subroutine regrid(iopt,mx,x,my,y,z,xb,xe,yb,ye,kx,ky,s,
* nxest,nyest,nx,tx,ny,ty,c,fp,wrk,lwrk,iwrk,kwrk,ier)
c given the set of values z(i,j) on the rectangular grid (x(i),y(j)),
c i=1,...,mx;j=1,...,my, subroutine regrid 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 regrid calculates the least-squares spline according
c 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 ((z(i,j)-s(x(i),y(j))))**2) be <= s, with s a given non-negative
c 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 regrid(iopt,mx,x,my,y,z,xb,xe,yb,ye,kx,ky,s,nxest,nyest,
c * nx,tx,ny,ty,c,fp,wrk,lwrk,iwrk,kwrk,ier)
c
c parameters:
c iopt : integer flag. on entry iopt must specify whether a least-
c squares spline (iopt=-1) or a smoothing spline (iopt=0 or 1)
c 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 and
c s.ne.0.
c unchanged on exit.
c mx : integer. on entry mx must specify the number of grid points
c along the x-axis. mx > kx . unchanged on exit.
c x : real array of dimension at least (mx). before entry, x(i)
c must be set to the x-co-ordinate of the i-th grid point
c along the x-axis, for i=1,2,...,mx. these values must be
c supplied in strictly ascending order. unchanged on exit.
c my : integer. on entry my must specify the number of grid points
c along the y-axis. my > ky . unchanged on exit.
c y : real array of dimension at least (my). before entry, y(j)
c must be set to the y-co-ordinate of the j-th grid point
c along the y-axis, for j=1,2,...,my. these values must be
c supplied in strictly ascending order. unchanged on exit.
c z : real array of dimension at least (mx*my).
c before entry, z(my*(i-1)+j) must be set to the data value at
c the grid point (x(i),y(j)) for i=1,...,mx and j=1,...,my.
c 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,i=1,...,mx; yb<=y(j)<=ye,j=1,...,my.
c 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 = mx/2, nyest=my/2, will
c be sufficient. always large enough are nxest=mx+kx+1, nyest=
c my+ky+1, the number of knots needed for interpolation (s=0).
c see also further comments.
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 sum of squared
c residuals of the spline approximation returned.
c wrk : real array of dimension (lwrk). used as workspace.
c if the computation mode iopt=1 is used the values of wrk(1),
c ...,wrk(4) should be left unchanged 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.
c lwrk must not be too small.
c lwrk >= 4+nxest*(my+2*kx+5)+nyest*(2*ky+5)+mx*(kx+1)+
c my*(ky+1) +u
c where u is the larger of my and nxest.
c iwrk : integer array of dimension (kwrk). used as workspace.
c if the computation mode iopt=1 is used the values of iwrk(1),
c ...,iwrk(3) should be left unchanged between subsequent calls
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 >= 3+mx+my+nxest+nyest.
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 least-squares
c polynomial of degrees kx and ky. in this extreme case fp
c gives the upper bound for the smoothing factor s.
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 least-squares spline
c according to the current set of knots. the parameter fp
c gives the corresponding sum of 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 sum of squared residuals does not satisfy the condition
c 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 sum of squared residuals does not satisfy the condition
c 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<=kx,ky<=5, mx>kx, my>ky, nxest>=2*kx+2,
c nyest>=2*ky+2, kwrk>=3+mx+my+nxest+nyest,
c lwrk >= 4+nxest*(my+2*kx+5)+nyest*(2*ky+5)+mx*(kx+1)+
c my*(ky+1) +max(my,nxest),
c xb<=x(i-1)=0: s>=0
c if s=0 : nxest>=mx+kx+1, nyest>=my+ky+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 regrid does not allow individual weighting of the data-values.
c so, if these were determined to widely different accuracies, then
c perhaps the general data set routine surfit should rather be used
c in spite of efficiency.
