Efficient and Fast Simulation of RF Circuits and Systems via Spectral Method

Transcription

1 Efficien and Fas Simulain f RF Circuis and Sysems via Specral Mehd 1. Prjec Summary The prpsed research will resul in a new specral algrihm, preliminary simular based n he new algrihm will be subsanially mre efficien in simulain f RF circuis and sysems. Wrk dae indicaes ha he algrihm is inherenly parallel and hus will be suiable fr implemenain in parallel cmpuing sysems. Such analyses are very impran in imprving prduciviy f design f RF pwer amplifiers hrugh deerminain f sabiliy regins. Sabiliy is a significan prblem in design. Currenly design f RF pwer amplifier requires apprximaely 2 weeks, bu is sabilizain requires 6 mnhs because i invlves high degree bench experimens. The wrk will be based n he specral algrihm develped earlier by he auhrs f his prpsal. The cmpuer implemenain f he algrihm will be made available be esed by praciiners. The fllwing asks are prpsed: 1. Fr bh he specral algrihm and harmnic balance mehds errr analysis will be perfrmed deermine he degree and he number f erms be used in apprximaing he sluin wihin a given errr lerance. 2. The errr analysis will be accmpanied by a sabiliy analysis in rder deermine he effec f errr during he enire cmpuain prcess. 3. A new and very efficien linear equain sysem slver will be develped ha uilizes he special marix srucures being characerisic fr specral algrihms. The slver will use prblem riened precndiiners. 4. Ieraive algrihms will be cmbined wih cefficien exraplain reduce he size f he linear equains be slved. 5. The algrihms will be implemened using an bjec riened apprach, cded in C++. Visual C++ develpmen sysem will be uilized in cnsrucing he sfware srucure. The cncep f dynamic link libraries will be uilized in building he simular assure is flexibiliy, cnvenience in use, and adapabiliy applicains. The mdule srucure f he sfware will allw is inegrain in simulars currenly used in indusry. 6. The simular will be geared ward suppr fr design f cmmunicain equipmens, in paricular i will include faciliies fr evaluain f sabiliy regins f pwer amplifiers. The sfware sysem be develped will be mdular and i is anicipaed ha he individual mdules will be made available he users as sn as hey are develped. The mdular srucure will make mdules available her applicains if needed. I is anicipaed ha sme mdules will be general in naure and will be applicable in slving her dynamic prblems. 1

2 2. Table f Cnens 3. Prjec Descripin 3.1. Inrducin 3.2. Gals f Prpsed Research 3.3. Prjec Research Resuls frm prir wrk Specral and harmnic balance algrihms Circui simulain sudies Simulain f ransmissin lines Cmpuer implemenain 3.4. Relaed wrk elsewhere 3.5. Errr cnrl 3.6. Refinemen f algrihms 3.7. Cmpuer implemenain 4. Plan f wrk and respnsibiliies 5. Appendix - Errr analysis in SPEC 6. References 7. Budge 2

3 3. Prjec Descripin 3.1 Inrducin Curren circui simulars need imprvemen in he speed f simulain, beer errr cnrl, and ls allwing cnrl f muliple simulain runs fr pimizain, sensiiviy analysis, and generain f design curves such as sabiliy regins f RF pwer amplifiers. Previus research has yielded an algrihm based n Chebyshev plynmials which is faser and cnains beer errr cnrl han exising circui simulars. This algrihm, which we call he specral algrihm, has been successfully applied sysems cnaining ransmissin lines and ransisr circuis. The specral algrihm has many advanages ver he mehds used in currenly available circui simulars. In ime dmain simulains, nly lcal runcain errr (LTE) is cnrlled. LTE is defined under he assumpin ha here was n errr in previus ierains. This assumpin is invalid as cmpuain prceeds in a sequence f perains. Cnrl f LTE des n direcly limi he glbal errr and simulain resuls may n be accurae. The specral algrihm cnrls he glbal errr hus guaraneeing sluin accuracy. Anher advanage is he use f analyical ransfrmains which lead simple linear equains fr numerical sluins. These ransfrmains speed-up he execuin f simulain. The specral algrihm has als a grea penial fr parallel implemenain which is f increasing imprance. Harmnic balance echniques presen a useful alernaive fr specral mehds, when Chebyshev plynmials are replaced by he rignmeric funcin sysem and herefre insead f Chebyshev series, harmnic series are used apprximae he sluin. I is well knwn ha Chebyshev plynmials give he bes minimax funcin apprximain. Trignmeric series apprximain als have advanageus cnvergence prperies as well as nice and simple srucure. A sysemaic cmparisn f he w echniques will be very impran fr cnsrucin f an algrihm expliing apprpriaely he advanages f bh mehds. 3.2 Gals f Prpsed Research The verall gals f he prpsed research are refinemen f he specral and harmnic balance algrihms develped previusly and heir implemenain in he frm f a prype simular. This simular will be available fr simulain f inegraed circuis and in paricular applied cmpnens f wireless cmmunicains sysems such as pwer amplifiers fr analg and digial mdulain in ransmiers. Experience dae indicaes ha a simular based n specral echniques will require subsanially less CPU imes han currenly available simulars, will be suiable fr simulain f high-speed digial circuis and inercnnecins, and will be superir in applicain analg, and mixed analg-digial circuis which represen he ype f circuis used in prable cmmunicains equipmen. 3

