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A matrix Euclidean algorithm for minimal partial realization

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A matrix Euclidean algorithm for minimal partial realization
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  A matrixEuclideanalgorithmforminimalpartialrealization MarcVanBarelAdhemarBultheel KatholiekeUniversiteitLeuvenDepartmentofComputerScienceCelestijnenlaan200AB-3030Leuven(Heverlee)BelgiumRUNNINGTITLE:AmatrixEuclideanalgorithmSubmittedasasynopsisofatalkheldattheInternationalConferenceonLinearAlgebraandApplicationsSeptember28-30,1987UniversidadPolitecnicadeValenciaValencia,Spain.  -1- AmatrixEuclideanalgorithmforminimalpartialrealization. MarcVanBarelandAdhemarBultheel Abstract GiventheMarkovparametersof M(z) = T(z)-lQ(z), with p(z) E KPXm[z] and Q(z) E KmXm[z] polynomialmatricesoversomefield K, thealgorithmweshalldescribewillcomputerightminimalpartialrealizationsoftheseparameters.Itcanalsobeusedtocomputeagreatestcommonleftdivisorof T(z) and Q(z).   Introduction Inreport [10] welookedatanextensionoftheBerlekamp-Masseyalgorithm.TheBerlekamp-Masseyalgorithm[3,4,9]solvestheminimalpartialrealizationproblemforscalarsequencesofMarkovparameters.ThematrixcaseissolvedbyanalgorithmdescribedbyDickinson,MorfandKailath[5]andAnderson,BraschandLopresti [1]. OurversionisalgorithmAofsection12inreport[10].In[2]Antoulasalsogivesanalgorithmtosolvethematrixcasesuchthatthesolutionisintheformofamatrixcontinuedfraction.HecallshisalgorithmamatrixalgorithmofEuclideswhileitisinfactamatrixBerlekamp-Masseyalgorithm.In [12] werelatedouralgorithmAtotheoneofAntoulas.ThealgorithmofEuc1idesiswellknownforfindingthegreatestcommondivisoroftwo(scalar)polynomials.Butthealgorithmhasotherusefulapplicationssuchasthesolutionofthescalarminimalpartialrealizationproblem.ThesolutionisgivenintheformofacontinuedfractionbutincontrastwiththematrixcontinuedfractionofAntoulaseachconvergenthasanotherstructurei.e.thenextconvergenthasmorestatesthanthepreviousoneorthedegreeofthedenominatorpolynomialoftheconvergentincreases.TocomputethatpartofthecontinuedfractionthatdeterminesthenextconvergentthealgorithmofEuc1idesusesmorethanoneMarkovparameterincontrastwiththealgorithmofAntoulaswhichusesjustonematrixMarkovparameterateachstep.Whatwewanttogiveinthenextsectionisanalgorithmthatconstructsamatrixcontinuedfractionsuchthatthenextconvergentofthecontinuedfractionhasanotherstructurethanthepreviousonethusmaintainingasmuchdetailaspossible. 2.AmatrixEuclideanalgorithm Throughoutthistextweshallfreelyusethedefinitionsandnotationsof[10].Asin[11]weshalltaketheshiftparameter k = -1, whichcorrespondstotheclassicalminimalpartialrealizationproblem.Werecallthefollowing:Giventhe p XmMarkovparameters Ml,Mz,M3' ... E KP Xm (K an   arbitraryfield),considerpolynomialvectors a(z) = r, aizi, a ,~ 0whichsolve i=O M1Mz ... ~ +11 r 0 .. laoal0 = I, a ~ n .. Mn+l\ I a: , rl ... n+zrz rl E KP possiblynonzeroiscalledtheresidualand rl.rZ' ... iscalledtheresidualsequence. n isthelevelandallsolutionsofthissystemaredenotedas S(n.a). Introducetheoperators R and RS by Ra = r 1 and RSa = [rfr~ ... ]T. Theresidualseriesisdenotedby RSa(z) = r,:lriz-n+oc-i. Thenotionofdegreewillbeextendedtoseriesin z-l tomeanthelargestexponentof z withanonzerocoefficient.Thusthedegreeof RSa(z) aboveis -n+a-l if rl~ O. Weshallusethenotation dRSa todenotethedegreeof RSa(z) andweshallalsorefer  (2) -2- toitasthedegreeofthecorrespondingresidualsequence.Wecallitresidualdegreeforshort.Solutionvectors ai(z) ••.