subroutine kvtx_known_lifetime(ktrk, update_track, vtx0, Vvtx0,
     *                               vtx, ctau, sig_ctau, chisq, error)
*
* begin_doc
*
*  Given a track and a starting point with a known covariance matrix,
*  do a vertex fit that forces the track to pass through the starting
*  point. The starting point is updated and c*tau (tau = proper time)
*  of the particle and its error are returned.
*
*  Note 1: You can get the complete 4x4 error matrix of the starting
*          point + ctau together by calling kget_lifetime_covar.
*
*  Note 2: If the track is not updated here, you can update it later by
*          calling kfit_update_tracks.
*
*  Input Parameters:
*  ktrk          integer variable
*                Track number to fit
*
*  update_track  integer variable
*                 0 ==> Calculate chisquare only (do not update track)
*                 1 ==> Calculate chisquare, update track
*                 2 ==> Calculate chisquare, update track & cov matrix
*
*  vtx0(3)       double precision array
*                Prior starting point (not updated)
*
*  Vvtx0(3,3)    double precision array
*                Covariance matrix of vtx0(1-3)
*
*  Output Parameters:
*  vtx(3)        double precision array
*                Updated starting point
*
*  ctau          double precision variable
*                c*tau of the particle
*
*  sig_ctau      double precision variable
*                Error on ctau
*
*  chisq         double precision variable
*                Chisquare of fit
*
*  error         integer variable
*                 0 ==> All OK
*                >0 ==> Error
*
*  Other routines:
*
*  Notes:
*
*  >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
*   The constraint is imposed by the method of Lagrange multipliers. For
*   this case I define the alpha vector as
*
*            alpha = (w,z)
*
*   i.e., the track and starting point coordinates, a total of 10
*   components.
*
*   I use the three equations of motion as the constraint equations from
*   which one unknown, ctau, must be determined, giving 2 degrees of
*   freedom. At the end I construct the 4x4 covariance matrix of
*   (z,ctau) and store it for future reference.
*  >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
*
*  Author:   Paul Avery      Created:  Fri Aug 29 16:12:52 EDT 1997
*
*  Major revisions:
*
*
* end_doc