4 The fllwing asks are prpsed: Errr Cnrl: Errr cnrl echniques will be develped fr scillary/ analg sysems in rder bain sluins wih given lerances and wih minimal cmpuainal effr. This will allw aumaic selecin f inegrain parameers such as he number f windws and he degree f plynmial r rignmeric series used represen circui variables. Sabiliy Analysis: The effec f rund-ff errrs and errr cummulain will be examined. A repeaed sabiliy analysis n differen mehd varians makes us able find he ms apprpriae mehds fr applicains. Refinemen f Algrihms: A new and very efficien slver f linear sysems fr he specral and harmnic balance simulars will be develped. Specific marix srucures which were discvered be characerisic fr hese algrihms [21] and which are impran fr building ieraive slvers will be explred. The mehd will use a specific precndiiner avid insabiliy. Cefficien exraplain and ieraive algrihms will be used reduce he final size f he linear sysems. These algrihms will be impran in an efficien implemenain f errr cnrl. Cmpuer Implemenain: The algrihms will be implemened using an bjec riened apprach, cded in he C++ prgramming language. They will be assembled in a specral simular which will frm a basis fr fuure cmmercial develpmens and applicains. The special mdular srucure f he newly develped sfware makes i able be inegraed in simulars currenly being used by he indusry. Simular Tesing: The simular will be applied in develping suppr fr design f cmmunicains equipmen. The fllwing specific applicains are prpsed: - pwer amplifiers - use f muli-level SPEC prgram families fr generain f design curves. - ransmiers sysem/subsysems wih digial mdulain - evaluain f feasibiliy f simulain. 3.3 Prpsed Research Resuls frm Previus Wrks f he Auhrs Specral and Harmnic Balance Algrihms 4

5 Research dae resuled in develpmen f he specral algrihm fr simulain f elecrnic circuis and sysems [24, 28, 29]. The algrihm differs subsanially frm hse used in currenly available simulars in he way ha circui equains are linearized, sluins are represened, and apprximain errrs are calculaed. Apprximain echniques based n harmnic series are knwn as harmnic balance mehds. A cmprehensive summary f he echnical issues cncerning his mehdlgy is given, fr example, in [37]. The circui linearizain is firs perfrmed alng a rajecry (Newn-Kanrvich apprach) yielding a sysem f linear ime varying differenial equains. The sluin is deermined in an ieraive prcess wih he secnd rder apprximain. Circui variables such as ndal vlages, branch currens, capacir charges, and inducr fluxes can be represened in he frm f a Chebyshev series, r as an harmnic series n v n k kr r r= = ψ T ( x) vk = ( φkr cs( rτx) + `φ kr sin( rτx)) r= 0 where v k represens he k-h variable, T ( r ) is he Chebyshev plynmial f degree ψ kr are he unknwn cefficiens, and n is he rder f expansin. In he harmnic series case, τ is a given perid, ψ kr and ψ kr are he unknwn cefficiens. The circui equains are ransfrmed in algebraic equains fr he series cefficiens which are cnsruced uilizing numerus prperies f he Chebyshev plynmials [32,30] and rignmeric funcins [37]. The exensive use f Chebyshev plynmial and rignmeric funcin prperies in develpmen f equains fr expansin cefficiens yields very simple, well srucured final equains fr cmpuer sluins and is ne f he reasns fr superir numerical efficiency f he mehd. The use f Chebyshev expansin is suggesed by is cnvenience as well as by is gd apprximaing prperies. If fec 1 [-1,1] and C n is is nh rder Chebyshev expansin, hen 4 U n (f) 11f - C n 11 U n (f) π byn, were U n(f) is he supremum nrm f he difference f f and is bes nh degree plynmial apprximain. Therefre Chebyshev series apprximains are n much wrse han he bes unifrm plynmial apprximains f he same degree. 5

