subroutine polar(iopt,m,x,y,z,w,rad,s,nuest,nvest,eps,nu,tu,
* nv,tv,u,v,c,fp,wrk1,lwrk1,wrk2,lwrk2,iwrk,kwrk,ier)
c subroutine polar fits a smooth function f(x,y) to a set of data
c points (x(i),y(i),z(i)) scattered arbitrarily over an approximation
c domain x**2+y**2 <= rad(atan(y/x))**2. through the transformation
c x = u*rad(v)*cos(v) , y = u*rad(v)*sin(v)
c the approximation problem is reduced to the determination of a bi-
c cubic spline s(u,v) fitting a corresponding set of data points
c (u(i),v(i),z(i)) on the rectangle 0<=u<=1,-pi<=v<=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) -pi <=v<= pi.
c
c d s(0,v)
c (2) -------- = rad(v)*(cos(v)*g(1,0)+sin(v)*g(0,1))
c d u
c -pi <=v<= pi
c 2
c d s(0,v) 2 2 2
c (3) -------- = rad(v)*(cos(v)*g(2,0)+sin(v)*g(0,2)+sin(2*v)*g(1,1))
c 2
c d u -pi <=v<= pi
c
c moreover, s(u,v) must be periodic in the variable v, i.e.
c
c j j
c d s(u,-pi) d s(u,pi)
c (4) ---------- = --------- 0 <=u<= 1, j=0,1,2
c j j
c d v d v
c
c if iopt(1) < 0 circle calculates a weighted least-squares spline
c according to a given set of knots in u- and v- direction.
c if iopt(1) >=0, the number of knots in each direction and their pos-
c ition tu(j),j=1,2,...,nu ; tv(j),j=1,2,...,nv are chosen automatical-
c ly by the routine. the smoothness of s(u,v) 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(u,v) is determined by
c the condition that fp = sum((w(i)*(z(i)-s(u(i),v(i))))**2) be <= s,
c with s a given non-negative constant.
c the bicubic spline is given in its standard b-spline representation
c and the corresponding function f(x,y) can be evaluated by means of
c function program evapol.
c
c calling sequence:
c call polar(iopt,m,x,y,z,w,rad,s,nuest,nvest,eps,nu,tu,
c * nv,tv,u,v,wrk1,lwrk1,wrk2,lwrk2,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 weighted
c least-squares polar spline (iopt(1)=-1) or a smoothing
c polar spline (iopt(1)=0 or 1) must be determined.
c if iopt(1)=0 the routine will start with an initial set of
c knots tu(i)=0,tu(i+4)=1,i=1,...,4;tv(i)=(2*i-9)*pi,i=1,...,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,
c if iopt(2)=1 conditions (1)+(2) must be fulfilled and
c if iopt(2)=2 conditions (1)+(2)+(3) 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 m : integer. on entry m must specify the number of data points.
c m >= 4-iopt(2)-iopt(3) unchanged on exit.
c x : real array of dimension at least (m).
c y : real array of dimension at least (m).
c z : real array of dimension at least (m).
c before entry, x(i),y(i),z(i) must be set to the co-ordinates
c of the i-th data point, for i=1,...,m. the order of the data
c 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 rad : real function subprogram defining the boundary of the approx-
c imation domain, i.e x = rad(v)*cos(v) , y = rad(v)*sin(v),
c -pi <= v <= pi.
c must be declared external in the calling (sub)program.
c s : real. on entry (in case iopt(1) >=0) s must specify the
c smoothing 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 resp.
c these numbers will also determine the storage space needed by
c the routine. nuest >= 8, nvest >= 8.
c in most practical situation nuest = nvest = 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 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
c is used, the value of nu should be left unchanged between
c subsequent calls.
