e02daf

Purpose

Synopsis

[lamda,mu,dl,c,sigma,rank,ifail] = e02daf(x,y,f,lamda,mu,point<,w,nc,epslon,...
ifail>)

Description

 
 This routine determines a bicubic spline fit s(x,y) to the set of
 data points (x ,y ,f ) with weights w , for r=1,2,...,m. The two 
               r  r  r                r                          
 sets of internal knots of the spline, {(lambda)} and {(mu)}, 
 associated with the variables x and y respectively, are 
 prescribed by the user. These knots can be thought of as dividing
 the data region of the (x,y) plane into panels. A bicubic spline 
 consists of a separate bicubic polynomial in each panel, the 
 polynomials joining together with continuity up to the second 
 derivative across the panel boundaries.
 
 s(x,y) has the property that (Sigma), the sum of squares of its 
 weighted residuals (rho) , for r=1,2,...,m, where
                         r                   
 
                   (rho) =w (s(x ,y )-f ),                     (1)
                        r  r    r  r   r                     
 
 is as small as possible for a bicubic spline with the given knot 
 sets. The routine produces this minimized value of (Sigma) and 
 the coefficients c   in the B-spline representation of s(x,y). 
                   ij                                           
 E02DEF and E02DFF are available to compute values of the fitted 
 spline from the coefficients c  .
                               ij
 
 The least-squares criterion is not always sufficient to determine
 the bicubic spline uniquely: there may be a whole family of 
 splines which have the same minimum sum of squares. In these 
 cases, the routine selects from this family the spline for which 
 the sum of squares of the coefficients c   is smallest: in other 
                                         ij                      
 words, the minimal least-squares solution. This choice, although 
 arbitrary, reduces the risk of unwanted fluctuations in the 
 spline fit.

Parameters

e02daf

Required Input Arguments:

x (:)                                 real
y (:)                                 real
f (:)                                 real
lamda (:)                             real
mu (:)                                real
point (:)                             integer

Optional Input Arguments:                       <Default>

w (:)                                 real     ones(length(x),1)
nc                                    integer  (length(lamda)-...
                                        ...    4)*(length(mu)-4)
epslon                                real     eps
ifail                                 integer  -1

Output Arguments:

lamda (:)                             real
mu (:)                                real
dl (nc)                               real
c (nc)                                real
sigma                                 real
rank                                  integer
ifail                                 integer