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subroutine fpopdi(ifsu,ifsv,ifbu,ifbv,u,mu,v,mv,z,mz,z0,dz,
* iopt,ider,tu,nu,tv,nv,nuest,nvest,p,step,c,nc,fp,fpu,fpv,
* nru,nrv,wrk,lwrk)
c given the set of function values z(i,j) defined on the rectangular
c grid (u(i),v(j)),i=1,2,...,mu;j=1,2,...,mv, fpopdi determines a
c smooth bicubic spline approximation with given knots tu(i),i=1,..,nu
c in the u-direction and tv(j),j=1,2,...,nv in the v-direction. this
c spline sp(u,v) will be periodic in the variable v and will satisfy
c the following constraints
c
c s(tu(1),v) = dz(1) , tv(4) <=v<= tv(nv-3)
c
c and (if iopt(2) = 1)
c
c d s(tu(1),v)
c ------------ = dz(2)*cos(v)+dz(3)*sin(v) , tv(4) <=v<= tv(nv-3)
c d u
c
c and (if iopt(3) = 1)
c
c s(tu(nu),v) = 0 tv(4) <=v<= tv(nv-3)
c
c where the parameters dz(i) correspond to the derivative values g(i,j)
c as defined in subroutine pogrid.
c
c the b-spline coefficients of sp(u,v) are determined as the least-
c squares solution of an overdetermined linear system which depends
c on the value of p and on the values dz(i),i=1,2,3. the correspond-
c ing sum of squared residuals sq is a simple quadratic function in
c the variables dz(i). these may or may not be provided. the values
c dz(i) which are not given will be determined so as to minimize the
c resulting sum of squared residuals sq. in that case the user must
c provide some initial guess dz(i) and some estimate (dz(i)-step,
c dz(i)+step) of the range of possible values for these latter.
c
c sp(u,v) also depends on the parameter p (p>0) in such a way that
c - if p tends to infinity, sp(u,v) becomes the least-squares spline
c with given knots, satisfying the constraints.
c - if p tends to zero, sp(u,v) becomes the least-squares polynomial,
c satisfying the constraints.
c - the function f(p)=sumi=1,mu(sumj=1,mv((z(i,j)-sp(u(i),v(j)))**2)
c is continuous and strictly decreasing for p>0.
c
c ..scalar arguments..
integer ifsu,ifsv,ifbu,ifbv,mu,mv,mz,nu,nv,nuest,nvest,
* nc,lwrk
real*8 z0,p,step,fp
c ..array arguments..
integer ider(2),nru(mu),nrv(mv),iopt(3)
real*8 u(mu),v(mv),z(mz),dz(3),tu(nu),tv(nv),c(nc),fpu(nu),fpv(nv)
*,
* wrk(lwrk)
c ..local scalars..
real*8 res,sq,sqq,step1,step2,three
integer i,id0,iop0,iop1,i1,j,l,laa,lau,lav1,lav2,lbb,lbu,lbv,
* lcc,lcs,lq,lri,lsu,lsv,l1,l2,mm,mvnu,number
c ..local arrays..
integer nr(3)
real*8 delta(3),dzz(3),sum(3),a(6,6),g(6)
c ..function references..
integer max0
c ..subroutine references..
c fpgrdi,fpsysy
c ..
c set constant
three = 3
c we partition the working space
lsu = 1
lsv = lsu+4*mu
lri = lsv+4*mv
mm = max0(nuest,mv+nvest)
lq = lri+mm
mvnu = nuest*(mv+nvest-8)
lau = lq+mvnu
lav1 = lau+5*nuest
lav2 = lav1+6*nvest
lbu = lav2+4*nvest
lbv = lbu+5*nuest
laa = lbv+5*nvest
lbb = laa+2*mv
lcc = lbb+2*nvest
lcs = lcc+nvest
c we calculate the smoothing spline sp(u,v) according to the input
c values dz(i),i=1,2,3.
iop0 = iopt(2)
iop1 = iopt(3)
call fpgrdi(ifsu,ifsv,ifbu,ifbv,0,u,mu,v,mv,z,mz,dz,
* iop0,iop1,tu,nu,tv,nv,p,c,nc,sq,fp,fpu,fpv,mm,mvnu,
* wrk(lsu),wrk(lsv),wrk(lri),wrk(lq),wrk(lau),wrk(lav1),
* wrk(lav2),wrk(lbu),wrk(lbv),wrk(laa),wrk(lbb),
* wrk(lcc),wrk(lcs),nru,nrv)
id0 = ider(1)
if(id0.ne.0) go to 5
res = (z0-dz(1))**2
fp = fp+res
sq = sq+res
c in case all derivative values dz(i) are given (step<=0) or in case
c we have spline interpolation, we accept this spline as a solution.
