subroutine sphere(iopt,m,teta,phi,r,w,s,ntest,npest,eps,
* nt,tt,np,tp,c,fp,wrk1,lwrk1,wrk2,lwrk2,iwrk,kwrk,ier)
c subroutine sphere determines a smooth bicubic spherical spline
c approximation s(teta,phi), 0 <= teta <= pi ; 0 <= phi <= 2*pi
c to a given set of data points (teta(i),phi(i),r(i)),i=1,2,...,m.
c such a spline has the following specific properties
c
c (1) s(0,phi) = constant 0 <=phi<= 2*pi.
c
c (2) s(pi,phi) = constant 0 <=phi<= 2*pi
c
c j j
c d s(teta,0) d s(teta,2*pi)
c (3) ----------- = ------------ 0 <=teta<=pi, j=0,1,2
c j j
c d phi d phi
c
c d s(0,phi) d s(0,0) d s(0,pi/2)
c (4) ---------- = -------- *cos(phi) + ----------- *sin(phi)
c d teta d teta d teta
c
c d s(pi,phi) d s(pi,0) d s(pi,pi/2)
c (5) ----------- = ---------*cos(phi) + ------------*sin(phi)
c d teta d teta d teta
c
c if iopt =-1 sphere calculates a weighted least-squares spherical
c spline according to a given set of knots in teta- and phi- direction.
c if iopt >=0, the number of knots in each direction and their position
c tt(j),j=1,2,...,nt ; tp(j),j=1,2,...,np are chosen automatically by
c the routine. the smoothness of s(teta,phi) is then achieved by mini-
c malizing the discontinuity jumps of the derivatives of the spline
c at the knots. the amount of smoothness of s(teta,phi) is determined
c by the condition that fp = sum((w(i)*(r(i)-s(teta(i),phi(i))))**2)
c be <= s, with s a given non-negative constant.
c the spherical spline is given in the standard b-spline representation
c of bicubic splines and can be evaluated by means of subroutine bispev
c
c calling sequence:
c call sphere(iopt,m,teta,phi,r,w,s,ntest,npest,eps,
c * nt,tt,np,tp,c,fp,wrk1,lwrk1,wrk2,lwrk2,iwrk,kwrk,ier)
c
c parameters:
c iopt : integer flag. on entry iopt must specify whether a weighted
c least-squares spherical spline (iopt=-1) or a smoothing
c spherical spline (iopt=0 or 1) must be determined.
c if iopt=0 the routine will start with an initial set of knots
c tt(i)=0,tt(i+4)=pi,i=1,...,4;tp(i)=0,tp(i+4)=2*pi,i=1,...,4.
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 pre-
c ceded by another call with iopt=1 or iopt=0.
c unchanged on exit.
c m : integer. on entry m must specify the number of data points.
c m >= 2. unchanged on exit.
c teta : real array of dimension at least (m).
c phi : real array of dimension at least (m).
c r : real array of dimension at least (m).
c before entry,teta(i),phi(i),r(i) must be set to the spherical
c co-ordinates of the i-th data point, for i=1,...,m.the order
c of the data points is immaterial. unchanged on exit.
c w : real array of dimension at least (m). before entry, w(i) must
c be set to the i-th value in the set of weights. the w(i) must
c be strictly positive. unchanged on exit.
c s : real. on entry (in case 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 ntest : integer. unchanged on exit.
c npest : integer. unchanged on exit.
c on entry, ntest and npest must specify an upper bound for the
c number of knots required in the teta- and phi-directions.
c these numbers will also determine the storage space needed by
c the routine. ntest >= 8, npest >= 8.
c in most practical situation ntest = npest = 8+sqrt(m/2) will
c be sufficient. see also further comments.
c eps : real.
c on entry, eps must specify a threshold for determining the
c effective rank of an over-determined linear system of equat-
c ions. 0 < eps < 1. if the number of decimal digits in the
c computer representation of a real number is q, then 10**(-q)
c is a suitable value for eps in most practical applications.
c unchanged on exit.
c nt : integer.