6 Circui simulars ha are currenly in use can cnrl lcal runcain errrs (LTE) nly. Since LTE can accumulae, he glbal errr in such simulars can n be direcly cnrlled. In cnras, he specral echniques prvide a means fr cnrlling he glbal errr in sluins. This is very impran in all applicains Circui Simulain Sudies The specral mehd implemened in he frm f a preliminary simular, SPEC, was applied in he simulain f several MOS digial circuis. Simulain resuls were carefully cmpared wih hse prduced by SPICE. The inegrain parameers in he specral mehd and in SPICE were adjused such ha bh mehds prduced equivalen resuls. In he prcedure fr adjusmen f inegrain parameers, he differences beween he circui variables cmpued using SPEC and he crrespnding variables bained frm SPICE were mnired. The simulain resuls were cnsidered equivalen when all differences were wihin 0.1% f he maximum value f he crrespnding variables. The CPU imes needed fr generain f equivalen resuls n he same cmpuer were recrded. A sample f resuls is given in Table 1. Circui Spice CPU Time SPEC CPU Time 4 h rder linear circui 33.46s 0.67s CMOS flip-flp 398s 135s ne inverer 1.31s 0.5s shif regiser 135s 17.8s memry cell 32.31s 14.3s 32 inverers s 351.1s Table 1: Cmparisn f simulain using SPICE and SPEC The specral mehds were als applied in simple examples f RF circuis, and in he wavefrm relaxain framewrk fr circui [6, 7, 23] and pariined sysems [27] prducing even mre specacular savings in cmparisn wih SPICE as shwn fr he example adders in Figure 1. We expec similar efficiency in cmparisn digial simulars such as RELAX, SWEC, SLS, ec Simulain f Transmissin Lines Simulain f ransmissin lines geher wih circuis cnaining acive devices is required in analysis and design f high speed circuis, elecrnic packages [25, 26, 20], and cmmunicains equipmen [9]. Specral mehds prvide an excellen fundain develp efficien algrihms fr cmpuain f ransiens in nn-linear circuis inercnneced by ransmissin lines. Algrihms fr lssless ransmissin lines and lines wih DC lsses were develped [18, 19]. Preliminary wrk n an algrihm fr lines wih frequency dependen lsses was cnduced [5]. Cmpuainal experimens cnfirmed he 6

7 numerical supeririy f specral algrihms and cnvenience f heir implemenain in he SPEC package. Similar advanages are expeced in applying harmnic balance echniques Cmpuer Implemenain The SPEC circui simular currenly cnains he fllwing cmpleed mdules: an inelligen inpu file parser, a Chebyshev Transfrmain mdule, a mdule ha slves linear sysems resuling frm Chebyshev Transfrmains, a mdule cnaining implemenain f primiive circui elemens, and SPEC Pwer Tls. The inpu file parser allws fr easy inpu f new mdels wihu mdifying he inpu parser cde. Each elemen is enered in he inpu file by specifying is ype, cnnecin ndes, and pinal parameers. The inpu file sparser knws wha each enry f he elemen lis is by is psiin in he lis. I maches hese enries wih an exernally supplied lis ha defines he new mdel. Figure 1: CPU ime cmparisns fr SPICE and SPEC simulaing adders in he relaxain framewrk [6, 7] The Chebyshev Transfrmains and relaed algrihms have been implemened. Tw f he mre impran and ms fen used subruines cnsis f expansin ransfrmain f a funcin in Chebyshev series and evaluain f he values f an expanded (ransfrmed) funcin. The Chebyshev applied differenial equains lead large-sparse linear sysems f equains. The linear slver mdule akes advanage f he sparsiy f hese marices prvide a quicker sluin [17, 35]. In rder cnrl rund-ff errr cummulain, spliing and precndiiners will be applied n he sysem befre cmpuing he sluins. 7

