subroutine fppogr(iopt,ider,u,mu,v,mv,z,mz,z0,r,s,nuest,nvest, * tol,maxit,nc,nu,tu,nv,tv,c,fp,fp0,fpold,reducu,reducv,fpintu, * fpintv,dz,step,lastdi,nplusu,nplusv,lasttu,nru,nrv,nrdatu, * nrdatv,wrk,lwrk,ier) c .. c ..scalar arguments.. integer mu,mv,mz,nuest,nvest,maxit,nc,nu,nv,lastdi,nplusu,nplusv, * lasttu,lwrk,ier real z0,r,s,tol,fp,fp0,fpold,reducu,reducv,step c ..array arguments.. integer iopt(3),ider(2),nrdatu(nuest),nrdatv(nvest),nru(mu), * nrv(mv) real u(mu),v(mv),z(mz),tu(nuest),tv(nvest),c(nc),fpintu(nuest), * fpintv(nvest),dz(3),wrk(lwrk) c ..local scalars.. real acc,fpms,f1,f2,f3,p,per,pi,p1,p2,p3,vb,ve,zmax,zmin,rn,one, * con1,con4,con9 integer i,ich1,ich3,ifbu,ifbv,ifsu,ifsv,istart,iter,i1,i2,j,ju, * ktu,l,l1,l2,l3,l4,mpm,mumin,mu0,mu1,nn,nplu,nplv,npl1,nrintu, * nrintv,nue,numax,nve,nvmax c ..local arrays.. integer idd(2) real dzz(3) c ..function references.. real abs,atan2,fprati integer max0,min0 c ..subroutine references.. c fpknot,fpopdi c .. c set constants one = 1 con1 = 0.1e0 con9 = 0.9e0 con4 = 0.4e-01 c initialization ifsu = 0 ifsv = 0 ifbu = 0 ifbv = 0 p = -one mumin = 4-iopt(3) if(ider(1).ge.0) mumin = mumin-1 if(iopt(2).eq.1 .and. ider(2).eq.1) mumin = mumin-1 pi = atan2(0.,-one) per = pi+pi vb = v(1) ve = vb+per cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c part 1: determination of the number of knots and their position. c c **************************************************************** c c given a set of knots we compute the least-squares spline sinf(u,v) c c and the corresponding sum of squared residuals fp = f(p=inf). c c if iopt(1)=-1 sinf(u,v) is the requested approximation. c c if iopt(1)>=0 we check whether we can accept the knots: c c if fp <= s we will continue with the current set of knots. c c if fp > s we will increase the number of knots and compute the c c corresponding least-squares spline until finally fp <= s. c c the initial choice of knots depends on the value of s and iopt. c c if s=0 we have spline interpolation; in that case the number of c c knots in the u-direction equals nu=numax=mu+5+iopt(2)+iopt(3) c c and in the v-direction nv=nvmax=mv+7. c c if s>0 and c c iopt(1)=0 we first compute the least-squares polynomial,i.e. a c c spline without interior knots : nu=8 ; nv=8. c c iopt(1)=1 we start with the set of knots found at the last call c c of the routine, except for the case that s > fp0; then we c c compute the least-squares polynomial directly. c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc if(iopt(1).lt.0) go to 120 c acc denotes the absolute tolerance for the root of f(p)=s. acc = tol*s c numax and nvmax denote the number of knots needed for interpolation. numax = mu+5+iopt(2)+iopt(3) nvmax = mv+7 nue = min0(numax,nuest) nve = min0(nvmax,nvest) if(s.gt.0.) go to 100 c if s = 0, s(u,v) is an interpolating spline. nu = numax nv = nvmax c test whether the required storage space exceeds the available one. if(nu.gt.nuest .or. nv.gt.nvest) go to 420 c find the position of the knots in the v-direction. do 10 l=1,mv tv(l+3) = v(l) 10 continue tv(mv+4) = ve l1 = mv-2 l2 = mv+5 do 20 i=1,3 tv(i) = v(l1)-per tv(l2) = v(i+1)+per l1 = l1+1 l2 = l2+1 20 continue c if not all the derivative values g(i,j) are given, we will first c estimate these values by computing a least-squares spline idd(1) = ider(1) if(idd(1).eq.0) idd(1) = 1 if(idd(1).gt.0) dz(1) = z0 idd(2) = ider(2) if(ider(1).lt.0) go to 30 if(iopt(2).eq.0 .or. ider(2).ne.0) go to 70 c we set up the knots in the u-direction for computing the least-squares c spline. 30 i1 = 3 i2 = mu-2 nu = 4 do 40 i=1,mu if(i1.gt.i2) go to 50 nu = nu+1 tu(nu) = u(i1) i1 = i1+2 40 continue 50 do 60 i=1,4 tu(i) = 0. nu = nu+1 tu(nu) = r 60 continue c we compute the least-squares spline for estimating the derivatives. call fpopdi(ifsu,ifsv,ifbu,ifbv,u,mu,v,mv,z,mz,z0,dz,iopt,idd, * tu,nu,tv,nv,nuest,nvest,p,step,c,nc,fp,fpintu,fpintv,nru,nrv, * wrk,lwrk) ifsu = 0 c if all the derivatives at the origin are known, we compute the c interpolating spline. c we set up the knots in the u-direction, needed for interpolation. 