e02baf

Purpose

Synopsis

[lamda,c,ss,ifail] = e02baf(x,y,lamda<,w,ifail>)

Description

 
 This routine determines a least-squares cubic spline 
 approximation s(x) to the set of data points (x ,y ) with
                                                r  r       
 weights w , for r=1,2,...,m. The values of the knots 
          r                                                           
 (lambda) ,(lambda) ,...,(lambda)_  , interior to the data 
         5         6             n+3                        
 interval, are prescribed by the user.
 
 s(x) has the property that it minimizes (theta), the sum of 
 squares of the weighted residuals (epsilon) , for r=1,2,...,m, 
                                            r                  
 where
 
                     (epsilon) =w (y -s(x )).
                              r  r  r    r
 
 The routine produces this minimizing value of (theta) and the 
                                    _                   
 coefficients c ,c ,...,c , where q=n+3, in the B-spline 
               1  2      q                              
 representation
 
                               q      
                               --     
                         s(x)= >  c N (x).
                               --  i i
                               i=1    
 
 Here N (x) denotes the normalised B-spline of degree 3 defined 
       i                                                       
 upon the knots (lambda) ,(lambda)   ,...,(lambda)   .
                        i         i+1             i+4
 
 In order to define the full set of B-splines required, eight 
 additional knots (lambda) ,(lambda) ,(lambda) ,(lambda)  and 
                          1         2         3         4    
 (lambda)_  ,(lambda)-  ,(lambda)_  ,(lambda)_   are inserted 
         n+4         n+5         n+6         n+7             
 automatically by the routine. The first four of these are set 
 equal to the smallest x  and the last four to the largest x .
                        r                                   r
 
 The representation of s(x) in terms of B-splines is the most 
                                    _                            
 compact form possible in that only n+3 coefficients, in addition 
        _                           
 to the n+7 knots, fully define s(x).
 
 
 Subsequent evaluation of s(x) from its B-spline representation 
 may be carried out using E02BBF. If derivatives of s(x) are also 
 required, E02BCF may be used. E02BDF can be used to compute the 
 definite integral of s(x).
 

Parameters

e02baf

Required Input Arguments:

x (:)                                 real
y (:)                                 real
lamda (:)                             real

Optional Input Arguments:                       <Default>

w (:)                                 real     ones(length(x),1)
ifail                                 integer  -1

Output Arguments:

lamda (:)                             real
c (:)                                 real
ss                                    real
ifail                                 integer