竖线型结点组上的插值及向高维情形的推广

Vol.18(1998)No.4J.ofMath.(PRC)X(,343009)R2,,Rs(s2).MR(1991)41A0565D05O241.3O241.61[1][2]11.1Am+1C0,,Cm,n+1,A,X^m,n.AX^m,n={xi:i=1,,(m+1)(n+1)},,,xj(n+1)+1,,x(j+1)(n+1)Cj,j=0,,m[2],w=(u,v)R2,Cjuj(j=0,,m).U1,U2,U3,:1,v,,vn;u,uv,,uvn;u2,u2v,,u2vn;ALagrange:VDm,nU1,,U(m+1)(n+1)x1,x(m+1)(n+1)=det515(m+1)(n+1)5i=U1(xi),,U(m+1)(n+1)(xi)T.Ci(j=0,,m),A:1.2BA,m+1X:1996206215©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.Cj:u=uj(j=0,,m).xj(n+1)+1,,x(j+1)(n+1)Cj,j=0,,mCj:xj(n+1)+1,,x(j+1)(n+1)=yj1,,yj1lj1,,yjkj,,yjkjljkjlj1++ljkj=n+1,j=0,,m.CjCjDkvfHermite(Dov)I).Dkv5i=[DkvU1(xi),,DkvU(m+1)(n+1)(xi)]T,BX^m,nHermiteVandermondeHDm,nU1,,U(m+1)(n+1)x1,,x(m+1)(n+1)=det515j1Dv5j1Dlj1-1v5j15jkjDlj1-1v5jkjDljkj-1v5jkjCjC0,C1,,Cm,A:1.3CA,X^m,nx1,,x(m+1)(n+1),C0,C1,,Cm=B1,,B1m1,,Bd,,Bdmd,m1++md=m+1,B1,,Bd1C,Dku5i=[DkuU1(xi),,DkuU(m+1)(n+1)(xi)]T,Dmr-k-1u(p-f)(xi)=0,0kmr-1,i=(m1++mr-1+k)(n+1)+1,,(m1+mr-1+k+1)(n+1),r=1,,d.r=1m1++mr-1=0.f,P.Birkhoff,,,.BDm,n,VandermondeBDm,nU1,,U(m+1)(n+1)x1,,x(m+1)(n+1)=det[Dmr-1u5(m1++mr-1)(n+1)+1Dmr-1u5(m1++mr-1+1)(n+1)k=05(m1++mr-1)(n+1)+15(m1++mr)(n+1)k=mr-1]Br.22.1X^m,nA,Dm,nU1,,Um(n+1),Um(n+1)+1,,U(m+1)(n+1)x1,,xm(n+1),xm(n+1)+1,,x(m+1)(n+1)493Vol.18©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.=cõ7m-1k=0[d(Ck,Cm)]n+1õ7m(n+1)ij(m+1)(n+1)d(xi,xj)õVDm-1,nU1,,Um(n+1)x1,,xm(n+1)c=1-1,d(Ci,Cj)CiCj,d(xi,xj)xixj1,m=2,n=3.Cju=uj(j=0,,m),Cjxj(n+1)+1,,x(j+1)(n+1),vj0,,vjn(j=0,,m).P[i,j(R)](12)jRi,VD2,3U1,,U12x1,,x12=det(P[5,1(-u2)]P[6,2(-u2)]P[7,3(-u2)]P[8,4(-u2)]P[9,5(-u2)]P[10,6(-u2)]P[11,7(-u2)]P[12,8(-u2)])õVD2.3U1,,U12x1,,x12=cõ(u0-u2)4(u1-u2)4õ70pq3(v2q-v2p)õVD1,3U1,,U8x1,,x8..2.1,VDo,nVandermonde,2.1X^m,n={xi:i=1,,(m+1)(n+1)}R2A,VDm,nU1,,U(m+1)(n+1)x1,,x(m+1)(n+1)=cõ7mj=17j-1k=0[d(Ck,Cj)]n+1õ7mp=07p(n+1)ij(p+1)(n+1)d(xi,xj)c=1-11VDm,n0,APm,n(Pm,n(u)m,(v)n,).BA,2.1,,2.2X^m,n={xi:i=1,,(m+1)(n+1)}R2B,HermiteHDm,nU1,,U(m+1)(n+1)x1,,x(m+1)(n+1)=cõ7mj=17j-1k=0[d(Ck,Cj)]n+1õ7mp=071stkp[d(yps,ypt)]lpslpc7kpu=17lpu-1q=1q!c=1-1.C.Bd,Lagrange:(f-P)(xi)=0,i=m(n+1)+1,,(m+1)(n+1)BdBd,:55u(f-P)(xi)=0,i=(m-1)(n+1)+1,,m(n+1).,Bd,md-1:593No.4©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.5md-15umd-1(f-P)(xi)=0i=(m-md+1)(n+1)+1,,(m-md+2)(n+1).Bd-1,,B1,BDm,n.2.12.3X^m,n={xi:i=1,,(m+1)(n+1)}R2C,BDm,nU1,,U(m+1)(n+1)x1,,x(m+1)(n+1)=cõ{7dp=271qp[d(Bq,Bp)]mq(n+1).7dp=17mp-1s=1(s!)n+1}õ7mp=07p(n+1)ij(p+1)(n+1)d(xi,xj)c=1-1.HDm,n0BDm,n0,Pm,n,BCHermiteBirkhoff.3,,Cju=uj(j=0,,m),Cjxj(n+1)+1,,x(j+