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