subroutine surfit(iopt,m,x,y,z,w,xb,xe,yb,ye,kx,ky,s,nxest,nyest,
* nmax,eps,nx,tx,ny,ty,c,fp,wrk1,lwrk1,wrk2,lwrk2,iwrk,kwrk,ier)
c given the set of data points (x(i),y(i),z(i)) and the set of positive
c numbers w(i),i=1,...,m, subroutine surfit determines a smooth bivar-
c iate spline approximation s(x,y) of degrees kx and ky on the rect-
c angle xb <= x <= xe, yb <= y <= ye.
c if iopt = -1 surfit calculates the weighted least-squares spline
c according to a given set of knots.
c if iopt >= 0 the total numbers nx and ny of these knots and their
c position tx(j),j=1,...,nx and ty(j),j=1,...,ny are chosen automatic-
c ally by the routine. the smoothness of s(x,y) is then achieved by
c minimalizing the discontinuity jumps in the derivatives of s(x,y)
c across the boundaries of the subpanels (tx(i),tx(i+1))*(ty(j),ty(j+1).
c the amounth of smoothness is determined by the condition that f(p) =
c sum ((w(i)*(z(i)-s(x(i),y(i))))**2) be <= s, with s a given non-neg-
c ative constant, called the smoothing factor.
c the fit is given in the b-spline representation (b-spline coefficients
c c((ny-ky-1)*(i-1)+j),i=1,...,nx-kx-1;j=1,...,ny-ky-1) and can be eval-
c uated by means of subroutine bispev.
c
c calling sequence:
c call surfit(iopt,m,x,y,z,w,xb,xe,yb,ye,kx,ky,s,nxest,nyest,
c * nmax,eps,nx,tx,ny,ty,c,fp,wrk1,lwrk1,wrk2,lwrk2,iwrk,kwrk,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=0
c or 1) must be determined.
c if iopt=0 the routine will start with an initial set of knots
c tx(i)=xb,tx(i+kx+1)=xe,i=1,...,kx+1;ty(i)=yb,ty(i+ky+1)=ye,i=
c 1,...,ky+1. 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 pre-
c ceded 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 >= (kx+1)*(ky+1). unchanged on exit.
c x : real array of dimension at least (m).
c y : real array of dimension at least (m).
c z : real array of dimension at least (m).
c before entry, x(i),y(i),z(i) must be set to the co-ordinates
c of the i-th data point, for i=1,...,m. the order of the data
c points is immaterial. unchanged on exit.
c w : real array of dimension at least (m). before entry, w(i) must
c be set to the i-th value in the set of weights. the w(i) must
c be strictly positive. unchanged on exit.
c xb,xe : real values. on entry xb,xe,yb and ye must specify the bound-
c yb,ye aries of the rectangular approximation domain.
c xb<=x(i)<=xe,yb<=y(i)<=ye,i=1,...,m. unchanged on exit.
c kx,ky : integer values. on entry kx and ky must specify the degrees
c of the spline. 1<=kx,ky<=5. it is recommended to use bicubic
c (kx=ky=3) splines. 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 nxest : integer. unchanged on exit.
c nyest : integer. unchanged on exit.
c on entry, nxest and nyest must specify an upper bound for the
c number of knots required in the x- and y-directions respect.
c these numbers will also determine the storage space needed by
c the routine. nxest >= 2*(kx+1), nyest >= 2*(ky+1).
c in most practical situation nxest = kx+1+sqrt(m/2), nyest =
c ky+1+sqrt(m/2) will be sufficient. see also further comments.
c nmax : integer. on entry nmax must specify the actual dimension of
c the arrays tx and ty. nmax >= nxest, nmax >=nyest.
c unchanged on exit.
c eps : real.
c on entry, eps must specify a threshold for determining the
c effective rank of an over-determined linear system of equat-
c ions. 0 < eps < 1. if the number of decimal digits in the
c computer representation of a real number is q, then 10**(-q)
c is a suitable value for eps in most practical applications.
c unchanged on exit.
c nx : integer.