5 if(step.le.0. .or. sq.le.0.) return
dzz(1) = dz(1)
dzz(2) = dz(2)
dzz(3) = dz(3)
c number denotes the number of derivative values dz(i) that still must
c be optimized. let us denote these parameters by g(j),j=1,...,number.
number = 0
if(id0.gt.0) go to 10
number = 1
nr(1) = 1
delta(1) = step
10 if(iop0.eq.0) go to 20
if(ider(2).ne.0) go to 20
step2 = step*three/tu(5)
nr(number+1) = 2
nr(number+2) = 3
delta(number+1) = step2
delta(number+2) = step2
number = number+2
20 if(number.eq.0) return
c the sum of squared residuals sq is a quadratic polynomial in the
c parameters g(j). we determine the unknown coefficients of this
c polymomial by calculating (number+1)*(number+2)/2 different splines
c according to specific values for g(j).
do 30 i=1,number
l = nr(i)
step1 = delta(i)
dzz(l) = dz(l)+step1
call fpgrdi(ifsu,ifsv,ifbu,ifbv,1,u,mu,v,mv,z,mz,dzz,
* iop0,iop1,tu,nu,tv,nv,p,c,nc,sum(i),fp,fpu,fpv,mm,mvnu,
* wrk(lsu),wrk(lsv),wrk(lri),wrk(lq),wrk(lau),wrk(lav1),
* wrk(lav2),wrk(lbu),wrk(lbv),wrk(laa),wrk(lbb),
* wrk(lcc),wrk(lcs),nru,nrv)
if(id0.eq.0) sum(i) = sum(i)+(z0-dzz(1))**2
dzz(l) = dz(l)-step1
call fpgrdi(ifsu,ifsv,ifbu,ifbv,1,u,mu,v,mv,z,mz,dzz,
* iop0,iop1,tu,nu,tv,nv,p,c,nc,sqq,fp,fpu,fpv,mm,mvnu,
* wrk(lsu),wrk(lsv),wrk(lri),wrk(lq),wrk(lau),wrk(lav1),
* wrk(lav2),wrk(lbu),wrk(lbv),wrk(laa),wrk(lbb),
* wrk(lcc),wrk(lcs),nru,nrv)
if(id0.eq.0) sqq = sqq+(z0-dzz(1))**2
a(i,i) = (sum(i)+sqq-sq-sq)/step1**2
if(a(i,i).le.0.) go to 80
g(i) = (sqq-sum(i))/(step1+step1)
dzz(l) = dz(l)
30 continue
if(number.eq.1) go to 60
do 50 i=2,number
l1 = nr(i)
step1 = delta(i)
dzz(l1) = dz(l1)+step1
i1 = i-1
do 40 j=1,i1
l2 = nr(j)
step2 = delta(j)
dzz(l2) = dz(l2)+step2
call fpgrdi(ifsu,ifsv,ifbu,ifbv,1,u,mu,v,mv,z,mz,dzz,
* iop0,iop1,tu,nu,tv,nv,p,c,nc,sqq,fp,fpu,fpv,mm,mvnu,
* wrk(lsu),wrk(lsv),wrk(lri),wrk(lq),wrk(lau),wrk(lav1),
* wrk(lav2),wrk(lbu),wrk(lbv),wrk(laa),wrk(lbb),
* wrk(lcc),wrk(lcs),nru,nrv)
if(id0.eq.0) sqq = sqq+(z0-dzz(1))**2
a(i,j) = (sq+sqq-sum(i)-sum(j))/(step1*step2)
dzz(l2) = dz(l2)
40 continue
dzz(l1) = dz(l1)
50 continue
c the optimal values g(j) are found as the solution of the system
c d (sq) / d (g(j)) = 0 , j=1,...,number.
60 call fpsysy(a,number,g)
do 70 i=1,number
l = nr(i)
dz(l) = dz(l)+g(i)
70 continue
c we determine the spline sp(u,v) according to the optimal values g(j).
80 call fpgrdi(ifsu,ifsv,ifbu,ifbv,0,u,mu,v,mv,z,mz,dz,
* iop0,iop1,tu,nu,tv,nv,p,c,nc,sq,fp,fpu,fpv,mm,mvnu,
* wrk(lsu),wrk(lsv),wrk(lri),wrk(lq),wrk(lau),wrk(lav1),
* wrk(lav2),wrk(lbu),wrk(lbv),wrk(laa),wrk(lbb),
* wrk(lcc),wrk(lcs),nru,nrv)
if(id0.eq.0) fp = fp+(z0-dz(1))**2
return
end
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