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subroutine fpopsp(ifsu,ifsv,ifbu,ifbv,u,mu,v,mv,r,mr,r0,r1,dr,
* 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 r(i,j) defined on the rectangular
c grid (u(i),v(j)),i=1,2,...,mu;j=1,2,...,mv, fpopsp 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) = dr(1) , tv(4) <=v<= tv(nv-3)
c
c s(tu(nu),v) = dr(4) , tv(4) <=v<= tv(nv-3)
c
c and (if iopt(2) = 1)
c
c d s(tu(1),v)
c ------------ = dr(2)*cos(v)+dr(3)*sin(v) , tv(4) <=v<= tv(nv-3)
c d u
c
c and (if iopt(3) = 1)
c
c d s(tu(nu),v)
c ------------- = dr(5)*cos(v)+dr(6)*sin(v) , tv(4) <=v<= tv(nv-3)
c d u
c
c where the parameters dr(i) correspond to the derivative values at the
c poles as defined in subroutine spgrid.
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 dr(i),i=1,...,6. the correspond-
c ing sum of squared residuals sq is a simple quadratic function in
c the variables dr(i). these may or may not be provided. the values
c dr(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 dr(i) and some estimate (dr(i)-step,
c dr(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((r(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,mr,nu,nv,nuest,nvest,
* nc,lwrk
real*8 r0,r1,p,fp
c ..array arguments..
integer ider(4),nru(mu),nrv(mv),iopt(3)
real*8 u(mu),v(mv),r(mr),dr(6),tu(nu),tv(nv),c(nc),fpu(nu),fpv(nv)
*,
* wrk(lwrk),step(2)
c ..local scalars..
real*8 sq,sqq,sq0,sq1,step1,step2,three
integer i,id0,iop0,iop1,i1,j,l,lau,lav1,lav2,la0,la1,lbu,lbv,lb0,
* lb1,lc0,lc1,lcs,lq,lri,lsu,lsv,l1,l2,mm,mvnu,number
c ..local arrays..
integer nr(6)
real*8 delta(6),drr(6),sum(6),a(6,6),g(6)
c ..function references..
integer max0
c ..subroutine references..
c fpgrsp,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
la0 = lbv+5*nvest
la1 = la0+2*mv
lb0 = la1+2*mv
lb1 = lb0+2*nvest
lc0 = lb1+2*nvest
lc1 = lc0+nvest
lcs = lc1+nvest
c we calculate the smoothing spline sp(u,v) according to the input
c values dr(i),i=1,...,6.
iop0 = iopt(2)
iop1 = iopt(3)
id0 = ider(1)
id1 = ider(3)
call fpgrsp(ifsu,ifsv,ifbu,ifbv,0,u,mu,v,mv,r,mr,dr,
* 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(la0),wrk(la1),wrk(lb0),
* wrk(lb1),wrk(lc0),wrk(lc1),wrk(lcs),nru,nrv)
sq0 = 0.
sq1 = 0.
if(id0.eq.0) sq0 = (r0-dr(1))**2
if(id1.eq.0) sq1 = (r1-dr(4))**2
sq = sq+sq0+sq1
c in case all derivative values dr(i) are given (step<=0) or in case
c we have spline interpolation, we accept this spline as a solution.
if(sq.le.0.) return
if(step(1).le.0. .and. step(2).le.0.) return
do 10 i=1,6
drr(i) = dr(i)
10 continue
c number denotes the number of derivative values dr(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 20
number = 1
nr(1) = 1
delta(1) = step(1)
20 if(iop0.eq.0) go to 30
if(ider(2).ne.0) go to 30
step2 = step(1)*three/(tu(5)-tu(4))
nr(number+1) = 2
nr(number+2) = 3
delta(number+1) = step2
delta(number+2) = step2
number = number+2
30 if(id1.gt.0) go to 40
number = number+1
nr(number) = 4
delta(number) = step(2)
40 if(iop1.eq.0) go to 50
if(ider(4).ne.0) go to 50
step2 = step(2)*three/(tu(nu)-tu(nu-4))
nr(number+1) = 5
nr(number+2) = 6
delta(number+1) = step2
delta(number+2) = step2
number = number+2
50 if(number.eq.0) return
c the sum of squared residulas 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 60 i=1,number
l = nr(i)
step1 = delta(i)
drr(l) = dr(l)+step1
call fpgrsp(ifsu,ifsv,ifbu,ifbv,1,u,mu,v,mv,r,mr,drr,
* 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(la0),wrk(la1),wrk(lb0),
* wrk(lb1),wrk(lc0),wrk(lc1),wrk(lcs),nru,nrv)
if(id0.eq.0) sq0 = (r0-drr(1))**2
if(id1.eq.0) sq1 = (r1-drr(4))**2
sum(i) = sum(i)+sq0+sq1
drr(l) = dr(l)-step1
call fpgrsp(ifsu,ifsv,ifbu,ifbv,1,u,mu,v,mv,r,mr,drr,
* 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(la0),wrk(la1),wrk(lb0),
* wrk(lb1),wrk(lc0),wrk(lc1),wrk(lcs),nru,nrv)
if(id0.eq.0) sq0 = (r0-drr(1))**2
if(id1.eq.0) sq1 = (r1-drr(4))**2
sqq = sqq+sq0+sq1
drr(l) = dr(l)
a(i,i) = (sum(i)+sqq-sq-sq)/step1**2
if(a(i,i).le.0.) go to 110
g(i) = (sqq-sum(i))/(step1+step1)
60 continue
if(number.eq.1) go to 90
do 80 i=2,number
l1 = nr(i)
step1 = delta(i)
drr(l1) = dr(l1)+step1
i1 = i-1
do 70 j=1,i1
l2 = nr(j)
step2 = delta(j)
drr(l2) = dr(l2)+step2
call fpgrsp(ifsu,ifsv,ifbu,ifbv,1,u,mu,v,mv,r,mr,drr,
* 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(la0),wrk(la1),wrk(lb0),
* wrk(lb1),wrk(lc0),wrk(lc1),wrk(lcs),nru,nrv)
if(id0.eq.0) sq0 = (r0-drr(1))**2
if(id1.eq.0) sq1 = (r1-drr(4))**2
sqq = sqq+sq0+sq1
a(i,j) = (sq+sqq-sum(i)-sum(j))/(step1*step2)
drr(l2) = dr(l2)
70 continue
drr(l1) = dr(l1)
80 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.
90 call fpsysy(a,number,g)
do 100 i=1,number
l = nr(i)
dr(l) = dr(l)+g(i)
100 continue
c we determine the spline sp(u,v) according to the optimal values g(j).
110 call fpgrsp(ifsu,ifsv,ifbu,ifbv,0,u,mu,v,mv,r,mr,dr,
* 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(la0),wrk(la1),wrk(lb0),
* wrk(lb1),wrk(lc0),wrk(lc1),wrk(lcs),nru,nrv)
if(id0.eq.0) sq0 = (r0-dr(1))**2
if(id1.eq.0) sq1 = (r1-dr(4))**2
sq = sq+sq0+sq1
return
end
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