MEMO:                     GCVSPL software package
          (C) COPYRIGHT 1985, 1986: H.J. Woltring
                                    Philips Medical Systems Division, Eindhoven
                                    University of Nijmegen (The Netherlands)

          DATE:                     1986-05-12

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          NB:  This software is copyrighted, and may be  copied  for  excercise,
          study  and  use  without authorization from the copyright owner(s), in
          compliance with paragraph 16b of  the  Dutch  Copyright  Act  of  1912
          ("Auteurswet  1912").  Within the constraints of this legislation, all
          forms of academic and research-oriented excercise, study, and use  are
          allowed,  including  any  necessary modifications.  Copying and use as
          object for commercial exploitation are not allowed without  permission
          of  the  copyright owners, including those upon whose work the package
          is based.

          A full description of the package is provided in:

          H.J. Woltring (1986), A FORTRAN package for generalized, cross-valida-
          tory  spline  smoothing  and differentiation.  Advances in Engineering
          Software 8(2):104-113 (U.K.).

          For large datasets (N >> 0) and  negligible  boundary  artefacts,  the
          behaviour  of a natural spline approximates that of a periodic spline.
          For the latter case, the frequency characteristic in the equidistantly
          sampled,  uniformly weighted case is that of a double, phase-symmetric
          Butterworth filter, with transfer function H(w) = [1 +  (w/wo)^2M]^-1,
          where  w  is  the  frequency, wo = (p*T)^(-0.5/M) the filter's cut-off
          frequency, p the smoothing parameter, T the sampling interval, and  2M
          the  order  of the spline.  If T is expressed in seconds, the frequen-
          cies are expressed in radians/second.

          It has been found empirically, that the effective number of  estimated
          spline  parameters  Np is related to the Butterworth cut-off frequency
          wo as Np ~ M/2 + KM * wo * N * T, where Np ranges between M and N, and
          where  KM  is  the integral over x from 0 to infinity of (1 + x^2M)^-1
          divided by PI.  For large M, KM approaches 1/PI from above; values for
          small  M  are:  K1 = 1/2, K2 = 1/V8, K3 = 1/3.  This relation has also
          been found to apply for uniformly  weighted  data  which  are  sampled
          slightly  anequidistantly, with T taken as the average sampling inter-
          val.  For large Np, the relation with wo * N * T becomes nonlinear.

          A simple test-programme GCV is provided with the  package.   The  test
          data (a simple parabolic time signal) are not entirely appropriate for
          smoothing and differentiation by means of natural splines:   for  low-
          order  splines,  boundary artefacts prevail, while low-frequency noise
          and the low-frequency signal are confounded  for  high-order  splines.
          Smoothing  via the optimization criteria in the GCVSPL package assumes
          a sufficiently large signal-to-noise ratio and number of measurements,
          a  sufficiently strong m-th signal derivative, and wide-band, uncorre-
          lated noise; these conditions are not fully  met  by  the  given  test
          data.   Perusal  of  GCV for different spline orders, smoothing modes,
          and numbers of observations will allow the user  to  acquaint  himself
          with the GCVSPL package.