subroutine clocur(iopt,ipar,idim,m,u,mx,x,w,k,s,nest,n,t,nc,c,fp, * wrk,lwrk,iwrk,ier) c given the ordered set of m points x(i) in the idim-dimensional space c with x(1)=x(m), and given also a corresponding set of strictly in- c creasing values u(i) and the set of positive numbers w(i),i=1,2,...,m c subroutine clocur determines a smooth approximating closed spline c curve s(u), i.e. c x1 = s1(u) c x2 = s2(u) u(1) <= u <= u(m) c ......... c xidim = sidim(u) c with sj(u),j=1,2,...,idim periodic spline functions of degree k with c common knots t(j),j=1,2,...,n. c if ipar=1 the values u(i),i=1,2,...,m must be supplied by the user. c if ipar=0 these values are chosen automatically by clocur as c v(1) = 0 c v(i) = v(i-1) + dist(x(i),x(i-1)) ,i=2,3,...,m c u(i) = v(i)/v(m) ,i=1,2,...,m c if iopt=-1 clocur calculates the weighted least-squares closed spline c curve 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 clocur(iopt,ipar,idim,m,u,mx,x,w,k,s,nest,n,t,nc,c, c * fp,wrk,lwrk,iwrk,ier) c c parameters: c iopt : integer flag. on entry iopt must specify whether a weighted c least-squares closed spline curve (iopt=-1) or a smoothing c closed spline curve (iopt=0 or 1) must be determined. if c iopt=0 the routine will start with an initial set of knots c t(i)=u(1)+(u(m)-u(1))*(i-k-1),i=1,2,...,2*k+2. if iopt=1 the c routine will continue with the knots found at the last call. 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 ipar : integer flag. on entry ipar must specify whether (ipar=1) c the user will supply the parameter values u(i),or whether c (ipar=0) these values are to be calculated by clocur. 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 > 1. unchanged on exit. c u : real array of dimension at least (m). in case ipar=1,before c entry, u(i) must be set to the i-th value of the parameter c variable u for i=1,2,...,m. these values must then be c supplied in strictly ascending order and will be unchanged c on exit. in case ipar=0, on exit,the array will contain the c values u(i) as determined by clocur. c mx : integer. on entry mx must specify the actual dimension of c the array x as declared in the calling (sub)program. mx must c 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. since first and last data point must coincide it c means that x(j)=x(idim*(m-1)+j),j=1,2,...,idim. c 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. w(m) is not used. c unchanged on exit. see also further comments. c k : integer. on entry k must specify the degree of the splines. 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 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+2*k, 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 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)=u(1) and u(m)=t(n-k),...,t(n) 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 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*(7+idim+5*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 close curve returned has a residual c sum of squares fp such that abs(fp-s)/s <= tol with tol a c relative tolerance set to 0.001 by the program. c ier=-1 : normal return. the curve returned is an interpolating c spline curve (fp=0). c ier=-2 : normal return. the curve returned is the weighted least- c squares point,i.e. each spline sj(u) is a constant. in c this extreme case fp gives the upper bound fp0 for the c 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 closed 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 curve 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 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, 1<=k<=5, m>1, nest>2*k+2, w(i)>0,i=1,2,...,m c 0<=ipar<=1, 0=(k+1)*m+nest*(7+idim+5*k), c nc>=nest*idim, x(j)=x(idim*(m-1)+j), j=1,2,...,idim c if ipar=0: sum j=1,idim (x(i*idim+j)-x((i-1)*idim+j))**2>0 c i=1,2,...,m-1. c if ipar=1: u(1)=0: s>=0 c if s=0 : nest >= m+2*k 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 weighted least-squares point if s is very large. c between these extremes, a properly chosen s will result in a good c 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 weighted c least-squares point 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 clocur 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 clocur once more with the selected value for s but now with iopt=0. c indeed, clocur 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 of clocur(in case ipar=0), i.e. 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 c other subroutines required: c fpbacp,fpbspl,fpchep,fpclos,fpdisc,fpgivs,fpknot,fprati,fprota 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,ipar,idim,m,mx,k,nest,n,nc,lwrk,ier c ..array arguments.. real u(m),x(mx),w(m),t(nest),c(nc),wrk(lwrk) integer iwrk(nest) c ..local scalars.. real per,tol,dist integer i,ia1,ia2,ib,ifp,ig1,ig2,iq,iz,i1,i2,j1,j2,k1,k2,lwest, * maxit,m1,nmin,ncc,j c ..function references.. real sqrt 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(ipar.lt.0 .or. ipar.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 k2 = k1+1 nmin = 2*k1 if(m.lt.2 .or. nest.lt.nmin) go to 90 ncc = nest*idim if(mx.lt.m*idim .or. nc.lt.ncc) go to 90 lwest = m*k1+nest*(7+idim+5*k) if(lwrk.lt.lwest) go to 90 i1 = idim i2 = m*idim do 5 j=1,idim if(x(i1).ne.x(i2)) go to 90 i1 = i1-1 i2 = i2-1 5 continue if(ipar.ne.0 .or. iopt.gt.0) go to 40 i1 = 0 i2 = idim u(1) = 0. do 20 i=2,m dist = 0. do 10 j1=1,idim i1 = i1+1 i2 = i2+1 dist = dist+(x(i2)-x(i1))**2 10 continue u(i) = u(i-1)+sqrt(dist) 20 continue if(u(m).le.0.) go to 90 do 30 i=2,m u(i) = u(i)/u(m) 30 continue u(m) = 0.1e+01 40 if(w(1).le.0.) go to 90 m1 = m-1 do 50 i=1,m1 if(u(i).ge.u(i+1) .or. w(i).le.0.) go to 90 50 continue if(iopt.ge.0) go to 70 if(n.le.nmin .or. n.gt.nest) go to 90 per = u(m)-u(1) j1 = k1 t(j1) = u(1) i1 = n-k t(i1) = u(m) j2 = j1 i2 = i1 do 60 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 60 continue call fpchep(u,m,t,n,k,ier) if(ier) 90,80,90 70 if(s.lt.0.) go to 90 if(s.eq.0. .and. nest.lt.(m+2*k)) go to 90 ier = 0 c we partition the working space and determine the spline approximation. 80 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) 90 return end