pure subroutine clocur(iopt,ipar,idim,m,u,mx,x,w,k,s,nest,n,t,nc,c,fp,wrk,lwrk,iwrk,ier)
!
! calling sequence:
! call clocur(iopt,ipar,idim,m,u,mx,x,w,k,s,nest,n,t,nc,c,fp,wrk,lwrk,iwrk,ier)
!
! parameters:
! iopt : integer flag. on entry iopt must specify whether a weighted least-squares closed
! spline curve (iopt=-1) or a smoothing closed spline curve (iopt=0 or 1) must be
! determined. if iopt=0 the routine will start with an initial set of knots
! t(i)=u(1)+(u(m)-u(1))*(i-k-1),i=1,2,...,2*k+2. if iopt=1 the routine will continue
! with the knots found at the last call. attention: a call with iopt=1 must always be
! immediately preceded by another call with iopt=1 or iopt=zero.
! unchanged on exit.
! ipar : integer flag. on entry ipar must specify whether (ipar=1) the user will supply the
! parameter values u(i),or whether (ipar=0) these values are to be calculated by clocur.
! unchanged on exit.
! idim : integer. on entry idim must specify the dimension of the curve. 0 < idim <= MAX_IDIM.
! unchanged on exit.
! m : integer. on entry m must specify the number of data points. m>1. unchanged on exit.
! u : real array of dimension at least (m). in case ipar=1,before entry, u(i) must be set
! to the i-th value of the parameter variable u for i=1,2,...,m. these values must then
! be supplied in strictly ascending order and will be unchanged on exit. in case ipar=0,
! on exit,the array will contain the values u(i) as determined by clocur.
! mx : integer. on entry mx must specify the actual dimension of the array x as declared in
! the calling (sub)program. mx must not be too small (see x). unchanged on exit.
! x : real array of dimension at least idim*m.
! before entry, x(idim*(i-1)+j) must contain the j-th coordinate of the i-th data point
! for i=1,2,...,m and j=1,2,...,idim. since first and last data point must coincide it
! means that x(j)=x(idim*(m-1)+j),j=1,2,...,idim. unchanged on exit.
! w : real array of dimension at least (m). before entry, w(i) must be set to the i-th value
! in the set of weights. the w(i) must be strictly positive. w(m) is not used.
! unchanged on exit. see also further comments.
! k : integer. on entry k must specify the degree of the splines. 1<=k<=5. it is recommended
! to use cubic splines (k=3). the user is strongly dissuaded from choosing k even,
! together with a small s-value. unchanged on exit.
! s : real.on entry (in case iopt>=0) s must specify the smoothing factor. s >=zero
! unchanged on exit. for advice on the choice of s see further comments.
! nest : integer. on entry nest must contain an over-estimate of the total number of knots of
! the splines returned, to indicate the storage space available to the routine.
! nest >=2*k+2. in most practical situation nest=m/2 will be sufficient. always large
! enough is nest=m+2*k, the number of knots needed for interpolation (s=0).
! unchanged on exit.
! n : integer. unless ier = 10 (in case iopt >=0), n will contain the total number of knots
! of the smoothing spline curve returned if the computation mode iopt=1 is used this
! value of n should be left unchanged between subsequent calls. in case iopt=-1, the
! value of n must be specified on entry.
! t : real array of dimension at least (nest). on successful exit, this array will contain
! the knots of the spline curve,i.e. the position of the interior knots t(k+2),
! t(k+3),..,t(n-k-1) as well as the position of the additional t(1),t(2),..,t(k+1)=u(1)
! and u(m)=t(n-k),...,t(n) needed for the b-spline representation.
! if the computation mode iopt=1 is used, the values of t(1),t(2),...,t(n) should be
! left unchanged between subsequent calls. if the computation mode iopt=-1 is used, the
! values t(k+2),...,t(n-k-1) must be supplied by the user, before entry. see also the
! restrictions (ier=10).
! nc : integer. on entry nc must specify the actual dimension of the array c as declared in
! the calling (sub)program. nc must not be too small (see c). unchanged on exit.
! c : real array of dimension at least (nest*idim). on successful exit, this array will
! contain the coefficients in the b-spline representation of the spline curve s(u),i.e.
! the b-spline coefficients of the spline sj(u) will be given in c(n*(j-1)+i),i=1,2,...,
! n-k-1 for j=1,2,...,idim.
! fp : real. unless ier = 10, fp contains the weighted sum of squared residuals of the spline
! curve returned.
! wrk : real array of dimension at least m*(k+1)+nest*(7+idim+5*k). used as working space.
! if the computation mode iopt=1 is used, the values wrk(1),...,wrk(n) should be left
! unchanged between subsequent calls.
! lwrk : integer. on entry,lwrk must specify the actual dimension of the array wrk as declared
! in the calling (sub)program. lwrk must not be too small (see wrk). unchanged on exit.
! iwrk : integer array of dimension at least (nest). used as working space. if the computation
! mode iopt=1 is used,the values iwrk(1),...,iwrk(n) should be left unchanged
! between subsequent calls.