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 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 accuracy of the data values.
c if the user has an idea of the statistical errors on the data, he
c can also find a proper estimate for s. for, by assuming that, if he
c specifies the right s, regrid will return a spline s(x,y) which
c exactly reproduces the function underlying the data he can evaluate
c the sum((z(i,j)-s(x(i),y(j)))**2) to find a good estimate for this s
c for example, if he knows that the statistical errors on his z(i,j)-
c values is not greater than 0.1, he may expect that a good s should
c have a value not larger than mx*my*(0.1)**2.
c if nothing is known about the statistical error in z(i,j), s must
c be determined by trial and error, taking account of the comments
c above. the best is then to start with a very large value of s (to
c determine the least-squares polynomial and the corresponding upper
c bound fp0 for s) and then to progressively decrease the value of s
c ( say by a factor 10 in the beginning, i.e. s=fp0/10,fp0/100,...
c and more carefully as the approximation shows more detail) to
c 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 regrid 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 regrid once more with the selected value for s but now with iopt=0.
c indeed, regrid 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 regrid 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,fpregr,fpdisc,fpgivs,fpgrre,fprati,fprota,fpchec,
c fpknot
c
c references:
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. : a fast algorithm for smoothing data on a rectangular
c grid while using spline functions, report tw53, dept.
c 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 1989
c
c ..
c ..scalar arguments..
real xb,xe,yb,ye,s,fp
integer iopt,mx,my,kx,ky,nxest,nyest,nx,ny,lwrk,kwrk,ier
c ..array arguments..
real x(mx),y(my),z(mx*my),tx(nxest),ty(nyest),
* c((nxest-kx-1)*(nyest-ky-1)),wrk(lwrk)
integer iwrk(kwrk)
c ..local scalars..
real tol
integer i,j,jwrk,kndx,kndy,knrx,knry,kwest,kx1,kx2,ky1,ky2,
* lfpx,lfpy,lwest,lww,maxit,nc,nminx,nminy,mz
c ..function references..
integer max0
c ..subroutine references..
c fpregr,fpchec
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(kx.le.0 .or. kx.gt.5) go to 70
kx1 = kx+1
kx2 = kx1+1
if(ky.le.0 .or. ky.gt.5) go to 70
ky1 = ky+1
ky2 = ky1+1
if(iopt.lt.(-1) .or. iopt.gt.1) go to 70
nminx = 2*kx1
if(mx.lt.kx1 .or. nxest.lt.nminx) go to 70
nminy = 2*ky1
if(my.lt.ky1 .or. nyest.lt.nminy) go to 70
mz = mx*my
nc = (nxest-kx1)*(nyest-ky1)
lwest = 4+nxest*(my+2*kx2+1)+nyest*(2*ky2+1)+mx*kx1+
* my*ky1+max0(nxest,my)
kwest = 3+mx+my+nxest+nyest
if(lwrk.lt.lwest .or. kwrk.lt.kwest) go to 70
if(xb.gt.x(1) .or. xe.lt.x(mx)) go to 70
do 10 i=2,mx
if(x(i-1).ge.x(i)) go to 70
10 continue
if(yb.gt.y(1) .or. ye.lt.y(my)) go to 70
do 20 i=2,my
if(y(i-1).ge.y(i)) go to 70
20 continue
if(iopt.ge.0) go to 50
if(nx.lt.nminx .or. nx.gt.nxest) go to 70
j = nx
do 30 i=1,kx1
tx(i) = xb
tx(j) = xe
j = j-1
30 continue
call fpchec(x,mx,tx,nx,kx,ier)
if(ier.ne.0) go to 70
if(ny.lt.nminy .or. ny.gt.nyest) go to 70
j = ny
do 40 i=1,ky1
ty(i) = yb
ty(j) = ye
j = j-1
40 continue
call fpchec(y,my,ty,ny,ky,ier)
if(ier) 70,60,70
50 if(s.lt.0.) go to 70
if(s.eq.0. .and. (nxest.lt.(mx+kx1) .or. nyest.lt.(my+ky1)) )
* go to 70
ier = 0
c we partition the working space and determine the spline approximation
60 lfpx = 5
lfpy = lfpx+nxest
lww = lfpy+nyest
jwrk = lwrk-4-nxest-nyest
knrx = 4
knry = knrx+mx
kndx = knry+my
kndy = kndx+nxest
call fpregr(iopt,x,mx,y,my,z,mz,xb,xe,yb,ye,kx,ky,s,nxest,nyest,
* tol,maxit,nc,nx,tx,ny,ty,c,fp,wrk(1),wrk(2),wrk(3),wrk(4),
* wrk(lfpx),wrk(lfpy),iwrk(1),iwrk(2),iwrk(3),iwrk(knrx),
* iwrk(knry),iwrk(kndx),iwrk(kndy),wrk(lww),jwrk,ier)
70 return
end