•• a~(z) oflevel n aresolutionsfoundonlevel n oflowestpossibledegreeandwithlinearindependenthighestdegreecoefficients.Auxiliaryvectorsaresolutionvectorsoflevel < n whoseresidualsspanthespace {R(S(l.a)].l = 1.2..... n-1; a = 0.1.2•...• n ofresiduals.Anauxiliaryvectororasolutionvectorissaidtohavepotentialdegree 71 (w.r.t.level n) ifitisasolutionvectorofdegree 71 -j forlevel n-j forsome j.j=O.1....• 71 . Ifthesolutionvectorsoflevel n areassembledasthecolumnsofa m X m matrix A(z). then A(z) istherightdenominatorofaminimalpartialrealizationoforder n forthegivenMarkovsequence.BecausewewanttoconstructamatrixcontinuedfractionalaAntoulas.thatisassimpleaspossiblebutwhoseconvergentssolveminimalpartialrealizationproblemsofincreasingorders.weconstructsolutionvectorssuchthattheycanbeusedatasmuchfurtherlevelsaspossible.Tofindthesesolutionvectors.weusethefactthatweknowallvectorsof S(n.a). Itwasshownin[12]thatabasisforthemisgivenbythesolutionvectorsandauxiliaryvectors(possiblyshifted)ofthatlevel.Moreprecisely a(z) E Sen. a) hastheform mOlds min(n.OI). a(z) = LL 1Il,dZ-«Iai(z) + LL f.Li,)ZJ~IXi(Z) (1) l=ld=OIIi=l )=7T1 where ai(z) •.•.• a~(z) arethesolutionvectorsofdegrees a1 ~ az ~...~ am and Xj(z). i = 1.2..... s aretheauxiliaryvectorsofpotentialdegrees 71 l.71 Z ..• 71 s forlevel n. Usingformula(1).wewanttotransformthesolutionvectors a7. makingthemsolutionvectorsoflevelsasdeepaspossible.Le.withleastpossibleresidualdegree.Indoingsowecanuse a7 notonlyatlevel n butpossiblyalsoatdeeperlevels.E.g.startingfromabasisforS(O.O)consistingof m linearlyindependentconstantvectors a~•...•a~ E Km• instepD.1.1.ofthealgorithmDbelow.weonlyneedtoconsidercombinationsoftheform i-1 ,0~0 j = ai + L,  Yi,kak k=l Moregenerally.instepD.2.4.1..wearegiven m solutionvectors ai+1.az+z•...•a~+1 ofdegrees a1 ~ az ~.•.~ aT; Byconsideringcombinationsoftheform i-lOll s OIl a . +1 +~~ 11zd-«lan+1 +~~ f.Lz)~A:x L,L, l,dl L,L, k.)kl=ld=OIIk=l)=7TA: where Xk aretheauxiliaryvectorsofpotentialdegree 71 k. weshallfind m solutionvectorswithlinearindependenthighestdegreecoefficientsandwithresidualdegreesassmallaspossiblewithoutincreasingtheirdegrees.AlgorithmDwearegoingtodescribenowisanadaptationofalgorithmAgiveninsection12of[10]oritsreformulationgiveninsection2.3of[11].Thelatterrelatesittoamatrixcontinuedfractionasin[2].Moredetailsandsomeexamplescanbefoundin[13].Somecommentsonthenotationused:Forsuccessivelevels n = 0.1.2.....thesolutionvectorsarecomputed(D.2.1.)andthen.usingformula(1).theirresidualdegreesarem( .deassmallaspossible(D.2.4.).Eachtimeseveralmatricesareupdated:Thenumeratormatrices (NM). thedenominatormatrices (DM) andtheresidualmatrices (RSM). Atalltime M(z)DM(z)NM(z) = RSM(z). Alsoinformationaboutdegreesandpotentialdegrees(vector dM) isupdated.Eachmatrixhastwoparts:TheA-partwhichreferstotheauxiliaryvectorsandtheS-partwhichreferstothesolutionvectors.Alltransformationsofthesematricescanbedescribedintermsofelementaryoperationsappliedtothecolumns.Le.multiplicationsfromtherightwithelementarymatrices