! ier : integer. unless the routine detects an error, ier contains a non-positive value on
! exit, i.e.
! ier=0 : normal return. the close curve returned has a residual sum of squares fp such that
! abs(fp-s)/s <= tol with tol a relative tolerance set to 0.001 by the program.
! ier=-1 : normal return. the curve returned is an interpolating spline curve (fp=0).
! ier=-2 : normal return. the curve returned is the weighted least-squares point,i.e. each
! spline sj(u) is a constant. in this extreme case fp gives the upper bound fp0 for
! the smoothing factor s.
! ier=1 : error. the required storage space exceeds the available storage space, as specified
! by the parameter nest. likely causes : nest too small. if nest is already large (say
! nest > m/2), it may also indicate that s is too small. the approximation returned is
! the least-squares closed curve according to the knots t(1),t(2),...,t(n). (n=nest)
! the parameter fp gives the corresponding weighted sum of squared residuals (fp>s).
! ier=2 : error. a theoretically impossible result was found during the iteration process for
! finding a smoothing curve with fp = s. probably causes : s too small.
! there is an approximation returned but the corresponding weighted sum of squared
! residuals does not satisfy the condition abs(fp-s)/s < tol.
! ier=3 : error. the maximal number of iterations maxit (set to 20 by the program) allowed for
! finding a smoothing curve with fp=s has been reached. probably causes : s too small
! there is an approximation returned but the corresponding weighted sum of squared
! residuals does not satisfy the condition abs(fp-s)/s < tol.
! ier=10 : error. on entry, the input data are controlled on validity the following
! restrictions must be satisfied.
! -1<=iopt<=1, 1<=k<=5, m>1, nest>2*k+2, w(i)>0,i=1,2,...,m
! 0<=ipar<=1, 0<idim<=10, lwrk>=(k+1)*m+nest*(7+idim+5*k),
! nc>=nest*idim, x(j)=x(idim*(m-1)+j), j=1,2,...,idim
! if ipar=0: sum j=1,idim (x(i*idim+j)-x((i-1)*idim+j))**2>0
! i=1,2,...,m-1.
! if ipar=1: u(1)<u(2)<...<u(m)
! if iopt=-1: 2*k+2<=n<=min(nest,m+2*k)
! u(1)<t(k+2)<t(k+3)<...<t(n-k-1)<u(m)
! (u(1)=0 and u(m)=1 in case ipar=0)
! the schoenberg-whitney conditions, i.e. there
! must be a subset of data points uu(j) with
! uu(j) = u(i) or u(i)+(u(m)-u(1)) such that
! t(j) < uu(j) < t(j+k+1), j=k+1,...,n-k-1
! if iopt>=0: s>=0
! if s=0 : nest >= m+2*k
! if one of these conditions is found to be violated,control is immediately repassed
! to the calling program. in that case there is no approximation returned.
!
! further comments:
! by means of the parameter s, the user can control the tradeoff between closeness of fit and
! smoothness of fit of the approximation. if s is too large, the curve will be too smooth and
! signal will be lost ; if s is too small the curve will pick up too much noise. in the extreme
! cases the program will return an interpolating curve if s=0 and the weighted least-squares
! point if s is very large. between these extremes, a properly chosen s will result in a good
! compromise between closeness of fit and smoothness of fit. to decide whether an approximation,
! corresponding to a certain s is satisfactory the user is highly recommended to inspect the
! fits graphically.
! recommended values for s depend on the weights w(i). if these are taken as 1/d(i) with d(i) an
! estimate of the standard deviation of x(i), a good s-value should be found in the range
! (m-sqrt(2*m),m+sqrt(2*m)). if nothing is known about the statistical error in x(i) each w(i)
! can be set equal to one and s determined by trial and error, taking account of the comments
! above. the best is then to start with a very large value of s ( to determine the weighted
! least-squares point and the upper bound fp0 for s) and then to progressively decrease the
! value of s ( say by a factor 10 in the beginning, i.e. s=fp0/10, fp0/100,...and more carefully
! as the approximating curve shows more detail) to obtain closer fits. to economize the search
! for a good s-value the program provides with different modes of computation. at the first call
! of the routine, or whenever he wants to restart with the initial set of knots the user must set
! iopt=zero.
! if iopt=1 the program will continue with the set of knots found at the last call of the
! routine. this will save a lot of computation time if clocur is called repeatedly for different
! values of s. the number of knots of the spline returned and their location will depend on the
! value of s and on the complexity of the shape of the curve underlying the data. but, if the
! computation mode iopt=1 is used, the knots returned may also depend on the s-values at
! previous calls (if these were smaller). therefore, if after a number of trials with different
! s-values and iopt=1, the user can finally accept a fit as satisfactory, it may be worthwhile
! for him to call clocur once more with the selected value for s but now with iopt=zero indeed,
! clocur may then return an approximation of the same quality of fit but with fewer knots and
! therefore better if data reduction is also an important objective for the user.
!
