subroutine curfit(iopt,m,x,y,w,xb,xe,k,s,nest,n,t,c,fp,
* wrk,lwrk,iwrk,ier)
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 successful 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 successful 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 process 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 ..
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.ApproxSpline.Curve1d.Type
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