c unless ier=10 (in case iopt >=0), nx will contain the total
c number of knots with respect to the x-variable, of the spline
c approximation returned. if the computation mode iopt=1 is
c used, the value of nx should be left unchanged between sub-
c sequent calls.
c in case iopt=-1, the value of nx should be specified on entry
c tx : real array of dimension nmax.
c on succesful exit, this array will contain the knots of the
c spline with respect to the x-variable, i.e. the position of
c the interior knots tx(kx+2),...,tx(nx-kx-1) as well as the
c position of the additional knots tx(1)=...=tx(kx+1)=xb and
c tx(nx-kx)=...=tx(nx)=xe needed for the b-spline representat.
c if the computation mode iopt=1 is used, the values of tx(1),
c ...,tx(nx) should be left unchanged between subsequent calls.
c if the computation mode iopt=-1 is used, the values tx(kx+2),
c ...tx(nx-kx-1) must be supplied by the user, before entry.
c see also the restrictions (ier=10).
c ny : integer.
c unless ier=10 (in case iopt >=0), ny will contain the total
c number of knots with respect to the y-variable, of the spline
c approximation returned. if the computation mode iopt=1 is
c used, the value of ny should be left unchanged between sub-
c sequent calls.
c in case iopt=-1, the value of ny should be specified on entry
c ty : real array of dimension nmax.
c on succesful exit, this array will contain the knots of the
c spline with respect to the y-variable, i.e. the position of
c the interior knots ty(ky+2),...,ty(ny-ky-1) as well as the
c position of the additional knots ty(1)=...=ty(ky+1)=yb and
c ty(ny-ky)=...=ty(ny)=ye needed for the b-spline representat.
c if the computation mode iopt=1 is used, the values of ty(1),
c ...,ty(ny) should be left unchanged between subsequent calls.
c if the computation mode iopt=-1 is used, the values ty(ky+2),
c ...ty(ny-ky-1) must be supplied by the user, before entry.
c see also the restrictions (ier=10).
c c : real array of dimension at least (nxest-kx-1)*(nyest-ky-1).
c on succesful exit, c contains the coefficients of the spline
c approximation s(x,y)
c fp : real. unless ier=10, fp contains the weighted sum of
c squared residuals of the spline approximation returned.
c wrk1 : real array of dimension (lwrk1). used as workspace.
c if the computation mode iopt=1 is used the value of wrk1(1)
c should be left unchanged between subsequent calls.
c on exit wrk1(2),wrk1(3),...,wrk1(1+(nx-kx-1)*(ny-ky-1)) will
c contain the values d(i)/max(d(i)),i=1,...,(nx-kx-1)*(ny-ky-1)
c with d(i) the i-th diagonal element of the reduced triangular
c matrix for calculating the b-spline coefficients. it includes
c those elements whose square is less than eps,which are treat-
c ed as 0 in the case of presumed rank deficiency (ier<-2).
c lwrk1 : integer. on entry lwrk1 must specify the actual dimension of
c the array wrk1 as declared in the calling (sub)program.
c lwrk1 must not be too small. let
c u = nxest-kx-1, v = nyest-ky-1, km = max(kx,ky)+1,
c ne = max(nxest,nyest), bx = kx*v+ky+1, by = ky*u+kx+1,
c if(bx.le.by) b1 = bx, b2 = b1+v-ky
c if(bx.gt.by) b1 = by, b2 = b1+u-kx then
c lwrk1 >= u*v*(2+b1+b2)+2*(u+v+km*(m+ne)+ne-kx-ky)+b2+1
c wrk2 : real array of dimension (lwrk2). used as workspace, but
c only in the case a rank deficient system is encountered.
c lwrk2 : integer. on entry lwrk2 must specify the actual dimension of
c the array wrk2 as declared in the calling (sub)program.
c lwrk2 > 0 . a save upper boundfor lwrk2 = u*v*(b2+1)+b2
c where u,v and b2 are as above. if there are enough data
c points, scattered uniformly over the approximation domain
c and if the smoothing factor s is not too small, there is a
c good chance that this extra workspace is not needed. a lot
c of memory might therefore be saved by setting lwrk2=1.
c (see also ier > 10)
c iwrk : integer array of dimension (kwrk). used as workspace.
c kwrk : integer. on entry kwrk must specify the actual dimension of
c the array iwrk as declared in the calling (sub)program.
c kwrk >= m+(nxest-2*kx-1)*(nyest-2*ky-1).
