subroutine pogrid(iopt,ider,mu,u,mv,v,z,z0,r,s,nuest,nvest,
* nu,tu,nv,tv,c,fp,wrk,lwrk,iwrk,kwrk,ier)
c subroutine pogrid fits a function f(x,y) to a set of data points
c z(i,j) given at the nodes (x,y)=(u(i)*cos(v(j)),u(i)*sin(v(j))),
c i=1,...,mu ; j=1,...,mv , of a radius-angle grid over a disc
c x ** 2 + y ** 2 <= r ** 2 .
c
c this approximation problem is reduced to the determination of a
c bicubic spline s(u,v) smoothing the data (u(i),v(j),z(i,j)) on the
c rectangle 0<=u<=r, v(1)<=v<=v(1)+2*pi
c in order to have continuous partial derivatives
c i+j
c d f(0,0)
c g(i,j) = ----------
c i j
c dx dy
c
c s(u,v)=f(x,y) must satisfy the following conditions
c
c (1) s(0,v) = g(0,0) v(1)<=v<= v(1)+2*pi
c
c d s(0,v)
c (2) -------- = cos(v)*g(1,0)+sin(v)*g(0,1) v(1)<=v<= v(1)+2*pi
c d u
c
c moreover, s(u,v) must be periodic in the variable v, i.e.
c
c j j
c d s(u,vb) d s(u,ve)
c (3) ---------- = --------- 0 <=u<= r, j=0,1,2 , vb=v(1),
c j j ve=vb+2*pi
c d v d v
c
c the number of knots of s(u,v) and their position tu(i),i=1,2,...,nu;
c tv(j),j=1,2,...,nv, is chosen automatically by the routine. the
c smoothness of s(u,v) is achieved by minimalizing the discontinuity
c jumps of the derivatives of the spline at the knots. the amount of
c smoothness of s(u,v) is determined by the condition that
c fp=sumi=1,mu(sumj=1,mv((z(i,j)-s(u(i),v(j)))**2))+(z0-g(0,0))**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 bispev. f(x,y) = s(u,v) can also be
c evaluated by means of function program evapol.
c
c calling sequence:
c call pogrid(iopt,ider,mu,u,mv,v,z,z0,r,s,nuest,nvest,nu,tu,
c * ,nv,tv,c,fp,wrk,lwrk,iwrk,kwrk,ier)
c
c parameters:
c iopt : integer array of dimension 3, specifying different options.
c unchanged on exit.
c iopt(1):on entry iopt(1) must specify whether a least-squares spline
c (iopt(1)=-1) or a smoothing spline (iopt(1)=0 or 1) must be
c determined.
c if iopt(1)=0 the routine will start with an initial set of
c knots tu(i)=0,tu(i+4)=r,i=1,...,4;tv(i)=v(1)+(i-4)*2*pi,i=1,.
c ...,8.
c if iopt(1)=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)=1 must always be immediately
c preceded by another call with iopt(1) = 1 or iopt(1) = 0.
c iopt(2):on entry iopt(2) must specify the requested order of conti-
c nuity for f(x,y) at the origin.
c if iopt(2)=0 only condition (1) must be fulfilled and
c if iopt(2)=1 conditions (1)+(2) must be fulfilled.
c iopt(3):on entry iopt(3) must specify whether (iopt(3)=1) or not
c (iopt(3)=0) the approximation f(x,y) must vanish at the
c boundary of the approximation domain.
c ider : integer array of dimension 2, specifying different options.
c unchanged on exit.
c ider(1):on entry ider(1) must specify whether (ider(1)=0 or 1) or not
c (ider(1)=-1) there is a data value z0 at the origin.
c if ider(1)=1, z0 will be considered to be the right function
c value, and it will be fitted exactly (g(0,0)=z0=c(1)).
c if ider(1)=0, z0 will be considered to be a data value just
c like the other data values z(i,j).
c ider(2):on entry ider(2) must specify whether (ider(2)=1) or not
c (ider(2)=0) f(x,y) must have vanishing partial derivatives
c g(1,0) and g(0,1) at the origin. (in case iopt(2)=1)
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-iopt(3)-ider(2) if ider(1)<0
c =3-iopt(3)-ider(2) if ider(1)>=0
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 positive and supplied in strictly ascending order.
c unchanged on exit.
c mv : integer. on entry mv must specify the number of grid points
c along the v-axis. mv > 3 . unchanged on exit.
