would crease the total degree of each equato (35 wth the degree of ths polyoal, (36: deg(pa {deg(f (,...,, }+(- a deg[c(,...,, + (D(,..., (36 ]} Ths observato shows that tug, the uber of easureets to be take to accout fro a gve set of easureets o a specfc crcut, fluece (crease/decrease the total degree of the equatos of the polyoal

18 80 Crsta Zoro, rcea Bodea, Ioa Rusu syste (35, whe keepg uodfed the crcut, ts atheatcal odel ad the coputg algorths for solvg the ath proble assocated to the etracto proble. The uber of easureets wll be used the et paragraph as a paraeter order to estate the power of coputg algorths whch ca solve the atheatcal proble, syste (35, assocated wth the etracto proble. Evaluato of dfferet sybolc algorth pleetatos for polyoal equato syste solvg, usg coplety tug The partcular for of the ath equato syste (35 for the etracto proble defed secto 0 showed that eve for a sple crcut, Fg., the sze of the polyoal equato syste posed the use of a ath-cad syste for solvg the ath assocated proble (see [] where the resultat elato resultat ethod was used. The aple [] ad Sgular [] ath-cad algebrac systes cota ther ow fuctos for coputg the resultat of two polyoals ad also for coputg the Gröber Bass of a deal f a set of geerators, the polyoals defg a equato syste s gve. Based o these fuctos, two ethods for varable elato were pleeted usg each of these CAD systes, ad ther perforaces sybolc coputg were evaluated for the case of the crcut of fgure Fg.. Resultat ethod Coputg the sybolc epresso of the resultat of two polyoals, [6], geerates a addtoal equato for whch oe of the varables s elated. Ths basc elato step ca be used a ore geeral algorth by applyg t to ay par of two equatos order to obta a tragular syste, []. Ths syste s the solved by usg uercal routes to fd the roots of the fal polyoals of oe varable ad replacg the step by step, a recursve way, the rest of the equatos. For a tal syste of two varables (the etracto proble descrbed secto 0 a elato step was requred [] (o varable elato strategy for two varables ad the au uber of easureets, or whch the elato could be ade a reasoable te was detered by tug the coplety of the tal polyoal syste, as show secto 0. The detered aal value for was, whe usg aple [] ad also, whe usg Sgular [].

19 Sybolc coputato tug ethod for the evaluato of algorths sall sgal para. 8 Gröber Bass ethod At least fro a theoretcal pot of vew, a alterate soluto to the varable elato part of the algorth descrbed secto 0 - Resultat ethod, s coputg a Gröber Bass wth lecographc orderg of ooals,.e. a equvalet syste havg a quas tragular for [7]. However, the case of our syste, coputg a Gröber Bass proved to be a too dffcult task for both aple s [] ad Sgular s [] trsc fuctos a reasoable uber or easured value,, were take to accout. Tug the uber of easureets, as show secto 0, order to reduce the coplety of the tal syste uder a lt that akes t solvable wth a reasoable coputatoal effort, showedro a practcal pot of vew, weaker coputatoal perforace tha resultat ethod as 3, whe usg Sgular [], ad whe usg aple []. au coputato te was lted to 3 hours both resultat ad Gröber Bass ethod ad eve f the aal value of for reas the sae whe usg the aple s [] CAD syste, the coputg perforace ( ters of coputg te was sgfcatly superor wth oe order of agtude whe usg the resultat ethod. 7. Coclusos Ths paper aalyses a sybolc ethod whch uses sybolc coputato to solve the ath proble assocated to the etracto proble, the case of a lear crcut. The ethod s based o coputg partal dervatves of the error fucto obtaed wth the least square ethod ad geerates a tal equato syste of ratoal fucto whch ca be reduced to a polyoal equato syste. Ths polyoal syste s solved wth a cobed sybolc/uerc ethod based o the resultat ethod or the Gröber bass ethod, [7]or varable elato. The process of trasforg the partal dervatves syste cosstg of ratoal fuctos a tal polyoal equato fuctos, s descrbed ad the effect o the total degree of the resultg equatos s aalyzed. A portat cocluso of ths aalyss s thator the sae etracto proble, the degree of the tal polyoal strogly depeds o the uber of easureets cosdered the etracto proble. Ths gves a ethod of tug the coplety of the assocated ath odel ador ay partcular etracto proble, t shows that t s possble to obta a upper lt for the uber of easureets take to accout, so that, below t, the varable elato algorth (cluded the ore geeral equato syste solvg ethod eds properly.

20 8 Crsta Zoro, rcea Bodea, Ioa Rusu Usg the coplety tug ethod, the etracto process for a partcular crcut s aalyzed, partcular the capablty of usg sybolc coputatos for polyoal syste varable elato. Evaluato of the coputg power of estg pleeted fuctos for calculatg the resultat, [6] ad the Gröber bass wth lecographc ooal orderg, [7], the case of two ath-cad systes, shows that, both cases, the pleetato for the resultat fucto perfors better tha the pleetato of the fucto whch calculates a Gröber bass. Ths ght be terpreted as the cosequece of the behavor of the classc Buchberger algorth [7], probably used to the pleetato of these trsc fuctos, because t has bee show, [7], [8], [0], that ths algorth ca geerate, as teredate results, huge ultvarate cople polyoals of degrees whch ca grow o a double epoetal law wth the degree of the put polyoals. R E F E R E C E S [] Waterloo aple - A atheetcal CAD Prgra, [] G.-. Greuel, G. Pfster, H. Schöea, Sgular - A Coputer Algebra Syste for Polyoal Coputatos, Cetre for Coputer Algebra, Uversty of Kaserslauter, 005 [3] C. Zoro, A BASH scrpt for covertg spce lke, schea descrpto - tet fles, to odfed odal ecuato atr wth sybolc eleets, Uversty POLITECHICA of Bucharest - SCIETIFIC BULLETI, Seres C: Electrcal Egeerg, Vol. 69, 007 o. 3, ISS 5-3, pag [] C. Zoro,.Bodea, I.Rusu, Sall Sgal Lear Crcut Paraeter Etracto ethod Usg Sybolc Coputato Proceedgs of the -th Iteratoal Coferece o Optzato of Electrcal ad Electroc Equpet - OPTI008,Vol. IV, pg. -,ISB , IEEE Catalog uber 08EX996, ISB [5] B.S. Rodask,. Hassou, Sybolc Aalyss, cap. The Crcuts ad Flters Hadbook, Secod Edto ISB [6] Erc W. Wesste, Resultat. Fro athworld-a Wolfra Web Resource CRC Press LLC, Wolfra Research, Ic. [7] D. Co, J. Lttle, D. O Shea, Coputer Algebra Systes : Ideals, Varetes ad Algorths- A Itroducto to Coputatoal Algebrac Geoetry ad Coutatve Algebra, Sprger-Verlag ew York Berl 99 ISB , SPI [8] E. ayr, A. eyer, The Coplety of The Word Proble for Coutatve Se-Groups ad Polyoal Ideals - Adv. aths. 6 (98, [9]. Gust, J. Hetz, Algorthes dsos rapdes pour la décoposto d ue varété algébrque e coposates rréductbles et équdesoelles - Effectve ethods Algebrac Geoetry (Proceedgs of EGA'90 T. ora-c.traverso eds. Progress ath. 9, Brkhäuser, 99, p69-93,avalable at [0]. Gust, Cobatoral Deso Theory of Algebrac Varetes - Coputatoal Aspects of Coutatve Algebra -specal ssue of the Joural of Sybolc Coputato, 6, Acadec Press, (988, 9-65.