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 degrees kx and ky. in this extreme
c case fp gives the upper bound for the smoothing factor s.
c ier<-2 : warning. the coefficients of the spline returned have been
c computed as the minimal norm least-squares solution of a
c (numerically) rank deficient system. (-ier) gives the rank.
c especially if the rank deficiency which can be computed as
c (nx-kx-1)*(ny-ky-1)+ier, is large the results may be inac-
c curate. they could also seriously depend on the value of
c eps.
c ier=1 : error. the required storage space exceeds the available
c storage space, as specified by the parameters nxest and
c nyest.
c probably causes : nxest or nyest too small. if these param-
c eters are already large, it may also indicate that s is
c too small
c the approximation returned is the weighted least-squares
c spline according to the current set of knots.
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 or badly chosen eps.
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=4 : error. no more knots can be added because the number of
c b-spline coefficients (nx-kx-1)*(ny-ky-1) already exceeds
c the number of data points m.
c probably causes : either s or m too small.
c the approximation returned is the weighted least-squares
c spline according to the current set of knots.
c the parameter fp gives the corresponding weighted sum of
c squared residuals (fp>s).
c ier=5 : error. no more knots can be added because the additional
c knot would (quasi) coincide with an old one.
c probably causes : s too small or too large a weight to an
c inaccurate data point.
c the approximation returned is the weighted least-squares
c spline according to the current set of knots.
c the parameter fp gives the corresponding weighted sum of
c squared residuals (fp>s).
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<=kx,ky<=5, m>=(kx+1)*(ky+1), nxest>=2*kx+2,
c nyest>=2*ky+2, 0=nxest, nmax>=nyest,
c xb<=x(i)<=xe, yb<=y(i)<=ye, w(i)>0, i=1,...,m
c lwrk1 >= u*v*(2+b1+b2)+2*(u+v+km*(m+ne)+ne-kx-ky)+b2+1
c kwrk >= m+(nxest-2*kx-1)*(nyest-2*ky-1)
c if iopt=-1: 2*kx+2<=nx<=nxest
c xb=0: s>=0
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 ier>10 : error. lwrk2 is too small, i.e. there is not enough work-
c space for computing the minimal least-squares solution of
c a rank deficient system of linear equations. ier gives the
c requested value for lwrk2. there is no approximation re-
c turned but, having saved the information contained in nx,
c ny,tx,ty,wrk1, and having adjusted the value of lwrk2 and
c the dimension of the array wrk2 accordingly, the user can
c continue at the point the program was left, by calling
c surfit with iopt=1.
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 (degrees kx,ky)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 z(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 z(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 choose s very small is strongly discouraged. this considerably
c increases computation time and memory requirements. it may also
c cause rank-deficiency (ier<-2) and endager numerical stability.
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 surfit 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. 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 surfit once more with the selected value for s but now with iopt=0.
c indeed, surfit 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 the number of knots may also depend on the upper bounds nxest and
c nyest. indeed, if at a certain stage in surfit the number of knots
c in one direction (say nx) has reached the value of its upper bound
c (nxest), then from that moment on all subsequent knots are added
c in the other (y) direction. this may indicate that the value of
c nxest is too small. on the other hand, it gives the user the option
c of limiting the number of knots the routine locates in any direction
c for example, by setting nxest=2*kx+2 (the lowest allowable value for
c nxest), the user can indicate that he wants an approximation which
c is a simple polynomial of degree kx in the variable x.
c
c other subroutines required:
c fpback,fpbspl,fpsurf,fpdisc,fpgivs,fprank,fprati,fprota,fporde
c
c references:
c dierckx p. : an algorithm for surface fitting with spline functions
c ima j. numer. anal. 1 (1981) 267-283.
c dierckx p. : an algorithm for surface fitting with spline functions
c report tw50, dept. computer science,k.u.leuven, 1980.
