subroutine concon(iopt,m,x,y,w,v,s,nest,maxtr,maxbin,n,t,c,sq, * sx,bind,wrk,lwrk,iwrk,kwrk,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 concon determines a cubic spline c approximation s(x) which satisfies the following local convexity c constraints s''(x(i))*v(i) <= 0, i=1,2,...,m. c the number of knots n and the position t(j),j=1,2,...n is chosen c automatically by the routine in a way that c sq = sum((w(i)*(y(i)-s(x(i))))**2) be <= s. c the fit is given in the b-spline representation (b-spline coef- c ficients c(j),j=1,2,...n-4) and can be evaluated by means of c subroutine splev. c c calling sequence: c c call concon(iopt,m,x,y,w,v,s,nest,maxtr,maxbin,n,t,c,sq, c * sx,bind,wrk,lwrk,iwrk,kwrk,ier) c c parameters: c iopt: integer flag. c if iopt=0, the routine will start with the minimal number of c knots to guarantee that the convexity conditions will be c satisfied. if iopt=1, the routine will continue with the set c of 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 m : integer. on entry m must specify the number of data points. c m > 3. 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 v : real array of dimension at least (m). before entry, v(i) c must be set to 1 if s(x) must be locally concave at x(i), c to (-1) if s(x) must be locally convex at x(i) and to 0 c if no convexity constraint is imposed at x(i). c s : real. on entry s must specify an over-estimate for the c the weighted sum of squared residuals sq of the requested c spline. s >=0. unchanged on exit. 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 >=8. c in most practical situation nest=m/2 will be sufficient. c always large enough is nest=m+4. unchanged on exit. c maxtr : integer. on entry maxtr must contain an over-estimate of the c total number of records in the used tree structure, to indic- c ate the storage space available to the routine. maxtr >=1 c in most practical situation maxtr=100 will be sufficient. c always large enough is c nest-5 nest-6 c maxtr = ( ) + ( ) with l the greatest c l l+1 c integer <= (nest-6)/2 . unchanged on exit. c maxbin: integer. on entry maxbin must contain an over-estimate of the c number of knots where s(x) will have a zero second derivative c maxbin >=1. in most practical situation maxbin = 10 will be c sufficient. always large enough is maxbin=nest-6. c unchanged on exit. c n : integer. c on exit with ier <=0, n will contain the total number of c knots of the spline approximation returned. if the comput- c ation mode iopt=1 is used this value of n should be left c unchanged between subsequent calls. c t : real array of dimension at least (nest). c on exit with ier<=0, this array will contain the knots of the c spline,i.e. the position of the interior knots t(5),t(6),..., c t(n-4) as well as the position of the additional knots c t(1)=t(2)=t(3)=t(4)=x(1) and t(n-3)=t(n-2)=t(n-1)=t(n)=x(m) c needed for the 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. 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-4) in the b-spline representation of s(x) c sq : real. unless ier>0 , sq contains the weighted sum of c squared residuals of the spline approximation returned. c sx : real array of dimension at least m. on exit with ier<=0 c this array will contain the spline values s(x(i)),i=1,...,m c if the computation mode iopt=1 is used, the values of sx(1), c sx(2),...,sx(m) should be left unchanged between subsequent c calls. c bind: logical array of dimension at least nest. on exit with ier<=0 c this array will indicate the knots where s''(x)=0, i.e. c s''(t(j+3)) .eq. 0 if bind(j) = .true. c s''(t(j+3)) .ne. 0 if bind(j) = .false., j=1,2,...,n-6 c if the computation mode iopt=1 is used, the values of bind(1) c ,...,bind(n-6) should be left unchanged between subsequent c calls. c wrk : real array of dimension at least (m*4+nest*8+maxbin*(maxbin+ c nest+1)). used as working space. 