70 nn = numax-8 if(nn.eq.0) go to 95 ju = 2-iopt(2) do 80 l=1,nn tu(l+4) = u(ju) ju = ju+1 80 continue nu = numax l = nu do 90 i=1,4 tu(i) = 0. tu(l) = r l = l-1 90 continue c we compute the interpolating spline. 95 call fpopdi(ifsu,ifsv,ifbu,ifbv,u,mu,v,mv,z,mz,z0,dz,iopt,idd, * tu,nu,tv,nv,nuest,nvest,p,step,c,nc,fp,fpintu,fpintv,nru,nrv, * wrk,lwrk) go to 430 c if s>0 our initial choice of knots depends on the value of iopt(1). 100 ier = 0 if(iopt(1).eq.0) go to 115 step = -step if(fp0.le.s) go to 115 c if iopt(1)=1 and fp0 > s we start computing the least-squares spline c according to the set of knots found at the last call of the routine. c we determine the number of grid coordinates u(i) inside each knot c interval (tu(l),tu(l+1)). l = 5 j = 1 nrdatu(1) = 0 mu0 = 2-iopt(2) mu1 = mu-2+iopt(3) do 105 i=mu0,mu1 nrdatu(j) = nrdatu(j)+1 if(u(i).lt.tu(l)) go to 105 nrdatu(j) = nrdatu(j)-1 l = l+1 j = j+1 nrdatu(j) = 0 105 continue c we determine the number of grid coordinates v(i) inside each knot c interval (tv(l),tv(l+1)). l = 5 j = 1 nrdatv(1) = 0 do 110 i=2,mv nrdatv(j) = nrdatv(j)+1 if(v(i).lt.tv(l)) go to 110 nrdatv(j) = nrdatv(j)-1 l = l+1 j = j+1 nrdatv(j) = 0 110 continue idd(1) = ider(1) idd(2) = ider(2) go to 120 c if iopt(1)=0 or iopt(1)=1 and s >= fp0,we start computing the least- c squares polynomial (which is a spline without interior knots). 115 ier = -2 idd(1) = ider(1) idd(2) = 1 nu = 8 nv = 8 nrdatu(1) = mu-3+iopt(2)+iopt(3) nrdatv(1) = mv-1 lastdi = 0 nplusu = 0 nplusv = 0 fp0 = 0. fpold = 0. reducu = 0. reducv = 0. c main loop for the different sets of knots.mpm=mu+mv is a save upper c bound for the number of trials. 120 mpm = mu+mv do 270 iter=1,mpm c find nrintu (nrintv) which is the number of knot intervals in the c u-direction (v-direction). nrintu = nu-7 nrintv = nv-7 c find the position of the additional knots which are needed for the c b-spline representation of s(u,v). i = nu do 130 j=1,4 tu(j) = 0. tu(i) = r i = i-1 130 continue l1 = 4 l2 = l1 l3 = nv-3 l4 = l3 tv(l2) = vb tv(l3) = ve do 140 j=1,3 l1 = l1+1 l2 = l2-1 l3 = l3+1 l4 = l4-1 tv(l2) = tv(l4)-per tv(l3) = tv(l1)+per 140 continue c find an estimate of the range of possible values for the optimal c derivatives at the origin. ktu = nrdatu(1)+2-iopt(2) if(nrintu.eq.1) ktu = mu if(ktu.lt.mumin) ktu = mumin if(ktu.eq.lasttu) go to 150 zmin = z0 zmax = z0 l = mv*ktu do 145 i=1,l if(z(i).lt.zmin) zmin = z(i) if(z(i).gt.zmax) zmax = z(i) 145 continue step = zmax-zmin lasttu = ktu c find the least-squares spline sinf(u,v). 150 call fpopdi(ifsu,ifsv,ifbu,ifbv,u,mu,v,mv,z,mz,z0,dz,iopt,idd, * tu,nu,tv,nv,nuest,nvest,p,step,c,nc,fp,fpintu,fpintv,nru,nrv, * wrk,lwrk) if(step.lt.0.) step = -step if(ier.eq.(-2)) fp0 = fp c test whether the least-squares spline is an acceptable solution. if(iopt(1).lt.0) go to 440 fpms = fp-s if(abs(fpms) .lt. acc) go to 440 c if f(p=inf) < s, we accept the choice of knots. if(fpms.lt.0.) go to 300 c if nu=numax and nv=nvmax, sinf(u,v) is an interpolating spline if(nu.eq.numax .and. nv.eq.nvmax) go to 430 c increase the number of knots. c if nu=nue and nv=nve we cannot further increase the number of knots c because of the storage capacity limitation. if(nu.eq.nue .and. nv.eq.nve) go to 420 if(ider(1).eq.0) fpintu(1) = fpintu(1)+(z0-c(1))**2 ier = 0 c adjust the parameter reducu or reducv according to the direction c in which the last added knots were located. if(lastdi) 160,155,170 155 nplv = 3 idd(2) = ider(2) fpold = fp go to 230 160 reducu = fpold-fp go to 175 170 reducv = fpold-fp c store the sum of squared residuals for the current set of knots. 