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 xb,xe,yb,ye,s,eps,fp
integer iopt,m,kx,ky,nxest,nyest,nmax,nx,ny,lwrk1,lwrk2,kwrk,ier
c ..array arguments..
real x(m),y(m),z(m),w(m),tx(nmax),ty(nmax),
* c((nxest-kx-1)*(nyest-ky-1)),wrk1(lwrk1),wrk2(lwrk2)
integer iwrk(kwrk)
c ..local scalars..
real tol
integer i,ib1,ib3,jb1,ki,kmax,km1,km2,kn,kwest,kx1,ky1,la,lbx,
* lby,lco,lf,lff,lfp,lh,lq,lsx,lsy,lwest,maxit,ncest,nest,nek,
* nminx,nminy,nmx,nmy,nreg,nrint,nxk,nyk
c ..function references..
integer max0
c ..subroutine references..
c fpsurf
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(eps.le.0. .or. eps.ge.1.) go to 70
if(kx.le.0 .or. kx.gt.5) go to 70
kx1 = kx+1
if(ky.le.0 .or. ky.gt.5) go to 70
ky1 = ky+1
kmax = max0(kx,ky)
km1 = kmax+1
km2 = km1+1
if(iopt.lt.(-1) .or. iopt.gt.1) go to 70
if(m.lt.(kx1*ky1)) go to 70
nminx = 2*kx1
if(nxest.lt.nminx .or. nxest.gt.nmax) go to 70
nminy = 2*ky1
if(nyest.lt.nminy .or. nyest.gt.nmax) go to 70
nest = max0(nxest,nyest)
nxk = nxest-kx1
nyk = nyest-ky1
ncest = nxk*nyk
nmx = nxest-nminx+1
nmy = nyest-nminy+1
nrint = nmx+nmy
nreg = nmx*nmy
ib1 = kx*nyk+ky1
jb1 = ky*nxk+kx1
ib3 = kx1*nyk+1
if(ib1.le.jb1) go to 10
ib1 = jb1
ib3 = ky1*nxk+1
10 lwest = ncest*(2+ib1+ib3)+2*(nrint+nest*km2+m*km1)+ib3
kwest = m+nreg
if(lwrk1.lt.lwest .or. kwrk.lt.kwest) go to 70
if(xb.ge.xe .or. yb.ge.ye) go to 70
do 20 i=1,m
if(w(i).le.0.) go to 70
if(x(i).lt.xb .or. x(i).gt.xe) go to 70
if(y(i).lt.yb .or. y(i).gt.ye) go to 70
20 continue
if(iopt.ge.0) go to 50
if(nx.lt.nminx .or. nx.gt.nxest) go to 70
nxk = nx-kx1
tx(kx1) = xb
tx(nxk+1) = xe
do 30 i=kx1,nxk
if(tx(i+1).le.tx(i)) go to 70
30 continue
if(ny.lt.nminy .or. ny.gt.nyest) go to 70
nyk = ny-ky1
ty(ky1) = yb
ty(nyk+1) = ye
do 40 i=ky1,nyk
if(ty(i+1).le.ty(i)) go to 70
40 continue
go to 60
50 if(s.lt.0.) go to 70
60 ier = 0
c we partition the working space and determine the spline approximation
kn = 1
ki = kn+m
lq = 2
la = lq+ncest*ib3
lf = la+ncest*ib1
lff = lf+ncest
lfp = lff+ncest
lco = lfp+nrint
lh = lco+nrint
lbx = lh+ib3
nek = nest*km2
lby = lbx+nek
lsx = lby+nek
lsy = lsx+m*km1
call fpsurf(iopt,m,x,y,z,w,xb,xe,yb,ye,kx,ky,s,nxest,nyest,
* eps,tol,maxit,nest,km1,km2,ib1,ib3,ncest,nrint,nreg,nx,tx,
* ny,ty,c,fp,wrk1(1),wrk1(lfp),wrk1(lco),wrk1(lf),wrk1(lff),
* wrk1(la),wrk1(lq),wrk1(lbx),wrk1(lby),wrk1(lsx),wrk1(lsy),
* wrk1(lh),iwrk(ki),iwrk(kn),wrk2,lwrk2,ier)
70 return
end