subroutine concur(iopt,idim,m,u,mx,x,xx,w,ib,db,nb,ie,de,ne,k,s,
* nest,n,t,nc,c,np,cp,fp,wrk,lwrk,iwrk,ier)
c given the ordered set of m points x(i) in the idim-dimensional space
c and given also a corresponding set of strictly increasing values u(i)
c and the set of positive numbers w(i),i=1,2,...,m, subroutine concur
c determines a smooth approximating spline curve s(u), i.e.
c x1 = s1(u)
c x2 = s2(u) ub = u(1) <= u <= u(m) = ue
c .........
c xidim = sidim(u)
c with sj(u),j=1,2,...,idim spline functions of odd degree k with
c common knots t(j),j=1,2,...,n.
c in addition these splines will satisfy the following boundary
c constraints (l)
c if ib > 0 : sj (u(1)) = db(idim*l+j) ,l=0,1,...,ib-1
c and (l)
c if ie > 0 : sj (u(m)) = de(idim*l+j) ,l=0,1,...,ie-1.
c if iopt=-1 concur calculates the weighted least-squares spline curve
c according to a given set of knots.
c if iopt>=0 the number of knots of the splines sj(u) and the position
c t(j),j=1,2,...,n is chosen automatically by the routine. the smooth-
c ness of s(u) is then achieved by minimalizing the discontinuity
c jumps of the k-th derivative of s(u) at the knots t(j),j=k+2,k+3,...,
c n-k-1. the amount of smoothness is determined by the condition that
c f(p)=sum((w(i)*dist(x(i),s(u(i))))**2) be <= s, with s a given non-
c negative constant, called the smoothing factor.
c the fit s(u) is given in the b-spline representation and can be
c evaluated by means of subroutine curev.
c
c calling sequence:
c call concur(iopt,idim,m,u,mx,x,xx,w,ib,db,nb,ie,de,ne,k,s,nest,n,
c * t,nc,c,np,cp,fp,wrk,lwrk,iwrk,ier)
c
c parameters:
c iopt : integer flag. on entry iopt must specify whether a weighted
c least-squares spline curve (iopt=-1) or a smoothing spline
c curve (iopt=0 or 1) must be determined.if iopt=0 the routine
c will start with an initial set of knots t(i)=ub,t(i+k+1)=ue,
c i=1,2,...,k+1. if iopt=1 the routine will continue with the
c knots 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 unchanged on exit.
c idim : integer. on entry idim must specify the dimension of the
c curve. 0 < idim < 11.
c unchanged on exit.
c m : integer. on entry m must specify the number of data points.
c m > k-max(ib-1,0)-max(ie-1,0). unchanged on exit.
c u : real array of dimension at least (m). before entry,
c u(i) must be set to the i-th value of the parameter variable
c u for i=1,2,...,m. these values must be supplied in
c strictly ascending order and will be unchanged on exit.
c mx : integer. on entry mx must specify the actual dimension of
c the arrays x and xx as declared in the calling (sub)program
c mx must not be too small (see x). unchanged on exit.
c x : real array of dimension at least idim*m.
c before entry, x(idim*(i-1)+j) must contain the j-th coord-
c inate of the i-th data point for i=1,2,...,m and j=1,2,...,
c idim. unchanged on exit.
c xx : real array of dimension at least idim*m.
c used as working space. on exit xx contains the coordinates
c of the data points to which a spline curve with zero deriv-
c ative constraints has been determined.
c if the computation mode iopt =1 is used xx should be left
c unchanged between calls.
c w : real array of dimension at least (m). before entry, w(i)
c must be set to the i-th value in the set of weights. the
c w(i) must be strictly positive. unchanged on exit.
c see also further comments.
c ib : integer. on entry ib must specify the number of derivative
c constraints for the curve at the begin point. 0<=ib<=(k+1)/2
c unchanged on exit.
c db : real array of dimension nb. before entry db(idim*l+j) must
c contain the l-th order derivative of sj(u) at u=u(1) for
c j=1,2,...,idim and l=0,1,...,ib-1 (if ib>0).
c unchanged on exit.
c nb : integer, specifying the dimension of db. nb>=max(1,idim*ib)
c unchanged on exit.
c ie : integer. on entry ie must specify the number of derivative
c constraints for the curve at the end point. 0<=ie<=(k+1)/2
c unchanged on exit.
c de : real array of dimension ne. before entry de(idim*l+j) must
c contain the l-th order derivative of sj(u) at u=u(m) for
c j=1,2,...,idim and l=0,1,...,ie-1 (if ie>0).
c unchanged on exit.
c ne : integer, specifying the dimension of de. ne>=max(1,idim*ie)
c unchanged on exit.
c k : integer. on entry k must specify the degree of the splines.
c k=1,3 or 5.
c 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 nest : integer. on entry nest must contain an over-estimate of the
c total number of knots of the splines returned, to indicate
c the storage space available to the routine. nest >=2*k+2.