c unless ier=10 (in case iopt >=0), nt will contain the total
c number of knots with respect to the teta-variable, of the
c spline approximation returned. if the computation mode iopt=1
c is used, the value of nt should be left unchanged between
c subsequent calls.
c in case iopt=-1, the value of nt should be specified on entry
c tt : real array of dimension at least ntest.
c on succesful exit, this array will contain the knots of the
c spline with respect to the teta-variable, i.e. the position
c of the interior knots tt(5),...,tt(nt-4) as well as the
c position of the additional knots tt(1)=...=tt(4)=0 and
c tt(nt-3)=...=tt(nt)=pi needed for the b-spline representation
c if the computation mode iopt=1 is used, the values of tt(1),
c ...,tt(nt) should be left unchanged between subsequent calls.
c if the computation mode iopt=-1 is used, the values tt(5),
c ...tt(nt-4) must be supplied by the user, before entry.
c see also the restrictions (ier=10).
c np : integer.
c unless ier=10 (in case iopt >=0), np will contain the total
c number of knots with respect to the phi-variable, of the
c spline approximation returned. if the computation mode iopt=1
c is used, the value of np should be left unchanged between
c subsequent calls.
c in case iopt=-1, the value of np (>=9) should be specified
c on entry.
c tp : real array of dimension at least npest.
c on succesful exit, this array will contain the knots of the
c spline with respect to the phi-variable, i.e. the position of
c the interior knots tp(5),...,tp(np-4) as well as the position
c of the additional knots tp(1),...,tp(4) and tp(np-3),...,
c tp(np) needed for the b-spline representation.
c if the computation mode iopt=1 is used, the values of tp(1),
c ...,tp(np) should be left unchanged between subsequent calls.
c if the computation mode iopt=-1 is used, the values tp(5),
c ...tp(np-4) must be supplied by the user, before entry.
c see also the restrictions (ier=10).
c c : real array of dimension at least (ntest-4)*(npest-4).
c on succesful exit, c contains the coefficients of the spline
c approximation s(teta,phi).
c fp : real. unless ier=10, fp contains the weighted sum of
c squared residuals of the spline approximation returned.
c wrk1 : real array of dimension (lwrk1). used as workspace.
c if the computation mode iopt=1 is used the value of wrk1(1)
c should be left unchanged between subsequent calls.
c on exit wrk1(2),wrk1(3),...,wrk1(1+ncof) will contain the
c values d(i)/max(d(i)),i=1,...,ncof=6+(np-7)*(nt-8)
c with d(i) the i-th diagonal element of the reduced triangular
c matrix for calculating the b-spline coefficients. it includes
c those elements whose square is less than eps,which are treat-
c ed as 0 in the case of presumed rank deficiency (ier<-2).
c lwrk1 : integer. on entry lwrk1 must specify the actual dimension of
c the array wrk1 as declared in the calling (sub)program.
c lwrk1 must not be too small. let
c u = ntest-7, v = npest-7, then
c lwrk1 >= 185+52*v+10*u+14*u*v+8*(u-1)*v**2+8*m
c wrk2 : real array of dimension (lwrk2). used as workspace, but
c only in the case a rank deficient system is encountered.
c lwrk2 : integer. on entry lwrk2 must specify the actual dimension of
c the array wrk2 as declared in the calling (sub)program.
c lwrk2 > 0 . a save upper bound for lwrk2 = 48+21*v+7*u*v+
c 4*(u-1)*v**2 where u,v are as above. if there are enough data
c points, scattered uniformly over the approximation domain
c and if the smoothing factor s is not too small, there is a
c good chance that this extra workspace is not needed. a lot
c of memory might therefore be saved by setting lwrk2=1.
c (see also ier > 10)
c iwrk : integer array of dimension (kwrk). used as workspace.
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 >= m+(ntest-7)*(npest-7).