8 The mdule cnaining implemenain f circui elemens cnsiss f nly simple elemens. These currenly include: resisrs, inducrs, capacirs, vlage surces, and curren surces. Simulain f a circui fen requires muliple runs wih he sweeping (r mre sphisicaed changes) f key parameers in he circui descripin. 3.4 Relaed Wrk Elsewhere The prblem f acceleraing simulain f inegraed circuis is f significan, pracical and hereical imprance and hus araced aenin f many researchers. Three her majr appraches, differen han he ne which is he subjec f his prpsal, can be disinguished: wavefrm relaxain (WR), piecewise linear apprximain (PWLA), and asympic wavefrm evaluain (AWE). Wavefrm relaxain is an elegan idea f exending ieraive echniques differenial equains [15]. Prpsed inegrain mehds were based n ime marching echniques and cnvergence prblems were encunered because f ime discreizain. Currenly, here appears be less aciviy relaed WR. Specral echniques will prvide a beer basis wihu ime discreizain fr furher imprvemen in WR [22]. PWLA is based n a series f linear apprximains. In applicain larger circuis wih many nn-lineariies, PLA becmes inefficien because many segmens fr linearizain are required accmplish a sluin wih reasnable errr. In addiin, errr esimaes are n adequae as hey d n accun fr errr prpagain frm segmen segmen. One idea in imprving his echnique is he applicain f piece-wise quadraic r cubic plynmials. Cubic splines are used in [38] apprximae circui characerisics, and a simulain sudy is presened find he pimal number f windws fr a given errr lerance. Recenly, AWE is aracing he aenin f several scieniss and here is nable publicain aciviy assciaed wih his apprach. AWE is based n a Pade apprximain he ransfer funcin f a linear circui. The apprximain uilizes nly he ms dminan zers and ples. The ransien analysis is equivalen evaluaing he apprximaed ransfer funcin. This apprach can dramaically speed up ransien analysis f a linear circui a he expense f accuracy [33, 13]. An ineresing leas squares prcedure fr deermining bes Pade apprximains is given in [39]. The main prblem ha AWE suffers frm is lack f gd linearizain. The nly scheme wrked u fr his mehd s far is piecewise linearizain. Hwever, he apprximain errr is difficul esimae. Besides, fr circuis wih many nnlineariies, piecewise linearizain (which is a linearizain a an peraing pin) will n be efficien as i will require many segmens and hus many repeiins f he AWE prcedure. Linearizain alng he rajecry (which is used in specral echniques) is n uilized in AWE. In addiin he research effr in AWE, as bserved in he publicains, fcuses n simulain f digial circuis, which are easier linearize wih he piecewise scheme [1, 11, 37]. Bu, 8

9 even in hese applicains a l f heurisics are used bain reasnable Pade apprximans. There are als cases when such an apprximain is n desirable a all. Sme srngly nnlinear scillary circuis which wrk wih large signals (especially RF circuis, fr example pwer amplifiers) may exhibi very cmplicaed ( chaic ) behavir cnrlled by he nnlineariies. An apprximain wih very dim errr esimaes may unpredicably effec he mdel behavir. Thus he applicain f AWE is raher limied circuis which are linear r easy linearize. Fr example, AWE culd be applied creae macrmdels f large, linear subnewrks reducing he cmplexiy f he enire sysem. Circui exracrs prduce subnewrks f passive, linear elemens which add smeimes hundreds if n husands f ndes. Macrmdels creaed by AWE culd be prcessed geher wih nnlinear cre by a general purpse (ime marching r specral) simular. 3.5 Errr Cnrl Exising simulars based n ime marching echniques d n have direc cnrl f glbal errr. The glbal errr resuls frm he LTE in each inegrain sep. A ypical simulain invlves husands f inegrain seps and LTE may accumulae. Specral echniques are able prvide very efficien glbal errr cnrl and allw fr selecin f inegrain parameers (degree f series, number f windws) which minimize he cmpuainal effr necessary fr simulain. Mre deailed discussin f errr cnrl and sme preliminary resuls are given in Appendix A. The errr and cmpuainal effr esimaes ha will be develped fr he specral algrihm will frm a basis fr aumaic adjusmen f inegrain parameers befre and during he simulain. In rder develp an efficien errr cnrl sysem we have cmbine he esimaes f errrs resuling frm differen surces. Firs, a nnlinear differenial equain is linearized. The linearizain errr is a quadraic funcin f he incremen f he sluin, where he cefficiens depend n he derivaives f he righ hand side funcins. During he Newn- Kanrvich ierain scheme we need beer and beer accuracy in he cnsecuive seps, where he linearizain errr becmes smaller. Therefre he number f erms in he Chebyshev and harmnic expressins has be increased sep by sep. Mre erms lead mre accurae sluin f ha sep, bu his is achieved wih a higher cmpuainal effr. Hwever mre accurae sluins f each ierain sep imply faser cnvergence wih smaller number f ierain seps bain a given accuracy f he final sluin. Therefre in minimizing he verall cmpuainal cs we have find a balance beween he accuracy f each sep and he rder f he Chebyshev r harmnic series. This errr analysis als has be cnained wih he errr resuling frm he ieraive sluin f he large-scale sysem f linear algebraic equains find he unknwn Chebyshev cefficiens. Beer accuracy f each needs mre erms wih higher cs f each ierain, hwever if he number f ierains decreases hen here is a decrease in he verall cmpuain cs as well. Higher rder apprximain als increases he size f he linear 9