c in most practical situation nest=m/2 will be sufficient.
c always large enough is nest=m+k+1+max(0,ib-1)+max(0,ie-1),
c the number of knots needed for interpolation (s=0).
c unchanged on exit.
c n : integer.
c unless ier = 10 (in case iopt >=0), n will contain the
c total number of knots of the smoothing spline curve returned
c if the computation mode iopt=1 is used this value of n
c should be left unchanged between subsequent calls.
c in case iopt=-1, the value of n must be specified on entry.
c t : real array of dimension at least (nest).
c on succesful exit, this array will contain the knots of the
c spline curve,i.e. the position of the interior knots t(k+2),
c t(k+3),..,t(n-k-1) as well as the position of the additional
c t(1)=t(2)=...=t(k+1)=ub and t(n-k)=...=t(n)=ue needed for
c the b-spline representation.
c if the computation mode iopt=1 is used, the values of t(1),
c t(2),...,t(n) should be left unchanged between subsequent
c calls. if the computation mode iopt=-1 is used, the values
c t(k+2),...,t(n-k-1) must be supplied by the user, before
c entry. see also the restrictions (ier=10).
c nc : integer. on entry nc must specify the actual dimension of
c the array c as declared in the calling (sub)program. nc
c must not be too small (see c). unchanged on exit.
c c : real array of dimension at least (nest*idim).
c on succesful exit, this array will contain the coefficients
c in the b-spline representation of the spline curve s(u),i.e.
c the b-spline coefficients of the spline sj(u) will be given
c in c(n*(j-1)+i),i=1,2,...,n-k-1 for j=1,2,...,idim.
c cp : real array of dimension at least 2*(k+1)*idim.
c on exit cp will contain the b-spline coefficients of a
c polynomial curve which satisfies the boundary constraints.
c if the computation mode iopt =1 is used cp should be left
c unchanged between calls.
c np : integer. on entry np must specify the actual dimension of
c the array cp as declared in the calling (sub)program. np
c must not be too small (see cp). unchanged on exit.
c fp : real. unless ier = 10, fp contains the weighted sum of
c squared residuals of the spline curve returned.
c wrk : real array of dimension at least m*(k+1)+nest*(6+idim+3*k).
c used as working space. if the computation mode iopt=1 is
c used, the values wrk(1),...,wrk(n) should be left unchanged
c between subsequent calls.
c lwrk : integer. on entry,lwrk must specify the actual dimension of
c the array wrk as declared in the calling (sub)program. lwrk
c must not be too small (see wrk). unchanged on exit.
c iwrk : integer array of dimension at least (nest).
c used as working space. if the computation mode iopt=1 is
c used,the values iwrk(1),...,iwrk(n) should be left unchanged
c between subsequent calls.
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 curve 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 curve returned is an interpolating
c spline curve, satisfying the constraints (fp=0).
c ier=-2 : normal return. the curve returned is the weighted least-
c squares polynomial curve of degree k, satisfying the
c constraints. in this extreme case fp gives the upper
c bound fp0 for the smoothing factor s.
c ier=1 : error. the required storage space exceeds the available
c storage space, as specified by the parameter nest.
c probably causes : nest too small. if nest is already
c large (say nest > m/2), it may also indicate that s is
c too small
c the approximation returned is the least-squares spline
c curve according to the knots t(1),t(2),...,t(n). (n=nest)
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 curve
c with fp = s. 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=3 : error. the maximal number of iterations maxit (set to 20
c by the program) allowed for finding a smoothing curve
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=10 : error. on entry, the input data are controlled on validity
c the following restrictions must be satisfied.
c -1<=iopt<=1, k = 1,3 or 5, m>k-max(0,ib-1)-max(0,ie-1),
c nest>=2k+2, 0=(k+1)*m+nest*(6+idim+3*k),
c nc >=nest*idim ,u(1)__0 i=1,2,...,m,
c mx>=idim*m,0<=ib<=(k+1)/2,0<=ie<=(k+1)/2,nb>=1,ne>=1,
c nb>=ib*idim,ne>=ib*idim,np>=2*(k+1)*idim,
c if iopt=-1:2*k+2<=n<=min(nest,mmax) with mmax = m+k+1+
c max(0,ib-1)+max(0,ie-1)
c u(1)=0: s>=0
c if s=0 : nest >=mmax (see above)
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 curve will be too smooth and signal will be
c lost ; if s is too small the curve will pick up too much noise. in
c the extreme cases the program will return an interpolating curve if
c s=0 and the least-squares polynomial curve of degree k 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 x(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 x(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 curve and the 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 approximating curve shows more detail) to 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 set of knots found at
c the last call of the routine. this will save a lot of computation
c time if concur 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 curve underlying the data. but, if the computation mode iopt=1 is
c used, the knots returned may also depend on the s-values at previous
c calls (if these were smaller). therefore, if after a number of
c 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 concur once more with the selected value for s but now with iopt=0.