c in case iopt(1)=-1,the value of nu must be 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
c of the interior knots tu(5),...,tu(nu-4) as well as the
c position of the additional knots tu(1)=...=tu(4)=0 and
c tu(nu-3)=...=tu(nu)=1 needed for the b-spline representation
c if the computation mode iopt(1)=1 is used,the values of
c tu(1),...,tu(nu) should be left unchanged between subsequent
c calls. if the computation mode iopt(1)=-1 is used,the values
c tu(5),...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
c is used, the value of nv should be left unchanged between
c subsequent 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
c tv(1),...,tv(nv) should be left unchanged between subsequent
c calls. if the computation mode iopt(1)=-1 is used,the values
c tv(5),...tv(nv-4) must be supplied by the user, before entry.
c see also the restrictions (ier=10).
c u : real array of dimension at least (m).
c v : real array of dimension at least (m).
c on succesful exit, u(i),v(i) contains the co-ordinates of
c the i-th data point with respect to the transformed rectan-
c gular approximation domain, for i=1,2,...,m.
c if the computation mode iopt(1)=1 is used the values of
c u(i),v(i) should be left unchanged between subsequent calls.
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 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)=1 is used the value of
c wrk1(1) 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=1+iopt(2)*(iopt(2)+3)/2+
c (nv-7)*(nu-5-iopt(2)-iopt(3)) with d(i) the i-th diagonal el-
c ement of the triangular matrix for calculating the b-spline
c coefficients.it includes those elements whose square is < eps
c which are treated as 0 in the case of 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 k = nuest-7, l = nvest-7, p = 1+iopt(2)*(iopt(2)+3)/2,
c q = k+2-iopt(2)-iopt(3) then
c lwrk1 >= 129+10*k+21*l+k*l+(p+l*q)*(1+8*l+p)+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 = (p+l*q+1)*(4*l+p)
c +p+l*q where p,l,q 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+(nuest-7)*(nvest-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 an interpolating
c 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 1+iopt(2)*(iopt(2)+3)/2+(nv-7)*(nu-5-iopt(2)-iopt(3))+ier
c 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 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 weighted least-squares
c polar 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 spline 1+iopt(2)*(iopt(2)+3)/2+(nv-7)*(nu-5-iopt(2)
c -iopt(3)) already exceeds 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 polar 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 polar 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)<=1 , 0<=iopt(2)<=2 , 0<=iopt(3)<=1 ,
c m>=4-iopt(2)-iopt(3) , nuest>=8 ,nvest >=8, 00, i=1,...,m
c lwrk1 >= 129+10*k+21*l+k*l+(p+l*q)*(1+8*l+p)+8*m
c kwrk >= m+(nuest-7)*(nvest-7)
c if iopt(1)=-1:9<=nu<=nuest,9+iopt(2)*(iopt(2)+1)<=nv<=nvest
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 nu,
c nv,tu,tv,wrk1,u,v and having adjusted the value of lwrk2
c and the dimension of the array wrk2 accordingly, the user
c can continue at the point the program was left, by calling
c polar with iopt(1)=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 z(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 z(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(1)=0.
c if iopt(1)=1 the program will continue with the set of knots found
c at the last call of the routine. this will save a lot of computation
c time if polar 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 polar once more with the selected value for s but now with iopt(1)=0
c indeed, polar 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 polar 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
c other subroutines required:
c fpback,fpbspl,fppola,fpdisc,fpgivs,fprank,fprati,fprota,fporde,
c fprppo
c
c references:
c dierckx p.: an algorithm for fitting data over a circle using tensor
c product splines,j.comp.appl.maths 15 (1986) 161-173.
c dierckx p.: an algorithm for fitting data on a circle using tensor
c product splines, report tw68, dept. computer science,
c k.u.leuven, 1984.