175 fpold = fp c find nplu, the number of knots we should add in the u-direction. nplu = 1 if(nu.eq.8) go to 180 npl1 = nplusu*2 rn = nplusu if(reducu.gt.acc) npl1 = rn*fpms/reducu nplu = min0(nplusu*2,max0(npl1,nplusu/2,1)) c find nplv, the number of knots we should add in the v-direction. 180 nplv = 3 if(nv.eq.8) go to 190 npl1 = nplusv*2 rn = nplusv if(reducv.gt.acc) npl1 = rn*fpms/reducv nplv = min0(nplusv*2,max0(npl1,nplusv/2,1)) c test whether we are going to add knots in the u- or v-direction. 190 if(nplu-nplv) 210,200,230 200 if(lastdi.lt.0) go to 230 210 if(nu.eq.nue) go to 230 c addition in the u-direction. lastdi = -1 nplusu = nplu ifsu = 0 istart = 0 if(iopt(2).eq.0) istart = 1 do 220 l=1,nplusu c add a new knot in the u-direction call fpknot(u,mu,tu,nu,fpintu,nrdatu,nrintu,nuest,istart) c test whether we cannot further increase the number of knots in the c u-direction. if(nu.eq.nue) go to 270 220 continue go to 270 230 if(nv.eq.nve) go to 210 c addition in the v-direction. lastdi = 1 nplusv = nplv ifsv = 0 do 240 l=1,nplusv c add a new knot in the v-direction. call fpknot(v,mv,tv,nv,fpintv,nrdatv,nrintv,nvest,1) c test whether we cannot further increase the number of knots in the c v-direction. if(nv.eq.nve) go to 270 240 continue c restart the computations with the new set of knots. 270 continue c test whether the least-squares polynomial is a solution of our c approximation problem. 300 if(ier.eq.(-2)) go to 440 cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c part 2: determination of the smoothing spline sp(u,v) c c ***************************************************** c c we have determined the number of knots and their position. we now c c compute the b-spline coefficients of the smoothing spline sp(u,v). c c this smoothing spline depends on the parameter p in such a way that c c f(p) = sumi=1,mu(sumj=1,mv((z(i,j)-sp(u(i),v(j)))**2) c c is a continuous, strictly decreasing function of p. moreover the c c least-squares polynomial corresponds to p=0 and the least-squares c c spline to p=infinity. then iteratively we have to determine the c c positive value of p such that f(p)=s. the process which is proposed c c here makes use of rational interpolation. f(p) is approximated by a c c rational function r(p)=(u*p+v)/(p+w); three values of p (p1,p2,p3) c c with corresponding values of f(p) (f1=f(p1)-s,f2=f(p2)-s,f3=f(p3)-s)c c are used to calculate the new value of p such that r(p)=s. c c convergence is guaranteed by taking f1 > 0 and f3 < 0. c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c initial value for p. p1 = 0. f1 = fp0-s p3 = -one f3 = fpms p = one dzz(1) = dz(1) dzz(2) = dz(2) dzz(3) = dz(3) ich1 = 0 ich3 = 0 c iteration process to find the root of f(p)=s. do 350 iter = 1,maxit c find the smoothing spline sp(u,v) and the corresponding sum f(p). call fpopdi(ifsu,ifsv,ifbu,ifbv,u,mu,v,mv,z,mz,z0,dzz,iopt,idd, * tu,nu,tv,nv,nuest,nvest,p,step,c,nc,fp,fpintu,fpintv,nru,nrv, * wrk,lwrk) c test whether the approximation sp(u,v) is an acceptable solution. fpms = fp-s if(abs(fpms).lt.acc) go to 440 c test whether the maximum allowable number of iterations has been c reached. if(iter.eq.maxit) go to 400 c carry out one more step of the iteration process. p2 = p f2 = fpms if(ich3.ne.0) go to 320 if((f2-f3).gt.acc) go to 310 c our initial choice of p is too large. p3 = p2 f3 = f2 p = p*con4 if(p.le.p1) p = p1*con9 + p2*con1 go to 350 310 if(f2.lt.0.) ich3 = 1 320 if(ich1.ne.0) go to 340 if((f1-f2).gt.acc) go to 330 c our initial choice of p is too small p1 = p2 f1 = f2 p = p/con4 if(p3.lt.0.) go to 350 if(p.ge.p3) p = p2*con1 + p3*con9 go to 350 c test whether the iteration process proceeds as theoretically c expected. 330 if(f2.gt.0.) ich1 = 1 340 if(f2.ge.f1 .or. f2.le.f3) go to 410 c find the new value of p. p = fprati(p1,f1,p2,f2,p3,f3) 350 continue c error codes and messages. 400 ier = 3 go to 440 410 ier = 2 go to 440 420 ier = 1 go to 440 430 ier = -1 fp = 0. 440 return end