1)(n+1).vj0,,v(j+1)(n+1).UR=(u-u0)(u-uR-1)VRL=(v-vR0)(v-vR1,(v-vR.L-1)U0=VR0=1.Steffensen[4]:fRL=f(u0,,uR;vR0,,vRL)Newtonf(u,v)=mR=0URf(u0,,uR;v)+Um+1f(u,u0,,um;v)f(u0,,uR;v)=nL=0VRLfRL+VR,n+1f(u0,uR;v,vR0,,vRn),AX^m,n:f(u,v)=mR=0nL=0URVRLfRL+r1(u,v)P(u,v)=mR=0nL=0URVRLfRL,r1(u,v)=Um+1f(u,u0,,um;v)+mR=0URVR,n+1f(u0,,uR;v,vR0,,vRn).BHermite,B,yjLvjL(L=1,,kj),HermiteH(u,v)=mR=0UR[f(u0,,uR;vR1)+(v-vR1)f(u0,,uR;vR1,vR1)693Vol.18©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.++(v-vR1)lR1-1f(u0,,uR;vR1,,vR1lR1)+(v-vR1)lR1f(u0,,uR;vR1,,vR1,vR2lR1)++(v-vR1)lR1(v-vR2)lR2-1f(u0,,uR;vR1,,vR1lR1,vR2,vR2lR2)++(v-vR1)lR1(v-vR2)lR2(v-vR,kR-1)lR1-1(v-vRkR)lRkR-1f(u0,,uR;vR1,,vR1lR1,,vRkR,,vRKRlRkR)]Hermiter2(u,v)=f(u,v)-H(u,v)=Um+1f(u,u0,,um;v)+mR=0UR(v-vR1)lR1(v-vRkR)lRkRf(u0,,uR;v,vR1,,vR1lR1,,vRkR,,vRkRlRkR)CBirkhOff,C,B1,,Bdu1,,ud,BirkhoffB(u,v)=dR=1mh=1(u-u1)m1(u-uR-1)mR-1(u-uR)h-1[nL=0Vm1++mR-h+1,Lf(u1,,u1m1,,uR-1,,uR-1mR-1,uR,,uRh;vm1++mR-h+1,0,,vm1++mR-h+1,L)]r3(u,v)=f(u,v)-B(u,v)=(u-u1)m1(u-ud)mdf(u,u1,,u1m1,,ud,,udmd;v)+dR=1mRh=1(u-u1)m1(u-uR-1)mR-1(u-uR)h-1Vm1++mR-h+1,n+1f(u1,,u1m1,,uR-1,uR-1mR-1,uR,,uRh;v,vm1++mR-h+1,0,,vm1++mR-h+1,n)4RS(S2)Rsw=(w1,,ws),Pn1,,ns,:Us1,Us2,,Us7st=1(nt+1)Usi=wK11wKss(0Ktnt,t=1,,s)ijUsiUsj.A:4.1(Rs)AX^n1,,ns=xi:i=1,,7st=1(nt+1)Rs793No.4©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.,Rsn1+1w1Ksi(i=0,,n1)xj7st=2(nt+1)+1,,xj+17st=2(nt+1)Ksj,j=0,,n1xj7st=2(nt+1)+1,,x(j+1)st=2(nt+1)Rs-1Rs-1A(R2AA),X^n1,,nsRsA.A4.1X^n1,,ns=xi:i=1,,7st=1(nt+1)A,VDn1,,nsU1,,Us7st=1(nt+1)x1,,xs7st=1(nt+1)=Cõ7n1j=17j-1l=0[d(Ksl,Ksj)]7st=2(nt+1)õ7n1p=0VDn2,,nsUs-11,,Us-17st=2(nt+1)xp7st=2(nt+1)+1,,x(p+1)7st=2(nt+1)ûKspc=1-1.VDn1,,ns0,X^n1n,,s.,s=3,n1=2,n2=2,n3=3.,s=3,R3x=(u,v,w).U31,,U336():1,w,w2,w3;v,vw,vw2,vw3;v2,v2w,v2w2,v2w3;u,uw,uw2,uw3;uv,uvw,uvw2,uvw3;uv2,uv2w,uv2w2,uv2w3;u2,u2w,u2w2,u2w3;u2v,u2vw,u2vw2,u2vw3;u2v2,u2v2w,u2v2w2,u2v2w3.A,xi=(u0,vi,wi)1i12,(u1,vi,wi)13i24,(u2,vi,wi)25i36.VD2,2,3,(12+i)(-u2)(24+i)(1i12),i(-u2)(12+i)(1i12).VD2,2,3U31,,U336x1,,x36=(u0-u2)12(u1-u2)12õVD2,3U21,,U212x25,,x36ûK32õVD1,2,3U31,,U324x1,,x24VD1,2,3i(-u1)(12+i)(1i12),VD2,2,3U31,,U336x1,,x36=(u0-u2)12(u1-u2)12(u0-u1)12õVD2,3U21,,U212x25,,x36ûK32õVD2,3U21,,U212x13,,x24ûK31893Vol.18©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.õVD2,3U21,,U212x1,,x12ûK30..1R2BCRs(s2),4.1.3Rs(s2).2RsAw1wi(2is),..1Chui,C.K.andLai,M.J.,VandermondedeterminantandLagrangeinterpolationinRs,inNonlinearandConvexAnalysis,EditedbyLin,B.L.,Marc