E whichhavethefollowingmeaning(see[11]) Ej(a) = unitmatrixwithcolumn i multipliedby a Ei) = unitmatrixwithcolumn i andcolumn j interchanged    Eij(a) = unitmatrixwithcolumn j replacedbycolumn j plus a timescolumn i. Allthesetransformationsarecollectedinunimodulartransformationmatrices Vi (D.2.5.)where i referstothelevel.Theycontainalltheinformationtoconstructamatrixcontinuedfractionwhoseconvergentsarethesuccessiveminimalpartialrealizations. AlgorithmD D.l{Initialisation} NM = [NMA I NMS] = [Ip I 0]E KPx(p+m)[z] (numeratormatrix) DM = [DMA I DMS] = [0 11m] E Kmx(p+m)[z] (denominatormatrix) RSM = [RSMA I RSMS] = [-Ip I M(z)] E KPx(p+m)[[z-l]] (residualmatrix) dM = [dMAIdMS] = [1...1 I o·.. 0]E K1x(p+m) (degreematrix)D.1.1 For i = 2.3•....mD.1.1.1Searchfor a = a? + 1:::~  i.la? with dRSa(z) aslowaspossible o] aia.DMS D.1.1.2Replace I 0 by[]In[] RSaiRSaRSMS D.2 For n =0.1.2....Supposeweknow NM = [Yl Yp I ci c~] DM = [x 1 xp I ai a~]RSM = [RSx 1•.. RSxp I RSai ... RSa~] dM = [7T•••7T I a .,. a]   1 m D.2.1 For i = 1.2.....mD.2.1.1Take Ra7 from RSMS (thecoefficientof z -n+C:Xj-l) D.2.1.2 If Ra7 = 0then a7+1 canbechosenas a7 and V(i) = Ip+m D.2.1.3If Ra'j ;c 0thenD.2.1.3.1Write Rai asalinearcombinationof RX1 •...• Rxp : p Rai =- 1: f.LjRXj j=l D.2.1.3.2{Updateforsolutionvector ai+1 basedon ai} D.2.1.3.2.1Take r tobetheindexoftheauxiliaryvectorwithhighestpotentialdegreewhichisusedactivelyinthislinearcombination.i.e.with f.Lr ;c 0(notethat ai < 7Tr) D.2.1.3.2.2Set V(i) - EE () II E ( Trr-71 J)E (Trr-~Yj) r.p+ip+if.Lrj.p+if.LjZr.p+iZ j¢.r D.2.1.3.2.3Replace NMNMDM by DM I V(i)RSMRSM D.2.1.3.2.4{Thecorrespondinginterchangeofdegrees}Replace dM by dM . Er.p+i  -4- D.2.2{Reorderingw.r.t. dMS} Definethepermutationmatrix PS suchthat dMS.PS hasitsentriesinnon-decreasingorderfromlefttorightandreorderallotherentriescorrespondinglyLe.replace NMSNMS.DMS b DMS I. PS.RSMS Y RSMSdMSdMS D.2.3{Prepareforthenextlevel}Increasetheentriesof dMA by1D.2.4{Usingformula(1).wecomputenowthefinalsolutionvectorsforlevel n +1}D.2.4.1For i = 1.2....• m D.2.4.1.1If ai+1 isnotasolutionvectoroflevel n thenD.2.4.1.1.1Findalinearcombination i-1 OIj dP  I  --rr a = a,:,+1 + VZ -oI/an+1 + fI ,zJ  x LLI,dILL r-k,Jk 1=1d=OIZk=1j=7T suchthat dRSa isassmallaspossible.(Notethattheremaybemorethan1solution).D.2.4.1.1.2Thepreviouslinearcombinationisassembledintheunimodularmatrix W(i): i-1 W(i) = IT Ep+I,p +i 1=1  I L VI,dZd-ol/ d=OI/ p IT Ek,p +i k=1 OIj L fJ.k· zj--rr J j=7T D.2.4.1.1.3Replace NMDM I by RSM NMDM I W(i) RSM D.2.4.1.2If ai+1 isasolutionvectoroflevel n thensetW(i) = Ip+m D.2.5Define v: 1 = [Yn+1 Cn+11 = VO) ... v(m) . P . W(t)••. w(m) n+Xn+1An+1 with P = I EB PS. Notes1.ItispossiblethatinstepD.2.4.1.1.1thereismorethanonesolution a(z). Itissufficienttotakeoneofthemasthenewsolutionvector ai(z). Thesolutiongeneratedisnotunique.2.InsteaJoftheinitializations NMA = Ip and DMS = 1m. wecouldhavetaken NMA = Yo E KPXp and DMS = A~ E KmXm tobeanynonsingularmatrices.If Ao isthetransformationof A~ asperformedinstepD.1.l.thenwecoulddefine IYo Co I. o = wlth Xo Ao Xo = Om,p andCo = 0p,m. tocompletethesetofunimodularmatrices
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