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 a spherical
c interpolating spline (fp=0).
c ier=-2 : normal return. the spline returned is the weighted least-
c squares constrained polynomial . in this extreme case
c fp gives the upper bound for the smoothing factor s.
c ier<-2 : warning. the coefficients of the spline returned have been
c computed as the minimal norm least-squares solution of a
c (numerically) rank deficient system. (-ier) gives the rank.
c especially if the rank deficiency which can be computed as
c 6+(nt-8)*(np-7)+ier, is large the results may be inaccurate
c they could also seriously depend on the value of eps.
c ier=1 : error. the required storage space exceeds the available
c storage space, as specified by the parameters ntest and
c npest.
c probably causes : ntest or npest 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 weighted least-squares
c spherical spline according to the current set of knots.
c the parameter fp gives the corresponding weighted sum of
c 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 or badly chosen eps.
c there is an approximation returned but the corresponding
c weighted sum of squared residuals does not satisfy the
c condition 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 weighted sum of squared residuals does not satisfy the
c condition abs(fp-s)/s < tol.
c ier=4 : error. no more knots can be added because the dimension
c of the spherical spline 6+(nt-8)*(np-7) already exceeds
c the number of data points m.
c probably causes : either s or m too small.
c the approximation returned is the weighted least-squares
c spherical spline according to the current set of knots.
c the parameter fp gives the corresponding weighted sum of
c squared residuals (fp>s).
c ier=5 : error. no more knots can be added because the additional
c knot would (quasi) coincide with an old one.
c probably causes : s too small or too large a weight to an
c inaccurate data point.
c the approximation returned is the weighted least-squares
c spherical spline according to the current set of knots.
c the parameter fp gives the corresponding weighted sum of
c squared residuals (fp>s).
c ier=10 : error. on entry, the input data are controlled on validity
c the following restrictions must be satisfied.
c -1<=iopt<=1, m>=2, ntest>=8 ,npest >=8, 00, i=1,...,m
c lwrk1 >= 185+52*v+10*u+14*u*v+8*(u-1)*v**2+8*m
c kwrk >= m+(ntest-7)*(npest-7)
c if iopt=-1: 8<=nt<=ntest , 9<=np<=npest
c 0`=0: s>=0
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 ier>10 : error. lwrk2 is too small, i.e. there is not enough work-
c space for computing the minimal least-squares solution of
c a rank deficient system of linear equations. ier gives the
c requested value for lwrk2. there is no approximation re-
c turned but, having saved the information contained in nt,
c np,tt,tp,wrk1, and having adjusted the value of lwrk2 and
c the dimension of the array wrk2 accordingly, the user can
c continue at the point the program was left, by calling
c sphere with iopt=1.
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 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 weighted least-squares polynomial 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 weights w(i). if these are
c taken as 1/d(i) with d(i) an estimate of the standard deviation of
c r(i), a good s-value should be found in the range (m-sqrt(2*m),m+
c sqrt(2*m)). if nothing is known about the statistical error in r(i)
c each w(i) can be set equal to one and s determined by trial and
c error, taking account of the comments above. the best is then to
c start with a very large value of s ( to determine the least-squares
c polynomial and the corresponding upper bound fp0 for s) and then to
c progressively decrease the value of s ( say by a factor 10 in the
c beginning, i.e. s=fp0/10, fp0/100,...and more carefully as the
c approximation shows more detail) to obtain closer fits.
c to choose s very small is strongly discouraged. this considerably
c increases computation time and memory requirements. it may also
c cause rank-deficiency (ier<-2) and endager numerical stability.
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 set of knots found at
c the last call of the routine. this will save a lot of computation
c time if sphere 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
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 sphere once more with the selected value for s but now with iopt=0.
c indeed, sphere 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 ntest and
c npest. indeed, if at a certain stage in sphere the number of knots
c in one direction (say nt) has reached the value of its upper bound
c (ntest), then from that moment on all subsequent knots are added
c in the other (phi) direction. this may indicate that the value of
c ntest 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 ntest=8 (the lowest allowable value for
c ntest), the user can indicate that he wants an approximation which
c is a cubic polynomial in the variable teta.
c
c other subroutines required:
c fpback,fpbspl,fpsphe,fpdisc,fpgivs,fprank,fprati,fprota,fporde,
c fprpsp
c
c references:
c dierckx p. : algorithms for smoothing data on the sphere with tensor
c product splines, computing 32 (1984) 319-342.
c dierckx p. : algorithms for smoothing data on the sphere with tensor
c product splines, report tw62, dept. computer science,
c k.u.leuven, 1983.
