subroutine curfit(iopt,m,x,y,w,xb,xe,k,s,nest,n,t,c,fp, * wrk,lwrk,iwrk,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 curfit determines a smooth spline c approximation of degree k on the interval xb <= x <= xe. c if iopt=-1 curfit calculates the weighted least-squares spline c according to a given set of knots. c if iopt>=0 the number of knots of the spline s(x) and the position c t(j),j=1,2,...,n is chosen automatically by the routine. the smooth- c ness of s(x) is then achieved by minimalizing the discontinuity c jumps of the k-th derivative of s(x) 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)*(y(i)-s(x(i))))**2) be <= s, with s a given non- c negative constant, called the smoothing factor. c the fit s(x) is given in the b-spline representation (b-spline coef- c ficients c(j),j=1,2,...,n-k-1) and can be evaluated by means of c subroutine splev. c c calling sequence: c call curfit(iopt,m,x,y,w,xb,xe,k,s,nest,n,t,c,fp,wrk, c * lwrk,iwrk,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= c 0 or 1) must be determined. if iopt=0 the routine will start c with an initial set of knots t(i)=xb, t(i+k+1)=xe, i=1,2,... c k+1. if iopt=1 the routine will continue with the knots c 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 > k. 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 see also further comments. c xb,xe : real values. on entry xb and xe must specify the boundaries c of the approximation interval. xb<=x(1), xe>=x(m). c unchanged on exit. c k : integer. on entry k must specify the degree of the spline. 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 spline 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+k+1, 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 spline approximation 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 successful exit, this array will contain the knots of the c spline,i.e. the position of the interior knots t(k+2),t(k+3) c ...,t(n-k-1) as well as the position of the additional knots c t(1)=t(2)=...=t(k+1)=xb and t(n-k)=...=t(n)=xe 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 c : real array of dimension at least (nest). c on successful exit, this array will contain the coefficients c c(1),c(2),..,c(n-k-1) in the b-spline representation of s(x) c fp : real. unless ier=10, fp contains the weighted sum of c squared residuals of the spline approximation returned. c wrk : real array of dimension at least (m*(k+1)+nest*(7+3*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 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 degree k. in this extreme case fp c gives the upper bound fp0 for the 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 weighted least-squares c spline 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 process for finding a smoothing spline 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 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=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>k, nest>2*k+2, w(i)>0,i=1,2,...,m c xb<=x(1)<x(2)<...<x(m)<=xe, lwrk>=(k+1)*m+nest*(7+3*k) c if iopt=-1: 2*k+2<=n<=min(nest,m+k+1) c xb<t(k+2)<t(k+3)<...<t(n-k-1)<xe c the schoenberg-whitney conditions, i.e. there c must be a subset of data points xx(j) such that c t(j) < xx(j) < t(j+k+1), j=1,2,...,n-k-1 c if iopt>=0: s>=0 c if s=0 : nest >= m+k+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 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 of degree k 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 y(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 y(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 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 curfit 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. but, 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 curfit once more with the selected value for s but now with iopt=0. c indeed, curfit 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 other subroutines required: c fpback,fpbspl,fpchec,fpcurf,fpdisc,fpgivs,fpknot,fprati,fprota c c references: c dierckx p. : an algorithm for smoothing, differentiation and integ- c ration of experimental data using spline functions, c j.comp.appl.maths 1 (1975) 165-184. 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. : an improved algorithm for curve fitting with spline c functions, report tw54, dept. computer science,k.u. c 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 ..

Name | Description |
---|---|

constructor | create a 1D-Spline object |

destructor | remove a 1D-Spline object |

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