10 equain sysem be slved numerically and as a resul he sluin becmes mre expensive. Higher sysem dimensins als resul in larger accumulaed errr. The errr and sabiliy analysis f he mehdlgy has cnsider hese errrs and relaed rade-ffs in a simulaed sysem and mus find he bes cmpuainal sraegy bain a required accuracy f he final sluin wih a minimum expense. In he prpsed research we will develp w errr cnrl sraegies. The firs ne give an pimal (r clse pimal) sraegy befre cmpuains acually sar. The secnd mehd will allw make adapable adjusmens. 3.6 Refinemen f Algrihms a) Develpmen f an ieraive linear slver: The use f dedicaed linear slver is ne f he bes ways imprve he perfrmance f a simular based n specral echniques because sluin f linear sysems is repeaed many imes during a simulain. The marix srucure ha is characerisic fr he specral mehd is very sparse and banded as shwn in Figure 2 and hus is suiable fr cnsrucin f an pimized slver. This srucure suggess he cnsrucin f a sparse marix slver wih a special permuain algrihm, cllapse he bands psiins arund he main diagnal, where he elemens magniude decrease wih he disance frm he main diagnal as fllws: 1 a ij c i j where a ij are elemens f he sysem marix and c > 1 [21]. Bunds fr he elemens [21] will be imprved and an ieraive algrihm will be cnsruced fr furher imprvemen f he slver. In rder reduce he rund-ff errr cummulain, muli-spliing ([40]), and/r an apprpriae precndiiner ([41],[42]) will be applied. b) Cefficien exraplain: The relain beween he zer and he exrema expansin cefficiens [Appendix A] will be used develp ieraive cefficien exraplain algrihms. These algrihms will have he feaure ha fr a given se f exrema cefficiens f rder n, i will be pssible exraplae he expansin f rder 2n frm he already expanded rder n wihu recalculaing he full new expansin f rder 2n. The exraplain will be based n he cmbinain f he curren exrema expansin and a new zer expansin f rder n. This algrihm will be he basis fr develping an efficien errr cnrlled expansin fr general funcins. 10

11 . 3.7 Cmpuer Implemenain Figure 2: Typical marix srucure in specral analysis The algrihms are implemened using an bjec riened apprach, cded wih he C++ prgramming language and are assembled in a specral simular wih an inerface cmpaible wih SPICE. The simular will have exensins (such as he mdel cmpiler, Furier and Lyapunv analysis mdules) which will make i mre cnvenien use. The sfware will be buil using Visual C++ develpmen sysem and he archiecure f he enire package will be based n newes sfware echnlgy used in indusry including he Micrsf s Cmpnen Objec Mdel. The cncep f dynamic link libraries and her sfware develpmen ls available frm Micrsf will als be uilized in he wrk. 4. Plan f Wrk and Respnsibiliies The verall develpmen f simular and in paricular user and mdel inerfaces will be supervised by Dr. O. Palusinski, ECE. Develpmen f numerical algrihms fr ieraive slver, errrs esimaes, and harmnic balance mehd will be supervised by Dr. F. Szidarvszky, SIE, wh als will have he respnsibiliy in he cnsrucin f mdule fr cmpuing Lyapunv expnens and Pincare maps. Year 1 (a) Develping he harmnic balance mdel (b) Imprving he specral mdel (c) Errr cnrl and sabiliy analysis -develping a general scheme fr errr cnrl and sabiliy analysis -refinemen f errr esimaes fr scillary sysems -develping esimaes fr cmpuainal effr in simulain f RF circuis Year 2 Refinemen f Algrihms (a) Develpmen f an ieraive slver wih an apprpriae spliing and precndiiner 11