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 -pi <= v(1) < pi , v(mv) < v(1)+2*pi.
c z : real array of dimension at least (mu*mv).
c before entry, z(mv*(i-1)+j) must be set to the data value at
c the grid point (u(i),v(j)) for i=1,...,mu and j=1,...,mv.
c unchanged on exit.
c z0 : real value. on entry (if ider(1) >=0 ) z0 must specify the
c data value at the origin. unchanged on exit.
c r : real value. on entry r must specify the radius of the disk.
c r>=u(mu) (>u(mu) if iopt(3)=1). unchanged on exit.
c s : real. on entry (if iopt(1)>=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+5+iopt(2)+
c iopt(3), nvest = mv+7, the number of knots needed for
c interpolation (s=0). see also further comments.
c nu : integer.
c unless ier=10 (in case iopt(1)>=0), nu will contain the total
c number of knots with respect to the u-variable, of the spline
c approximation returned. if the computation mode iopt(1)=1 is
c used, the value of nu should be left unchanged between sub-
c sequent calls. in case iopt(1)=-1, the value of nu should be
c specified on entry.
c tu : real array of dimension at least (nuest).
c on succesful exit, this array will contain the knots of the
c spline 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)=0 and tu(nu-3)=...=
c tu(nu)=r needed for the b-spline representation.
c if the computation mode iopt(1)=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)=-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(1)>=0), nv will contain the total
c number of knots with respect to the v-variable, of the spline
c approximation returned. if the computation mode iopt(1)=1 is
c used, the value of nv should be left unchanged between sub-
c sequent calls. in case iopt(1) = -1, the value of nv should
c be specified on entry.
c tv : real array of dimension at least (nvest).
c on succesful exit, this array will contain the knots of the
c spline 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)=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)=-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).
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 approximation returned.
c wrk : real array of dimension (lwrk). used as workspace.
c if the computation mode iopt(1)=1 is used the values of
c wrk(1),...,wrk(8) 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 >= 8+nuest*(mv+nvest+3)+nvest*21+4*mu+6*mv+q
c where q is the larger of (mv+nvest) and nuest.
c iwrk : integer array of dimension (kwrk). used as workspace.
c if the computation mode iopt(1)=1 is used the values of
c iwrk(1),.,iwrk(4) 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 >= 4+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 spline 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 returned is an interpolating
c spline (fp=0).
c ier=-2 : normal return. the spline returned is the least-squares
c constrained polynomial. 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 spline
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 spline 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 spline
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)<=1, 0<=iopt(2)<=1, 0<=iopt(3)<=1,
c -1<=ider(1)<=1, 0<=ider(2)<=1, ider(2)=0 if iopt(2)=0.
c mu >= mumin (see above), mv >= 4, nuest >=8, nvest >= 8,
c kwrk>=4+mu+mv+nuest+nvest,
c lwrk >= 8+nuest*(mv+nvest+3)+nvest*21+4*mu+6*mv+
c max(nuest,mv+nvest)
c 0< u(i-1)__=0: s>=0
c if s=0: nuest>=mu+5+iopt(2)+iopt(3), nvest>=mv+7
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 pogrid does not allow individual weighting of the data-values.
c so, if these were determined to widely different accuracies, then
c perhaps the general data set routine polar should rather be used
c in spite of efficiency.
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 spline will be too smooth and signal will be
c lost ; if s is too small the spline will pick up too much noise. in
c the extreme cases the program will return an interpolating spline if
c s=0 and the constrained least-squares polynomial(degrees 3,0)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, pogrid will return a spline s(u,v) which
c exactly reproduces the function underlying the data he can evaluate
c the sum((z(i,j)-s(u(i),v(j)))**2) to find a good estimate for this s
c for example, if he knows that the statistical errors on his z(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 z(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 least-squares polynomial 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(1)=0.
c if iopt(1) = 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 pogrid is called repeatedly for different values of s.
c the number of knots of the spline returned and their location will
c depend on the value of s and on the complexity of the shape of the
c function underlying the data. if the computation mode iopt(1) = 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)=1,the user can finally
c accept a fit as satisfactory, it may be worthwhile for him to call
c pogrid once more with the chosen value for s but now with iopt(1)=0.
c indeed, pogrid 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 pogrid 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 which
c is a simple cubic polynomial in the variable u.
c
c other subroutines required:
c fppogr,fpchec,fpchep,fpknot,fpopdi,fprati,fpgrdi,fpsysy,fpback,
c fpbacp,fpbspl,fpcyt1,fpcyt2,fpdisc,fpgivs,fprota
c
c references:
c dierckx p. : fast algorithms for smoothing data over a disc or a
c sphere using tensor product splines, in "algorithms
c for approximation", ed. j.c.mason and m.g.cox,
c clarendon press oxford, 1987, pp. 51-65
c dierckx p. : fast algorithms for smoothing data over a disc or a
c sphere using tensor product splines, report tw73, dept.
c computer science,k.u.leuven, 1985.
c dierckx p. : curve and surface fitting with splines, monographs on
c numerical analysis, oxford university press, 1993.
