subroutine parsur(iopt,ipar,idim,mu,u,mv,v,f,s,nuest,nvest,
* nu,tu,nv,tv,c,fp,wrk,lwrk,iwrk,kwrk,ier)
c given the set of ordered points f(i,j) in the idim-dimensional space,
c corresponding to grid values (u(i),v(j)) ,i=1,...,mu ; j=1,...,mv,
c parsur determines a smooth approximating spline surface s(u,v) , i.e.
c f1 = s1(u,v)
c ... u(1) <= u <= u(mu) ; v(1) <= v <= v(mv)
c fidim = sidim(u,v)
c with sl(u,v), l=1,2,...,idim bicubic spline functions with common
c knots tu(i),i=1,...,nu in the u-variable and tv(j),j=1,...,nv in the
c v-variable.
c in addition, these splines will be periodic in the variable u if
c ipar(1) = 1 and periodic in the variable v if ipar(2) = 1.
c if iopt=-1, parsur determines the least-squares bicubic spline
c surface according to a given set of knots.
c if iopt>=0, the number of knots of s(u,v) and their position
c is chosen automatically by the routine. the smoothness of s(u,v) is
c achieved by minimalizing the discontinuity jumps of the derivatives
c of the splines at the knots. the amount of smoothness of s(u,v) is
c determined by the condition that
c fp=sumi=1,mu(sumj=1,mv(dist(f(i,j)-s(u(i),v(j)))**2))<=s,
c with s a given non-negative constant.
c the fit s(u,v) is given in its b-spline representation and can be
c evaluated by means of routine surev.
c
c calling sequence:
c call parsur(iopt,ipar,idim,mu,u,mv,v,f,s,nuest,nvest,nu,tu,
c * nv,tv,c,fp,wrk,lwrk,iwrk,kwrk,ier)
c
c parameters:
c iopt : integer flag. unchanged on exit.
c on entry iopt must specify whether a least-squares surface
c (iopt=-1) or a smoothing surface (iopt=0 or 1)must be
c determined.
c if iopt=0 the routine will start with the initial set of
c knots needed for determining the least-squares polynomial
c surface.
c if iopt=1 the routine will continue with the set of knots
c found at the last call of the routine.
c attention: a call with iopt=1 must always be immediately
c preceded by another call with iopt = 1 or iopt = 0.
c ipar : integer array of dimension 2. unchanged on exit.
c on entry ipar(1) must specify whether (ipar(1)=1) or not
c (ipar(1)=0) the splines must be periodic in the variable u.
c on entry ipar(2) must specify whether (ipar(2)=1) or not
c (ipar(2)=0) the splines must be periodic in the variable v.
c idim : integer. on entry idim must specify the dimension of the
c surface. 1 <= idim <= 3. unchanged on exit.
c mu : integer. on entry mu must specify the number of grid points
c along the u-axis. unchanged on exit.
c mu >= mumin where mumin=4-2*ipar(1)
c u : real array of dimension at least (mu). before entry, u(i)
c must be set to the u-co-ordinate of the i-th grid point
c along the u-axis, for i=1,2,...,mu. these values must be
c supplied in strictly ascending order. unchanged on exit.
c mv : integer. on entry mv must specify the number of grid points
c along the v-axis. unchanged on exit.
c mv >= mvmin where mvmin=4-2*ipar(2)
c v : real array of dimension at least (mv). before entry, v(j)
c must be set to the v-co-ordinate of the j-th grid point
c along the v-axis, for j=1,2,...,mv. these values must be
c supplied in strictly ascending order. unchanged on exit.
c f : real array of dimension at least (mu*mv*idim).
c before entry, f(mu*mv*(l-1)+mv*(i-1)+j) must be set to the
c l-th co-ordinate of the data point corresponding to the
c the grid point (u(i),v(j)) for l=1,...,idim ,i=1,...,mu
c and j=1,...,mv. unchanged on exit.
c if ipar(1)=1 it is expected that f(mu*mv*(l-1)+mv*(mu-1)+j)
c = f(mu*mv*(l-1)+j), l=1,...,idim ; j=1,...,mv
c if ipar(2)=1 it is expected that f(mu*mv*(l-1)+mv*(i-1)+mv)
c = f(mu*mv*(l-1)+mv*(i-1)+1), l=1,...,idim ; i=1,...,mu
c s : real. on entry (if iopt>=0) s must specify the smoothing
c factor. s >=0. unchanged on exit.
c for advice on the choice of s see further comments
c nuest : integer. unchanged on exit.
c nvest : integer. unchanged on exit.
c on entry, nuest and nvest must specify an upper bound for the
c number of knots required in the u- and v-directions respect.
