一类广义范德蒙矩阵的求逆公式及递推公式-陆全

34720047MATHEMATICSINPRACTICEANDTHEORYVol.34　No.7　July,2004　陆　全(,　710072):　,.:　;;1　　　:2003-03-26　　V=11…1T1T2…TnT21T22…T2nTn-11Tn-12…Tn-1n,　A=11…1T1T2…TnT21T22…T2nTn-21Tn-22…Tn-2nTn1Tn2…Tnn(1)Vn,An.、()、、.,[1—4].,.2　,AAD-ZTA=en-1fT+engT(2)Z=0T0In-10,D=diag(T1,T2,…,Tn),en-1enn-1n,f=(Tn-11-Tn1,…,Tn-1n-Tnn)T,　g=(Tn+11,…,Tn+1n)T(3)　　　An,Aw=en-1,Au=en,1)A;2)ATx=fATy=gx=(x1,x2,…,xn)T,　y=(y1,y2,…,yn)Tf,g(3),vj(j=1,2,…,n)A-1j,vn=u,　vn-1=wvj=Dvj+1-xj+1w-yj+1u　(j=n-2,…,2,1)(4)　　3)A-1A-1=-w1w2wnVTx2x3…xn-1-10x3x4-10xn-1-100-u1u2unVTy2y3…yn-10-1y3y40yn-100(5)V(1).　1)Aw=en-1,Au=en(2)ZTA=AD-en-1fT-engT=A[D-wfT-ugT](ZT)jA=(ZT)j-1[ZTA]=(ZT)j-1A[D-wfT-ugT]=…=A[D-wfT-ugT]j(6)(6)u,(ZT)jen=A[D-wfT-ugT]ju(7)zj=[D-wfT-ugT]ju,(ZT)jen=(ZT)j-1en-1=…=en-j,　j=0,1,…,n-1Z0=I.(7)Azj=(ZT)jen=en-j,　j=0,1,…,n-1X=(zn-1,zn-2,…,z0),AX=(Azn-1,Azn-2,…,Az0)=(e1,e2,…,en)=InA.2)(2)A-1,Aw=en-1,Au=en,　ATx=f,　ATy=gDA-1-A-1ZT=A-1en-1fTA-1+A-1engTA-1=wxT+uyT(8)(8)ej+1DA-1ej+1-A-1ZTej+1=xj+1w+yj+1u(9)ZTej+1=ej,A-1=(v1,v2,…,vn),(9)1337　:Dvj+1-vj=xj+1w+yj+1uvj=Dvj+1-xj+1w-yj+1u,　j=n-2,…,2,1vn=A-1en=u,　vn-1=A-1en-1=w(4).3)(5)jrj,j=1,2,…,n-2,rj=-w1w2wnVTxj+1xn-1-100-u1u2unVTyj+1yn-100=-w1(xj+1+T1xj+2+…+Tn-j-21xn-1-Tn-j-11)…wn(xj+1+Tnxj+2+…+Tn-j-2nxn-1-Tn-j-1n)-u1(yj+1+T1yj+2+…+Tn-j-21yn-1)…un(yj+1+Tnyj+2+…+Tn-j-2nyn-1)Drj+1=-w1T1(xj+2+T1xj+3+…+Tn-j-31xn-1-Tn-j-21)…wnTn(xj+2+Tnxj+3+…+Tn-j-3nxn-1-Tn-j-2n)-u1T1(yj+2+T1yj+3+…+Tn-j-31yn-1)…unTn(yj+2+Tnyj+3+…+Tn-j-3nyn-1)rj-Drj+1=-w1xj+1wnxj+1-u1yj+1unyj+1=-xj+1w-yj+1urj=Drj+1-xj+1w-yj+1u,　j=1,2,…,n-2(10)rn=-u1u2unVT-100=u(11)rn-1=-w1w2wnVT-100=w(12)134　　　　　　　342)(11)、(12)(13)(5)..(4)A,Aw=en-1,Au=en,　ATx=f,ATy=gA,[5]Gohberg-Kailath-Koltracht.:[1]　FinchT,HeinigG,RostK.AninversionformulaandfastalgorithmforCauchy-Vandermondematrices[J].LinearAlgebraAppl,1993,183:176—191.[2]　FinckT,RostK.FastinversionofCauchy-Vandarmondematrices[J].SeminarAnalysis:OperEquNumAnal,WeierstraUInstitut,Berlin,1989/1990.69—79.[3]　RostK,Vavrínz.RecursivesolutionofLwner-Vandermondesystemsofequations.I[J].LinearAlgebraAppl,1996,233:51—65.[4]　RostK,VavrínZ.RecursivesolutionofLwner-Vandermondesystemsofequations.II[J].LinearAlgebraAppl,1995,223/224:597—617.[5]　.[M].,1997.AnInversionFormulaandaRecurrenceAlgorithmfortheGeneralizedVandermondeMatrixLUQuan(NorthwesternPolytechnicalUniversity,Xi′anShanxi710072,China)Abstract:　Thegeneralizedvandermondematrixisinvertibleiftwogeneralizedvandermondee-quationsaresolvable.Aninversionformulaandarecurrencealgorithmfoirthegeneralizedvan-dermondematrixaregiven.Keywords:　generalizedvandermondematrix;inversionmatrix;recurrencealgorithm1357　: