l2appr Subroutine

The routine constructs the weighted discrete L2-approximation by 
splines of order K with knot sequence T(1:N+K) to 
given data points ( TAU(1:NTAU), GTAU(1:NTAU) ).

The B-spline coefficients BCOEF of the approximating spline are 
determined from the normal equations using Cholesky's method.

The B-spline coefficients of the L2-approximation are determined as the 
solution of the normal equations, for 1 <= I <= N:

  sum ( 1 <= J <= N ) ( B(I), B(J) ) * BCOEF(J) = ( B(I), G ).

Here, B(I) denotes the I-th B-spline, G denotes the function to
be approximated, and the inner product of two functions F and G 
is given by

  ( F, G ) = sum ( 1 <= I <= NTAU ) WEIGHT(I) * F(TAU(I)) * G(TAU(I)).

The arrays TAU and WEIGHT are given in common block DATA, as is the 
array GTAU(1:NTAU) = G(TAU(1:NTAU)).

The values of the B-splines B(1:N) are supplied by BSPLVB.

The coefficient matrix C, with

   C(I,J) = ( B(I), B(J) )

of the normal equations is symmetric and (2*K-1)-banded, therefore
can be specified by giving its K bands at or below the diagonal.

For I = 1:N and J = I:min(I+K-1,N), we store

  ( B(I), B(J) ) = C(I,J)

in

  Q(I-J+1,J),

and the right hand side

  ( B(I), G )

in

  BCOEF(I).

Since B-spline values are most efficiently generated by finding
simultaneously the value of every nonzero B-spline at one point,
the entries of C (that is, of Q), are generated by computing, for
each LL, all the terms involving TAU(LL) simultaneously and adding
them to all relevant entries.

14 February 2007

Carl DeBoor

Carl DeBoor,
A Practical Guide to Splines,
Springer, 2001,
ISBN: 0387953663,
LC: QA1.A647.v27.

Input, real ( kind = 8 ) T(N+K), the knot sequence.

Input, integer ( kind = 4 ) N, the dimension of the space of splines 
of order K with knots T.

Input, integer ( kind = 4 ) K, the order of the splines.

Workspace, real ( kind = 8 ) Q(K,N), used to store the K lower 
diagonals of the Gramian matrix C.

Workspace, real ( kind = 8 ) DIAG(N), used in BCHFAC.

Output, real ( kind = 8 ) BCOEF(N), the B-spline coefficients of
the L2 approximation to the data.