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 : june 1984
c latest update : march 1989
c
c ..
c ..scalar arguments..
real s,eps,fp
integer m,nuest,nvest,nu,nv,lwrk1,lwrk2,kwrk,ier
c ..array arguments..
real x(m),y(m),z(m),w(m),tu(nuest),tv(nvest),u(m),v(m),
* c((nuest-4)*(nvest-4)),wrk1(lwrk1),wrk2(lwrk2)
integer iopt(3),iwrk(kwrk)
c ..user specified function
real rad
c ..local scalars..
real tol,pi,dist,r,one
integer i,ib1,ib3,ki,kn,kwest,la,lbu,lcc,lcs,lro,j
* lbv,lco,lf,lff,lfp,lh,lq,lsu,lsv,lwest,maxit,ncest,ncc,nuu,
* nvv,nreg,nrint,nu4,nv4,iopt1,iopt2,iopt3,ipar,nvmin
c ..function references..
real atan2,sqrt
external rad
c ..subroutine references..
c fppola
c ..
c set up constants
one = 1
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 60
iopt1 = iopt(1)
if(iopt1.lt.(-1) .or. iopt1.gt.1) go to 60
iopt2 = iopt(2)
if(iopt2.lt.0 .or. iopt2.gt.2) go to 60
iopt3 = iopt(3)
if(iopt3.lt.0 .or. iopt3.gt.1) go to 60
if(m.lt.(4-iopt2-iopt3)) go to 60
if(nuest.lt.8 .or. nvest.lt.8) go to 60
nu4 = nuest-4
nv4 = nvest-4
ncest = nu4*nv4
nuu = nuest-7
nvv = nvest-7
ipar = 1+iopt2*(iopt2+3)/2
ncc = ipar+nvv*(nuest-5-iopt2-iopt3)
nrint = nuu+nvv
nreg = nuu*nvv
ib1 = 4*nvv
ib3 = ib1+ipar
lwest = ncc*(1+ib1+ib3)+2*nrint+ncest+m*8+ib3+5*nuest+12*nvest
kwest = m+nreg
if(lwrk1.lt.lwest .or. kwrk.lt.kwest) go to 60
if(iopt1.gt.0) go to 40
do 10 i=1,m
if(w(i).le.0.) go to 60
dist = x(i)**2+y(i)**2
u(i) = 0.
v(i) = 0.
if(dist.le.0.) go to 10
v(i) = atan2(y(i),x(i))
r = rad(v(i))
if(r.le.0.) go to 60
u(i) = sqrt(dist)/r
if(u(i).gt.one) go to 60
10 continue
if(iopt1.eq.0) go to 40
nuu = nu-8
if(nuu.lt.1 .or. nu.gt.nuest) go to 60
tu(4) = 0.
do 20 i=1,nuu
j = i+4
if(tu(j).le.tu(j-1) .or. tu(j).ge.one) go to 60
20 continue
nvv = nv-8
nvmin = 9+iopt2*(iopt2+1)
if(nv.lt.nvmin .or. nv.gt.nvest) go to 60
pi = atan2(0.,-one)
tv(4) = -pi
do 30 i=1,nvv
j = i+4
if(tv(j).le.tv(j-1) .or. tv(j).ge.pi) go to 60
30 continue
go to 50
40 if(s.lt.0.) go to 60
50 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
lbu = lh+ib3
lbv = lbu+5*nuest
lro = lbv+5*nvest
lcc = lro+nvest
lcs = lcc+nvest
lsu = lcs+nvest*5
lsv = lsu+m*4
call fppola(iopt1,iopt2,iopt3,m,u,v,z,w,rad,s,nuest,nvest,eps,tol,
* maxit,ib1,ib3,ncest,ncc,nrint,nreg,nu,tu,nv,tv,c,fp,wrk1(1),
* wrk1(lfp),wrk1(lco),wrk1(lf),wrk1(lff),wrk1(lro),wrk1(lcc),
* wrk1(lcs),wrk1(la),wrk1(lq),wrk1(lbu),wrk1(lbv),wrk1(lsu),
* wrk1(lsv),wrk1(lh),iwrk(ki),iwrk(kn),wrk2,lwrk2,ier)
60 return
end