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 creation date : july 1985
c latest update : march 1989
c
c ..
c ..scalar arguments..
real z0,r,s,fp
integer mu,mv,nuest,nvest,nu,nv,lwrk,kwrk,ier
c ..array arguments..
integer iopt(3),ider(2),iwrk(kwrk)
real u(mu),v(mv),z(mu*mv),c((nuest-4)*(nvest-4)),tu(nuest),
* tv(nvest),wrk(lwrk)
c ..local scalars..
real per,pi,tol,uu,ve,zmax,zmin,one,half,rn,zb
integer i,i1,i2,j,jwrk,j1,j2,kndu,kndv,knru,knrv,kwest,l,
* ldz,lfpu,lfpv,lwest,lww,m,maxit,mumin,muu,nc
c ..function references..
real atan2
integer max0
c ..subroutine references..
c fpchec,fpchep,fppogr
c ..
c set constants
one = 1
half = 0.5e0
pi = atan2(0.,-one)
per = pi+pi
ve = v(1)+per
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(1).lt.(-1) .or. iopt(1).gt.1) go to 200
if(iopt(2).lt.0 .or. iopt(2).gt.1) go to 200
if(iopt(3).lt.0 .or. iopt(3).gt.1) go to 200
if(ider(1).lt.(-1) .or. ider(1).gt.1) go to 200
if(ider(2).lt.0 .or. ider(2).gt.1) go to 200
if(ider(2).eq.1 .and. iopt(2).eq.0) go to 200
mumin = 4-iopt(3)-ider(2)
if(ider(1).ge.0) mumin = mumin-1
if(mu.lt.mumin .or. mv.lt.4) go to 200
if(nuest.lt.8 .or. nvest.lt.8) go to 200
m = mu*mv
nc = (nuest-4)*(nvest-4)
lwest = 8+nuest*(mv+nvest+3)+21*nvest+4*mu+6*mv+
* max0(nuest,mv+nvest)
kwest = 4+mu+mv+nuest+nvest
if(lwrk.lt.lwest .or. kwrk.lt.kwest) go to 200
if(u(1).le.0. .or. u(mu).gt.r) go to 200
if(iopt(3).eq.0) go to 10
if(u(mu).eq.r) go to 200
10 if(mu.eq.1) go to 30
do 20 i=2,mu
if(u(i-1).ge.u(i)) go to 200
20 continue
30 if(v(1).lt. (-pi) .or. v(1).ge.pi ) go to 200
if(v(mv).ge.v(1)+per) go to 200
do 40 i=2,mv
if(v(i-1).ge.v(i)) go to 200
40 continue
if(iopt(1).gt.0) go to 140
c if not given, we compute an estimate for z0.
if(ider(1).lt.0) go to 50
zb = z0
go to 70
50 zb = 0.
do 60 i=1,mv
zb = zb+z(i)
60 continue
rn = mv
zb = zb/rn
c we determine the range of z-values.
70 zmin = zb
zmax = zb
do 80 i=1,m
if(z(i).lt.zmin) zmin = z(i)
if(z(i).gt.zmax) zmax = z(i)
80 continue
wrk(5) = zb
wrk(6) = 0.
wrk(7) = 0.
wrk(8) = zmax -zmin
iwrk(4) = mu
if(iopt(1).eq.0) go to 140
if(nu.lt.8 .or. nu.gt.nuest) go to 200
if(nv.lt.11 .or. nv.gt.nvest) go to 200
j = nu
do 90 i=1,4
tu(i) = 0.
tu(j) = r
j = j-1
90 continue
l = 9
wrk(l) = 0.
if(iopt(2).eq.0) go to 100
l = l+1
uu = u(1)
if(uu.gt.tu(5)) uu = tu(5)
wrk(l) = uu*half
100 do 110 i=1,mu
l = l+1
wrk(l) = u(i)
110 continue
if(iopt(3).eq.0) go to 120
l = l+1
wrk(l) = r
120 muu = l-8
call fpchec(wrk(9),muu,tu,nu,3,ier)
if(ier.ne.0) go to 200
j1 = 4
tv(j1) = v(1)
i1 = nv-3
tv(i1) = ve
j2 = j1
i2 = i1
do 130 i=1,3
i1 = i1+1
i2 = i2-1
j1 = j1+1
j2 = j2-1
tv(j2) = tv(i2)-per
tv(i1) = tv(j1)+per
130 continue
l = 9
do 135 i=1,mv
wrk(l) = v(i)
l = l+1
135 continue
wrk(l) = ve
call fpchep(wrk(9),mv+1,tv,nv,3,ier)
if(ier) 200,150,200
140 if(s.lt.0.) go to 200
if(s.eq.0. .and. (nuest.lt.(mu+5+iopt(2)+iopt(3)) .or.
* nvest.lt.(mv+7)) ) go to 200
c we partition the working space and determine the spline approximation
150 ldz = 5
lfpu = 9
lfpv = lfpu+nuest
lww = lfpv+nvest
jwrk = lwrk-8-nuest-nvest
knru = 5
knrv = knru+mu
kndu = knrv+mv
kndv = kndu+nuest
call fppogr(iopt,ider,u,mu,v,mv,z,m,zb,r,s,nuest,nvest,tol,maxit,
* nc,nu,tu,nv,tv,c,fp,wrk(1),wrk(2),wrk(3),wrk(4),wrk(lfpu),
* wrk(lfpv),wrk(ldz),wrk(8),iwrk(1),iwrk(2),iwrk(3),iwrk(4),
* iwrk(knru),iwrk(knrv),iwrk(kndu),iwrk(kndv),wrk(lww),jwrk,ier)
200 return
end
__