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 1983
c latest update : march 1989
c
c ..
c ..scalar arguments..
real s,eps,fp
integer iopt,m,ntest,npest,nt,np,lwrk1,lwrk2,kwrk,ier
c ..array arguments..
real teta(m),phi(m),r(m),w(m),tt(ntest),tp(npest),
* c((ntest-4)*(npest-4)),wrk1(lwrk1),wrk2(lwrk2)
integer iwrk(kwrk)
c ..local scalars..
real tol,pi,pi2,one
integer i,ib1,ib3,ki,kn,kwest,la,lbt,lcc,lcs,lro,j
* lbp,lco,lf,lff,lfp,lh,lq,lst,lsp,lwest,maxit,ncest,ncc,ntt,
* npp,nreg,nrint,ncof,nt4,np4
c ..function references..
real atan
c ..subroutine references..
c fpsphe
c ..
c set constants
one = 0.1e+01
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(eps.le.0. .or. eps.ge.1.) go to 80
if(iopt.lt.(-1) .or. iopt.gt.1) go to 80
if(m.lt.2) go to 80
if(ntest.lt.8 .or. npest.lt.8) go to 80
nt4 = ntest-4
np4 = npest-4
ncest = nt4*np4
ntt = ntest-7
npp = npest-7
ncc = 6+npp*(ntt-1)
nrint = ntt+npp
nreg = ntt*npp
ncof = 6+3*npp
ib1 = 4*npp
ib3 = ib1+3
if(ncof.gt.ib1) ib1 = ncof
if(ncof.gt.ib3) ib3 = ncof
lwest = 185+52*npp+10*ntt+14*ntt*npp+8*(m+(ntt-1)*npp**2)
kwest = m+nreg
if(lwrk1.lt.lwest .or. kwrk.lt.kwest) go to 80
if(iopt.gt.0) go to 60
pi = atan(one)*4
pi2 = pi+pi
do 20 i=1,m
if(w(i).le.0.) go to 80
if(teta(i).lt.0. .or. teta(i).gt.pi) go to 80
if(phi(i) .lt.0. .or. phi(i).gt.pi2) go to 80
20 continue
if(iopt.eq.0) go to 60
ntt = nt-8
if(ntt.lt.0 .or. nt.gt.ntest) go to 80
if(ntt.eq.0) go to 40
tt(4) = 0.
do 30 i=1,ntt
j = i+4
if(tt(j).le.tt(j-1) .or. tt(j).ge.pi) go to 80
30 continue
40 npp = np-8
if(npp.lt.1 .or. np.gt.npest) go to 80
tp(4) = 0.
do 50 i=1,npp
j = i+4
if(tp(j).le.tp(j-1) .or. tp(j).ge.pi2) go to 80
50 continue
go to 70
60 if(s.lt.0.) go to 80
70 ier = 0
c we partition the working space and determine the spline approximation
kn = 1
ki = kn+m
lq = 2
la = lq+ncc*ib3
lf = la+ncc*ib1
lff = lf+ncc
lfp = lff+ncest
lco = lfp+nrint
lh = lco+nrint
lbt = lh+ib3
lbp = lbt+5*ntest
lro = lbp+5*npest
lcc = lro+npest
lcs = lcc+npest
lst = lcs+npest
lsp = lst+m*4
call fpsphe(iopt,m,teta,phi,r,w,s,ntest,npest,eps,tol,maxit,
* ib1,ib3,ncest,ncc,nrint,nreg,nt,tt,np,tp,c,fp,wrk1(1),wrk1(lfp),
* wrk1(lco),wrk1(lf),wrk1(lff),wrk1(lro),wrk1(lcc),wrk1(lcs),
* wrk1(la),wrk1(lq),wrk1(lbt),wrk1(lbp),wrk1(lst),wrk1(lsp),
* wrk1(lh),iwrk(ki),iwrk(kn),wrk2,lwrk2,ier)
80 return
end
`