c these numbers will also determine the storage space needed by
c the routine. nuest >= 8, nvest >= 8.
c in most practical situation nuest = mu/2, nvest=mv/2, will
c be sufficient. always large enough are nuest=mu+4+2*ipar(1),
c nvest = mv+4+2*ipar(2), the number of knots needed for
c interpolation (s=0). see also further comments.
c nu : integer.
c unless ier=10 (in case iopt>=0), nu will contain the total
c number of knots with respect to the u-variable, of the spline
c surface returned. if the computation mode iopt=1 is used,
c the value of nu should be left unchanged between subsequent
c calls. in case iopt=-1, the value of nu should be specified
c on entry.
c tu : real array of dimension at least (nuest).
c on succesful exit, this array will contain the knots of the
c splines with respect to the u-variable, i.e. the position of
c the interior knots tu(5),...,tu(nu-4) as well as the position
c of the additional knots tu(1),...,tu(4) and tu(nu-3),...,
c tu(nu) needed for the b-spline representation.
c if the computation mode iopt=1 is used,the values of tu(1)
c ...,tu(nu) should be left unchanged between subsequent calls.
c if the computation mode iopt=-1 is used, the values tu(5),
c ...tu(nu-4) must be supplied by the user, before entry.
c see also the restrictions (ier=10).
c nv : integer.
c unless ier=10 (in case iopt>=0), nv will contain the total
c number of knots with respect to the v-variable, of the spline
c surface returned. if the computation mode iopt=1 is used,
c the value of nv should be left unchanged between subsequent
c calls. in case iopt=-1, the value of nv should be specified
c on entry.
c tv : real array of dimension at least (nvest).
c on succesful exit, this array will contain the knots of the
c splines with respect to the v-variable, i.e. the position of
c the interior knots tv(5),...,tv(nv-4) as well as the position
c of the additional knots tv(1),...,tv(4) and tv(nv-3),...,
c tv(nv) needed for the b-spline representation.
c if the computation mode iopt=1 is used,the values of tv(1)
c ...,tv(nv) should be left unchanged between subsequent calls.
c if the computation mode iopt=-1 is used, the values tv(5),
c ...tv(nv-4) must be supplied by the user, before entry.
c see also the restrictions (ier=10).
c c : real array of dimension at least (nuest-4)*(nvest-4)*idim.
c on succesful exit, c contains the coefficients of the spline
c approximation s(u,v)
c fp : real. unless ier=10, fp contains the sum of squared
c residuals of the spline surface returned.
c wrk : real array of dimension (lwrk). used as workspace.
c if the computation mode iopt=1 is used the values of
c wrk(1),...,wrk(4) should be left unchanged between subsequent
c calls.
c lwrk : integer. on entry lwrk must specify the actual dimension of
c the array wrk as declared in the calling (sub)program.
c lwrk must not be too small.
c lwrk >= 4+nuest*(mv*idim+11+4*ipar(1))+nvest*(11+4*ipar(2))+
c 4*(mu+mv)+q*idim where q is the larger of mv and nuest.
c iwrk : integer array of dimension (kwrk). used as workspace.
c if the computation mode iopt=1 is used the values of
c iwrk(1),.,iwrk(3) should be left unchanged between subsequent
c calls.
c kwrk : integer. on entry kwrk must specify the actual dimension of
c the array iwrk as declared in the calling (sub)program.
c kwrk >= 3+mu+mv+nuest+nvest.
c ier : integer. unless the routine detects an error, ier contains a
c non-positive value on exit, i.e.
c ier=0 : normal return. the surface returned has a residual sum of
c squares fp such that abs(fp-s)/s <= tol with tol a relat-
c ive tolerance set to 0.001 by the program.
c ier=-1 : normal return. the spline surface returned is an
c interpolating surface (fp=0).
c ier=-2 : normal return. the surface returned is the least-squares
c polynomial surface. in this extreme case fp gives the
c upper bound for the smoothing factor s.
c ier=1 : error. the required storage space exceeds the available
c storage space, as specified by the parameters nuest and
c nvest.
c probably causes : nuest or nvest too small. if these param-
c eters are already large, it may also indicate that s is
c too small
c the approximation returned is the least-squares surface
c according to the current set of knots. the parameter fp
c gives the corresponding sum of squared residuals (fp>s).
c ier=2 : error. a theoretically impossible result was found during
c the iteration proces for finding a smoothing surface with
c fp = s. probably causes : s too small.
c there is an approximation returned but the corresponding
c sum of squared residuals does not satisfy the condition
c abs(fp-s)/s < tol.