! the form of the approximating curve can strongly be affected by the choice of the parameter
! values u(i). if there is no physical reason for choosing a particular parameter u, often good
! results will be obtained with the choice of clocur(in case ipar=0), i.e.
! v(1)=0, v(i)=v(i-1)+q(i), i=2,...,m, u(i)=v(i)/v(m), i=1,..,m
! where
! q(i)= sqrt(sum j=1,idim (xj(i)-xj(i-1))**2 )
! other possibilities for q(i) are
! q(i)= sum j=1,idim (xj(i)-xj(i-1))**2
! q(i)= sum j=1,idim abs(xj(i)-xj(i-1))
! q(i)= max j=1,idim abs(xj(i)-xj(i-1))
! q(i)= 1
!
!
! other subroutines required:
! fpbacp,fpbspl,fpchep,fpclos,fpdisc,fpgivs,fpknot,fprati,fprota
!
! references:
! dierckx p. : algorithms for smoothing data with periodic and parametric splines,
! computer graphics and image processing 20 (1982) 171-184.
! dierckx p. : algorithms for smoothing data with periodic and parametric splines,
! report tw55, dept. computer science, k.u.leuven, 1981.
! dierckx p. : curve and surface fitting with splines,
! monographs on numerical analysis, oxford university press, 1993.
!
! author:
! p.dierckx
! dept. computer science, k.u. leuven
! celestijnenlaan 200a, b-3001 heverlee, belgium.
! e-mail : Paul.Dierckx@cs.kuleuven.ac.be
!
! creation date : may 1979
!
! ..
! ..scalar arguments..
real(RKIND), intent(in) :: s
real(RKIND), intent(inout) :: fp
integer, intent(in) :: iopt,ipar,idim,m,mx,k,nest,nc,lwrk
integer, intent(inout) :: n,ier
! ..array arguments..
real(RKIND), intent(in) :: x(mx),w(m)
real(RKIND), intent(inout) :: u(m),t(nest),c(nc),wrk(lwrk)
integer, intent(inout) :: iwrk(nest)
! ..local scalars..
real(RKIND) :: per,dist
integer :: i,ia1,ia2,ib,ifp,ig1,ig2,iq,iz,i1,i2,j1,j2,k1,k2,lwest,m1,nmin,ncc
! we set up the parameters tol and maxit
integer, parameter :: maxit = 20
real(RKIND), parameter :: tol = smallnum03
! before starting computations a data check is made. if the input data
! are invalid, control is immediately repassed to the calling program.
ier = FITPACK_INPUT_ERROR
k1 = k+1
k2 = k1+1
m1 = m-1
nmin = 2*k1
ncc = nest*idim
lwest = m*k1+nest*(7+idim+5*k)
if (iopt<(-1) .or. iopt>1) return
if (ipar<0 .or. ipar>1) return
if (idim<=0 .or. idim>MAX_IDIM) return
if (k<=0 .or. k>5) return
if (m<2 .or. nest<nmin) return
if (mx<m*idim .or. nc<ncc) return
if (lwrk<lwest) return
! Check closed curve (1st and last points match)
if (any(not_equal(x(1:idim),x((m-1)*idim+1:m*idim)))) return
! Normalized cumulative length parameter coordinate along the curve
if (ipar==0 .and. iopt<=0) then
i1 = 0
i2 = idim
u(1) = zero
do i=2,m
dist = zero
do j1=1,idim
i1 = i1+1
i2 = i2+1
dist = dist+(x(i2)-x(i1))**2
end do
u(i) = u(i-1)+sqrt(dist)
end do
if (u(m)<=zero) return
u(2:m) = u(2:m)/u(m)
u(m) = one
endif
if (w(1)<=zero) return
if (any(u(1:m1)>=u(2:m) .or. w(1:m1)<=zero)) return
if (iopt>=0) then
if (s<zero) return
if (equal(s,zero) .and. nest<(m+2*k)) return
else
if (n<=nmin .or. n>nest) return
per = u(m)-u(1)
j1 = k1
t(j1) = u(1)
i1 = n-k
t(i1) = u(m)
j2 = j1
i2 = i1
do i=1,k
i1 = i1+1
i2 = i2-1
j1 = j1+1
j2 = j2-1
t(j2) = t(i2)-per
t(i1) = t(j1)+per
end do
ier = fpchep(u,m,t,n,k)
if (ier/=FITPACK_OK) return
end if
ier = FITPACK_OK
! we partition the working space and determine the spline approximation.
ifp = 1
iz = ifp+nest
ia1 = iz+ncc
ia2 = ia1+nest*k1
ib = ia2+nest*k
ig1 = ib+nest*k2
ig2 = ig1+nest*k2
iq = ig2+nest*k1
call fpclos(iopt,idim,m,u,mx,x,w,k,s,nest,tol,maxit,k1,k2,n,t,ncc,c,fp, &
wrk(ifp),wrk(iz),wrk(ia1),wrk(ia2),wrk(ib),wrk(ig1),wrk(ig2),wrk(iq),iwrk,ier)
return
end subroutine clocur