Function:     SKYFIT

Purpose:      SKYFIT is a routine for determining the sky-projection
              parameters. It takes a set of grid positions and a set
              of sky coordinates to fit the parameters.
              The method is a mixture of the steepest descent method and
              the Taylor method.

Category:     MATH

File:         skyfit.c

Author:       K.G. Begeman

Use:          INTEGER SKYFIT( X    ,    Input      DOUBLE PRECISION ARRAY
                              Y    ,    Input      DOUBLE PRECISION ARRAY
                              A    ,    Input      DOUBLE PRECISION ARRAY
                              D    ,    Input      DOUBLE PRECISION ARRAY
                              N    ,    Input      INTEGER
                              XF   ,    Output     DOUBLE PRECISION ARRAY
                              YF   ,    Output     DOUBLE PRECISION ARRAY
                              FPAR ,   In/Output   DOUBLE PRECISION ARRAY
                              EPAR ,    Output     DOUBLE PRECISION ARRAY
                              MPAR ,    Input      INTEGER ARRAY
                              TOL  ,    Input      REAL
                              ITS  ,    Input      INTEGER
                              LAB  ,    Input      REAL
                              PROJ )    Input      INTEGER

              SKYFIT   Returns number of iterations needed to achieve
                       convergence according to TOL. When this
                       number is negative, the fitting was not
                       continued because a fatal error occurred:
                       -1 Unknown projection.
                       -2 No free parameters.
                       -3 Not enough degrees of freedom.
                       -4 Maximum number of iterations too small to
                          obtain a solution which satisfies TOL.
                       -5 Diagonal of matrix contains elements which
                          are zero.
                       -6 Determinant of the coefficient matrix is zero.
                       -7 Square root of negative number.
              X        Contains x grid coordinates.
              Y        Contains y grid coordinates.
              XF       Contains fitted x grid coordinates.
              YF       Contains fitted x grid coordinates.
              A        Contains longitude coordinates.
              D        Contains latitude coordinates.
              N        Number of positions.
              FPAR     On input contains initial estimates of the
                       parameters, on output the fitted parameters.
                       There are seven parameters, so the dimension of
                       FPAR (and EPAR and MPAR) must be seven.
                       The order of the parameters is the following:
                       1: longitude projection centre
                       2: latitude projection centre
                       3: rotation angle
                       4: grid size x grid
                       5: grid size y grid
                       6: reference x grid position
                       7: reference y grid position
              EPAR     Contains estimates of errors in fitted
                       parameters.
              MPAR     Logical mask telling which parameters are
                       free (MPAR(J)=non-zero) and which parameters
                       are fixed (MPAR(J)=0).
              TOL      Relative tolerance. LSQFIT stops when
                       successive iterations fail to produce a
                       decrement in reduced chi-squared less than
                       TOL. If TOL is less than the minimum tolerance
                       possible, TOL will be set to this value. This
                       means that maximum accuracy can be obtained by
                       setting TOL=0.0.
              ITS      Maximum number of iterations.
              LAB      Mixing parameter, LAB determines the initial
                       weight of steepest descent method relative to
                       the Taylor method. LAB should be a small
                       value (i.e. 0.01).
              PROJ     Projection type.
                       2: Cylindrical projection.
                       3: Flat projection.
                       4: Gnomonic projection.
                       5: Orthographic projection.
                       6: Rectangular projection.
                       7: Global Sinusoidal projection.
                       8: North Celestial Pole projection.
                       9: Stereographic projection.

Updates:      Jul  6, 1991: KGB, Document created.
              Aug 23, 1993: VOG, Output of fitted grid coordinates.