c ier=3 : error. the maximal number of iterations maxit (set to 20
c by the program) allowed for finding a smoothing surface
c with fp=s has been reached. probably causes : s too small
c there is an approximation returned but the corresponding
c sum of squared residuals does not satisfy the condition
c abs(fp-s)/s < tol.
c ier=10 : error. on entry, the input data are controlled on validity
c the following restrictions must be satisfied.
c -1<=iopt<=1, 0<=ipar(1)<=1, 0<=ipar(2)<=1, 1 <=idim<=3
c mu >= 4-2*ipar(1),mv >= 4-2*ipar(2), nuest >=8, nvest >= 8,
c kwrk>=3+mu+mv+nuest+nvest,
c lwrk >= 4+nuest*(mv*idim+11+4*ipar(1))+nvest*(11+4*ipar(2))
c +4*(mu+mv)+max(nuest,mv)*idim
c u(i-1)__=0: s>=0
c if s=0: nuest>=mu+4+2*ipar(1)
c nvest>=mv+4+2*ipar(2)
c if one of these conditions is found to be violated,control
c is immediately repassed to the calling program. in that
c case there is no approximation returned.
c
c further comments:
c by means of the parameter s, the user can control the tradeoff
c between closeness of fit and smoothness of fit of the approximation.
c if s is too large, the surface will be too smooth and signal will be
c lost ; if s is too small the surface will pick up too much noise. in
c the extreme cases the program will return an interpolating surface
c if s=0 and the constrained least-squares polynomial surface if s is
c very large. between these extremes, a properly chosen s will result
c in a good compromise between closeness of fit and smoothness of fit.
c to decide whether an approximation, corresponding to a certain s is
c satisfactory the user is highly recommended to inspect the fits
c graphically.
c recommended values for s depend on the accuracy of the data values.
c if the user has an idea of the statistical errors on the data, he
c can also find a proper estimate for s. for, by assuming that, if he
c specifies the right s, parsur will return a surface s(u,v) which
c exactly reproduces the surface underlying the data he can evaluate
c the sum(dist(f(i,j)-s(u(i),v(j)))**2) to find a good estimate for s.
c for example, if he knows that the statistical errors on his f(i,j)-
c values is not greater than 0.1, he may expect that a good s should
c have a value not larger than mu*mv*(0.1)**2.
c if nothing is known about the statistical error in f(i,j), s must
c be determined by trial and error, taking account of the comments
c above. the best is then to start with a very large value of s (to
c determine the le-sq polynomial surface and the corresponding upper
c bound fp0 for s) and then to progressively decrease the value of s
c ( say by a factor 10 in the beginning, i.e. s=fp0/10,fp0/100,...
c and more carefully as the approximation shows more detail) to
c obtain closer fits.
c to economize the search for a good s-value the program provides with
c different modes of computation. at the first call of the routine, or
c whenever he wants to restart with the initial set of knots the user
c must set iopt=0.
c if iopt = 1 the program will continue with the knots found at
c the last call of the routine. this will save a lot of computation
c time if parsur is called repeatedly for different values of s.
c the number of knots of the surface returned and their location will
c depend on the value of s and on the complexity of the shape of the
c surface underlying the data. if the computation mode iopt = 1
c is used, the knots returned may also depend on the s-values at
c previous calls (if these were smaller). therefore, if after a number
c of trials with different s-values and iopt=1,the user can finally
c accept a fit as satisfactory, it may be worthwhile for him to call
c parsur once more with the chosen value for s but now with iopt=0.
c indeed, parsur may then return an approximation of the same quality
c of fit but with fewer knots and therefore better if data reduction
c is also an important objective for the user.
c the number of knots may also depend on the upper bounds nuest and
c nvest. indeed, if at a certain stage in parsur the number of knots
c in one direction (say nu) has reached the value of its upper bound
c (nuest), then from that moment on all subsequent knots are added
c in the other (v) direction. this may indicate that the value of
c nuest is too small. on the other hand, it gives the user the option
c of limiting the number of knots the routine locates in any direction
c for example, by setting nuest=8 (the lowest allowable value for
c nuest), the user can indicate that he wants an approximation with
c splines which are simple cubic polynomials in the variable u.
c
c other subroutines required:
c fppasu,fpchec,fpchep,fpknot,fprati,fpgrpa,fptrnp,fpback,
c fpbacp,fpbspl,fptrpe,fpdisc,fpgivs,fprota
c
c author:
c p.dierckx
c dept. computer science, k.u. leuven
c celestijnenlaan 200a, b-3001 heverlee, belgium.
c e-mail : Paul.Dierckx@cs.kuleuven.ac.be
c
c latest update : march 1989
c
c ..
c ..scalar arguments..
real s,fp
integer iopt,idim,mu,mv,nuest,nvest,nu,nv,lwrk,kwrk,ier
c ..array arguments..