12 (b) Cefficien exraplain mehds (c) Cnsrucin f cmpiler fr mdels f circui elemens given in analyical frm Year 3 (a) Cmparisn f specral and harmnic balance echniques wih digial simulars (b) Cmpuer implemenain -Develpmen f mdule fr elemen mdels given in he frm f lk-up ables -Finalizing he harmnic balance and specral mdules -Develpmen f mdule fr cmpuing Lyapunv expnens and Pincare maps 12

13 Appendix A Errr Cnrl As i has been saed in he main bdy f he prpsal, he firs kind f errr is he resul f linearizain. Le F be a wice Freche differeniable mapping frm a Banach space in iself, hen F( a) L ( a) sup F"( c). a a, a 2 (1) Where L a is he linear Taylr plynmial f F arund a and F" denes he secnd Freche derivaive f F. in ur case F denes he nnlinear righ-hand-sides f he differenial equains be slved, herefre F" can be easily bscured in a clsed frm, and is nrm can be esimaed. The difference beween he sluins f he nnlinear and linear differenial equains can be esimaed as fllws. Le x& = h(, x, x( ) = x z& = A( ) z + f ( ), z( ) = x be a pair f nnlinear and linear differenial equains, where A( ) is marix, x, h, z, are all vecrs. Dene ε(, x) = h(, x) [ A( ) x + f ( )], and nice ha he nrm f his vecr is esimaed in (1). Therefre he nnlinear equain can be rewrien as x& = A( ) x + f ( ) + ε (, x). The sluins f he nnlinear and linear equains can herefre be wrien as x( ) = Φ(, ) x + φ(, )[ f ( T) + ε( τ x( ))] dτ and z( ) = φ(, ) x + φ(, ) f ( ) dτ 1 where φ(, ) is he fundamenal marix f he linear equain (see fr example Szidarvszky and Bahill, 1997). Simple subracin slws pl 13

14 x( ) z( ) = φ (, ) ε ( τ, x( τ )) d τ herefre x( ) z( ) φ (, τ ). ε (, x( )) d τ esimaing he errr f linearizain.. The secnd kind f errr is he resul f esimaing he sluin wih a finie Chebyshev expansin. Le f dene he exac sluin and C n is nh rder Chebyshev series apprximain. The fllwing resuls are knwn frm he lieraure (see fr example, Judd, 1998): Fac 1. Assume ha fε ec[ 11, ], hen here exiss a finie cnsan B such ha fr all n2, f C n B lgn n 2 The resul shws hw fas C n cnverges f as n in he supremum nrm. In he weighed leas squares nrm an analgus saemen is he fllwing: Fac 2. Where C r is he cefficien f he nh rder Chebyshev plynmial T r is he infinie Chebyshev expansin f funcin f. In rder see hw fas he righ hand side cnverges zer, we need esimaes f he Chebyshev cefficiens c r. Such a resul is given nex. Fac 3. Assume ha hen here is a finie number C such ha fr all Cmbining Facs 2 and 3 we see ha shwing ha he cnvergence is very fas fr sufficienly smh funcins. The hird kind f errr is he resul f he numerical sluin f he large-scale linear equain sysem ge he unknwn cefficiens. In he ierain prcess he errr depends n he number I f ierain seps: The paricular frm f he errr funcin depends n he special ierain scheme be develped in he framewrk f he prpsed research. The 14

15 errr funcin will be an upper bund fr he larges errr is he Chebyshev cefficiens. The errr in he sluin is hen bunded by In each linearizain sep, he hree kinds f errrs add up give verall errr bund fr he apprximaing sluin. The required errr lerance in he cnsecuive linearizain seps shuld be decreasing. If... are he required errr lerances fr he cnsecuive linearizain seps, hen we have a cmplicaed nnlinear pimizain prblem: minimizing he enire cmpuain cs subjec he cnsrain ha he sum f he hree errr bunds is n greaer han in he nh sep, k=1,2,... In his pimizain prblem he bjecive funcin as well as he cnsrains depend n he seleced windw size, number f Chebyshev erms in each windw, as well as he number f ierain seps in slving he linear equains. In he prpsed research we plan develp he paricular frms fr each cs iem, and as gd pssible errr erms. We als plan develp a mehdlgy find an pimal (r clse pimal) cmpuainal sraegy befre cmpuains begin as well as an adapive adjusmen scheme fr aumaic errr cnrl during he cmpuain prcess. References Judd, K.L. (1999), Numerical Mehds in Ecnmics, MIT Press, Cambridge, MA. Szidarvsky, F. and T.A. Bahill (1997), Linear Sysems Thery, CRC Press, Bca Ran, FL (2 nd Ediin). 15