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 (maxtr*4+2*(maxbin+1)) c used as working space. c kwrk : integer. on entry,kwrk must specify the actual dimension of c the array iwrk as declared in the calling (sub)program. kwrk c must not be too small (see iwrk). unchanged on exit. c ier : integer. error flag c ier=0 : normal return, s(x) satisfies the concavity/convexity c constraints and sq <= s. c ier<0 : abnormal termination: s(x) satisfies the concavity/ c convexity constraints but sq > s. c ier=-3 : the requested 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 c is too small. c the approximation returned is the least-squares cubic c spline according to the knots t(1),...,t(n) (n=nest) c which satisfies the convexity constraints. c ier=-2 : the maximal number of knots n=m+4 has been reached. c probably causes: s too small. c ier=-1 : the number of knots n is less than the maximal number c m+4 but concon finds that adding one or more knots c will not further reduce the value of sq. c probably causes : s too small. c ier>0 : abnormal termination: no approximation is returned c ier=1 : the number of knots where s''(x)=0 exceeds maxbin. c probably causes : maxbin too small. c ier=2 : the number of records in the tree structure exceeds c maxtr. c probably causes : maxtr too small. c ier=3 : the algoritm finds no solution to the posed quadratic c programming problem. c probably causes : rounding errors. c ier=4 : the minimum number of knots (given by n) to guarantee c that the concavity/convexity conditions will be c satisfied is greater than nest. c probably causes: nest too small. c ier=5 : the minimum number of knots (given by n) to guarantee c that the concavity/convexity conditions will be c satisfied is greater than m+4. c probably causes: strongly alternating convexity and c concavity conditions. normally the situation can be c coped with by adding n-m-4 extra data points (found c by linear interpolation e.g.) with a small weight w(i) c and a v(i) number equal to zero. c ier=10 : on entry, the input data are controlled on validity. c the following restrictions must be satisfied c 0<=iopt<=1, m>3, nest>=8, s>=0, maxtr>=1, maxbin>=1, c kwrk>=maxtr*4+2*(maxbin+1), w(i)>0, x(i) < x(i+1), c lwrk>=m*4+nest*8+maxbin*(maxbin+nest+1) c if one of these restrictions is found to be violated c control is immediately repassed to the calling program c c further comments: c as an example of the use of the computation mode iopt=1, the c following program segment will cause concon to return control c each time a spline with a new set of knots has been computed. c ............. c iopt = 0 c s = 0.1e+60 (s very large) c do 10 i=1,m c call concon(iopt,m,x,y,w,v,s,nest,maxtr,maxbin,n,t,c,sq,sx, c * bind,wrk,lwrk,iwrk,kwrk,ier) c ...... c s = sq c iopt=1 c 10 continue c ............. c c other subroutines required: c fpcoco,fpcosp,fpbspl,fpadno,fpdeno,fpseno,fpfrno c c references: c dierckx p. : an algorithm for cubic spline fitting with convexity c constraints, computing 24 (1980) 349-371. c dierckx p. : an algorithm for least-squares cubic spline fitting c with convexity and concavity constraints, report tw39, c dept. computer science, k.u.leuven, 1978. 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 : march 1978 c latest update : march 1987. c c .. c ..scalar arguments.. real s,sq integer iopt,m,nest,maxtr,maxbin,n,lwrk,kwrk,ier c ..array arguments.. real x(m),y(m),w(m),v(m),t(nest),c(nest),sx(m),wrk(lwrk) integer iwrk(kwrk) logical bind(nest) c ..local scalars.. integer i,lwest,kwest,ie,iw,lww real one c .. c set constant one = 0.1e+01 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.0 .or. iopt.gt.1) go to 30 if(m.lt.4 .or. nest.lt.8) go to 30 if(s.lt.0.) go to 30 if(maxtr.lt.1 .or. maxbin.lt.1) go to 30 lwest = 8*nest+m*4+maxbin*(1+nest+maxbin) kwest = 4*maxtr+2*(maxbin+1) if(lwrk.lt.lwest .or. kwrk.lt.kwest) go to 30 if(iopt.gt.0) go to 20 if(w(1).le.0.) go to 30 if(v(1).gt.0.) v(1) = one if(v(1).lt.0.) v(1) = -one do 10 i=2,m if(x(i-1).ge.x(i) .or. w(i).le.0.) go to 30 if(v(i).gt.0.) v(i) = one if(v(i).lt.0.) v(i) = -one 10 continue 20 ier = 0 c we partition the working space and determine the spline approximation ie = 1 iw = ie+nest lww = lwrk-nest call fpcoco(iopt,m,x,y,w,v,s,nest,maxtr,maxbin,n,t,c,sq,sx, * bind,wrk(ie),wrk(iw),lww,iwrk,kwrk,ier) 30 return end