real u(mu),v(mv),f(mu*mv*idim),tu(nuest),tv(nvest),
* c((nuest-4)*(nvest-4)*idim),wrk(lwrk)
integer ipar(2),iwrk(kwrk)
c ..local scalars..
real tol,ub,ue,vb,ve,peru,perv
integer i,j,jwrk,kndu,kndv,knru,knrv,kwest,l1,l2,l3,l4,
* lfpu,lfpv,lwest,lww,maxit,nc,mf,mumin,mvmin
c ..function references..
integer max0
c ..subroutine references..
c fppasu,fpchec,fpchep
c ..
c we set up the parameters tol and maxit.
maxit = 20
tol = 0.1e-02
c before starting computations a data check is made. if the input data
c are invalid, control is immediately repassed to the calling program.
ier = 10
if(iopt.lt.(-1) .or. iopt.gt.1) go to 200
if(ipar(1).lt.0 .or. ipar(1).gt.1) go to 200
if(ipar(2).lt.0 .or. ipar(2).gt.1) go to 200
if(idim.le.0 .or. idim.gt.3) go to 200
mumin = 4-2*ipar(1)
if(mu.lt.mumin .or. nuest.lt.8) go to 200
mvmin = 4-2*ipar(2)
if(mv.lt.mvmin .or. nvest.lt.8) go to 200
mf = mu*mv
nc = (nuest-4)*(nvest-4)
lwest = 4+nuest*(mv*idim+11+4*ipar(1))+nvest*(11+4*ipar(2))+
* 4*(mu+mv)+max0(nuest,mv)*idim
kwest = 3+mu+mv+nuest+nvest
if(lwrk.lt.lwest .or. kwrk.lt.kwest) go to 200
do 10 i=2,mu
if(u(i-1).ge.u(i)) go to 200
10 continue
do 20 i=2,mv
if(v(i-1).ge.v(i)) go to 200
20 continue
if(iopt.ge.0) go to 100
if(nu.lt.8 .or. nu.gt.nuest) go to 200
ub = u(1)
ue = u(mu)
if (ipar(1).ne.0) go to 40
j = nu
do 30 i=1,4
tu(i) = ub
tu(j) = ue
j = j-1
30 continue
call fpchec(u,mu,tu,nu,3,ier)
if(ier.ne.0) go to 200
go to 60
40 l1 = 4
l2 = l1
l3 = nu-3
l4 = l3
peru = ue-ub
tu(l2) = ub
tu(l3) = ue
do 50 j=1,3
l1 = l1+1
l2 = l2-1
l3 = l3+1
l4 = l4-1
tu(l2) = tu(l4)-peru
tu(l3) = tu(l1)+peru
50 continue
call fpchep(u,mu,tu,nu,3,ier)
if(ier.ne.0) go to 200
60 if(nv.lt.8 .or. nv.gt.nvest) go to 200
vb = v(1)
ve = v(mv)
if (ipar(2).ne.0) go to 80
j = nv
do 70 i=1,4
tv(i) = vb
tv(j) = ve
j = j-1
70 continue
call fpchec(v,mv,tv,nv,3,ier)
if(ier.ne.0) go to 200
go to 150
80 l1 = 4
l2 = l1
l3 = nv-3
l4 = l3
perv = ve-vb
tv(l2) = vb
tv(l3) = ve
do 90 j=1,3
l1 = l1+1
l2 = l2-1
l3 = l3+1
l4 = l4-1
tv(l2) = tv(l4)-perv
tv(l3) = tv(l1)+perv
90 continue
call fpchep(v,mv,tv,nv,3,ier)
if(ier) 200,150,200
100 if(s.lt.0.) go to 200
if(s.eq.0. .and. (nuest.lt.(mu+4+2*ipar(1)) .or.
* nvest.lt.(mv+4+2*ipar(2))) )go to 200
ier = 0
c we partition the working space and determine the spline approximation
150 lfpu = 5
lfpv = lfpu+nuest
lww = lfpv+nvest
jwrk = lwrk-4-nuest-nvest
knru = 4
knrv = knru+mu
kndu = knrv+mv
kndv = kndu+nuest
call fppasu(iopt,ipar,idim,u,mu,v,mv,f,mf,s,nuest,nvest,
* tol,maxit,nc,nu,tu,nv,tv,c,fp,wrk(1),wrk(2),wrk(3),wrk(4),
* wrk(lfpu),wrk(lfpv),iwrk(1),iwrk(2),iwrk(3),iwrk(knru),
* iwrk(knrv),iwrk(kndu),iwrk(kndv),wrk(lww),jwrk,ier)
200 return
end
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