curfit.f

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      subroutine curfit(iopt,m,x,y,w,xb,xe,k,s,nest,n,t,c,fp,
     * wrk,lwrk,iwrk,ier)
      implicit none
c  given the set of data points (x(i),y(i)) and the set of positive
c  numbers w(i),i=1,2,...,m,subroutine curfit determines a smooth spline
c  approximation of degree k on the interval xb <= x <= xe.
c  if iopt=-1 curfit calculates the weighted least-squares spline
c  according to a given set of knots.
c  if iopt>=0 the number of knots of the spline s(x) and the position
c  t(j),j=1,2,...,n is chosen automatically by the routine. the smooth-
c  ness of s(x) is then achieved by minimalizing the discontinuity
c  jumps of the k-th derivative of s(x) 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)*(y(i)-s(x(i))))**2) be <= s, with s a given non-
c  negative constant, called the smoothing factor.
c  the fit s(x) is given in the b-spline representation (b-spline coef-
c  ficients c(j),j=1,2,...,n-k-1) and can be evaluated by means of
c  subroutine splev.
c
c  calling sequence:
c     call curfit(iopt,m,x,y,w,xb,xe,k,s,nest,n,t,c,fp,wrk,
c    * lwrk,iwrk,ier)
c
c  parameters:
c   iopt  : integer flag. on entry iopt must specify whether a weighted
c           least-squares spline (iopt=-1) or a smoothing spline (iopt=
c           0 or 1) must be determined. if iopt=0 the routine will start
c           with an initial set of knots t(i)=xb, t(i+k+1)=xe, i=1,2,...
c           k+1. if iopt=1 the routine will continue with the knots
c           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   m     : integer. on entry m must specify the number of data points.
c           m > k. unchanged on exit.
c   x     : real array of dimension at least (m). before entry, x(i)
c           must be set to the i-th value of the independent variable x,
c           for i=1,2,...,m. these values must be supplied in strictly
c           ascending order. unchanged on exit.
c   y     : real array of dimension at least (m). before entry, y(i)
c           must be set to the i-th value of the dependent variable y,
c           for i=1,2,...,m. unchanged on exit.
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   xb,xe : real values. on entry xb and xe must specify the boundaries
c           of the approximation interval. xb<=x(1), xe>=x(m).
c           unchanged on exit.
c   k     : integer. on entry k must specify the degree of the spline.
c           1<=k<=5. it is recommended to use cubic splines (k=3).
c           the user is strongly dissuaded from choosing k even,together
c           with a small s-value. 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 spline 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, the number of knots
c           needed for interpolation (s=0). 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 spline approximation 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,i.e. the position of the interior knots t(k+2),t(k+3)
c           ...,t(n-k-1) as well as the position of the additional knots
c           t(1)=t(2)=...=t(k+1)=xb and t(n-k)=...=t(n)=xe 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   c     : real array of dimension at least (nest).
c           on succesful exit, this array will contain the coefficients
c           c(1),c(2),..,c(n-k-1) in the b-spline representation of s(x)
c   fp    : real. unless ier=10, fp contains the weighted sum of
c           squared residuals of the spline approximation returned.
c   wrk   : real array of dimension at least (m*(k+1)+nest*(7+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 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 polynomial of degree k. in this extreme case fp
c             gives the upper 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 weighted least-squares
c             spline 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 with
c             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 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=10 : error. on entry, the input data are controlled on validity
c             the following restrictions must be satisfied.
c             -1<=iopt<=1, 1<=k<=5, m>k, nest>2*k+2, w(i)>0,i=1,2,...,m
c             xb<=x(1)<x(2)<...<x(m)<=xe, lwrk>=(k+1)*m+nest*(7+3*k)
c             if iopt=-1: 2*k+2<=n<=min(nest,m+k+1)
c                         xb<t(k+2)<t(k+3)<...<t(n-k-1)<xe
c                       the schoenberg-whitney conditions, i.e. there
c                       must be a subset of data points xx(j) such that
c                         t(j) < xx(j) < t(j+k+1), j=1,2,...,n-k-1
c             if iopt>=0: s>=0
c                         if s=0 : nest >= m+k+1
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 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 weighted least-squares polynomial 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   y(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 y(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 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 curfit 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. but, 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   curfit once more with the selected value for s but now with iopt=0.
c   indeed, curfit 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  other subroutines required:
c    fpback,fpbspl,fpchec,fpcurf,fpdisc,fpgivs,fpknot,fprati,fprota
c
c  references:
c   dierckx p. : an algorithm for smoothing, differentiation and integ-
c                ration of experimental data using spline functions,
c                j.comp.appl.maths 1 (1975) 165-184.
c   dierckx p. : a fast algorithm for smoothing data on a rectangular
c                grid while using spline functions, siam j.numer.anal.
c                19 (1982) 1286-1304.
c   dierckx p. : an improved algorithm for curve fitting with spline
c                functions, report tw54, dept. computer science,k.u.
c                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*8 xb,xe,s,fp
      integer iopt,m,k,nest,n,lwrk,ier
c  ..array arguments..
      real*8 x(m),y(m),w(m),t(nest),c(nest),wrk(lwrk)
      integer iwrk(nest)
c  ..local scalars..
      real*8 tol
      integer i,ia,ib,ifp,ig,iq,iz,j,k1,k2,lwest,maxit,nmin
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(k.le.0 .or. k.gt.5) go to 50
      k1 = k+1
      k2 = k1+1
      if(iopt.lt.(-1) .or. iopt.gt.1) go to 50
      nmin = 2*k1
      if(m.lt.k1 .or. nest.lt.nmin) go to 50
      lwest = m*k1+nest*(7+3*k)
      if(lwrk.lt.lwest) go to 50
      if(xb.gt.x(1) .or. xe.lt.x(m) .or. w(1).le.0.) go to 50
      do 10 i=2,m
         if(x(i-1).ge.x(i) .or. w(i).le.0.) go to 50
  10  continue
      if(iopt.ge.0) go to 30
      if(n.lt.nmin .or. n.gt.nest) go to 50
      j = n
      do 20 i=1,k1
         t(i) = xb
         t(j) = xe
         j = j-1
  20  continue
      call fpchec(x,m,t,n,k,ier)
      if(ier) 50,40,50
  30  if(s.lt.0.) go to 50
      if(s.eq.0. .and. nest.lt.(m+k1)) go to 50
      ier = 0
c we partition the working space and determine the spline approximation.
  40  ifp = 1
      iz = ifp+nest
      ia = iz+nest
      ib = ia+nest*k1
      ig = ib+nest*k2
      iq = ig+nest*k2
      call fpcurf(iopt,x,y,w,m,xb,xe,k,s,nest,tol,maxit,k1,k2,n,t,c,fp,
     * wrk(ifp),wrk(iz),wrk(ia),wrk(ib),wrk(ig),wrk(iq),iwrk,ier)
  50  return
      end