c indeed, concur 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
c the form of the approximating curve can strongly be affected by
c the choice of the parameter values u(i). if there is no physical
c reason for choosing a particular parameter u, often good results
c will be obtained with the choice
c v(1)=0, v(i)=v(i-1)+q(i), i=2,...,m, u(i)=v(i)/v(m), i=1,..,m
c where
c q(i)= sqrt(sum j=1,idim (xj(i)-xj(i-1))**2 )
c other possibilities for q(i) are
c q(i)= sum j=1,idim (xj(i)-xj(i-1))**2
c q(i)= sum j=1,idim abs(xj(i)-xj(i-1))
c q(i)= max j=1,idim abs(xj(i)-xj(i-1))
c q(i)= 1
c
c other subroutines required:
c fpback,fpbspl,fpched,fpcons,fpdisc,fpgivs,fpknot,fprati,fprota
c curev,fppocu,fpadpo,fpinst
c
c references:
c dierckx p. : algorithms for smoothing data with periodic and
c parametric splines, computer graphics and image
c processing 20 (1982) 171-184.
c dierckx p. : algorithms for smoothing data with periodic and param-
c etric splines, report tw55, dept. computer science,
c k.u.leuven, 1981.
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 : may 1979
c latest update : march 1987
c
c ..
c ..scalar arguments..
real s,fp
integer iopt,idim,m,mx,ib,nb,ie,ne,k,nest,n,nc,np,lwrk,ier
c ..array arguments..
real u(m),x(mx),xx(mx),db(nb),de(ne),w(m),t(nest),c(nc),wrk(lwrk)
real cp(np)
integer iwrk(nest)
c ..local scalars..
real tol,dist
integer i,ib1,ie1,ja,jb,jfp,jg,jq,jz,j,k1,k2,lwest,maxit,nmin,
* ncc,kk,mmin,nmax,mxx
c ..function references
integer max0
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 90
if(idim.le.0 .or. idim.gt.10) go to 90
if(k.le.0 .or. k.gt.5) go to 90
k1 = k+1
kk = k1/2
if(kk*2.ne.k1) go to 90
k2 = k1+1
if(ib.lt.0 .or. ib.gt.kk) go to 90
if(ie.lt.0 .or. ie.gt.kk) go to 90
nmin = 2*k1
ib1 = max0(0,ib-1)
ie1 = max0(0,ie-1)
mmin = k1-ib1-ie1
if(m.lt.mmin .or. nest.lt.nmin) go to 90
if(nb.lt.(idim*ib) .or. ne.lt.(idim*ie)) go to 90
if(np.lt.(2*k1*idim)) go to 90
mxx = m*idim
ncc = nest*idim
if(mx.lt.mxx .or. nc.lt.ncc) go to 90
lwest = m*k1+nest*(6+idim+3*k)
if(lwrk.lt.lwest) go to 90
if(w(1).le.0.) go to 90
do 10 i=2,m
if(u(i-1).ge.u(i) .or. w(i).le.0.) go to 90
10 continue
if(iopt.ge.0) go to 30
if(n.lt.nmin .or. n.gt.nest) go to 90
j = n
do 20 i=1,k1
t(i) = u(1)
t(j) = u(m)
j = j-1
20 continue
call fpched(u,m,t,n,k,ib,ie,ier)
if(ier) 90,40,90
30 if(s.lt.0.) go to 90
nmax = m+k1+ib1+ie1
if(s.eq.0. .and. nest.lt.nmax) go to 90
ier = 0
if(iopt.gt.0) go to 70
c we determine a polynomial curve satisfying the boundary constraints.
40 call fppocu(idim,k,u(1),u(m),ib,db,nb,ie,de,ne,cp,np)
c we generate new data points which will be approximated by a spline
c with zero derivative constraints.
j = nmin
do 50 i=1,k1
wrk(i) = u(1)
wrk(j) = u(m)
j = j-1
50 continue
c evaluate the polynomial curve
call curev(idim,wrk,nmin,cp,np,k,u,m,xx,mxx,ier)
c substract from the old data, the values of the polynomial curve
do 60 i=1,mxx
xx(i) = x(i)-xx(i)
60 continue
c we partition the working space and determine the spline curve.
70 jfp = 1
jz = jfp+nest
ja = jz+ncc
jb = ja+nest*k1
jg = jb+nest*k2
jq = jg+nest*k2
call fpcons(iopt,idim,m,u,mxx,xx,w,ib,ie,k,s,nest,tol,maxit,k1,
* k2,n,t,ncc,c,fp,wrk(jfp),wrk(jz),wrk(ja),wrk(jb),wrk(jg),wrk(jq),
* iwrk,ier)
c add the polynomial curve to the calculated spline.
call fpadpo(idim,t,n,c,ncc,k,cp,np,wrk(jz),wrk(ja),wrk(jb))
90 return
end
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