Appendix A. "Modification of the spectrum scaling method"

   In the monitoring programs of AGN, the spectral scaling scheme
described by van Groningen & Wanders (1992) is used. The
main idea of the algorithm is the creation of the difference
spectrum between an input  spectrum and a reference spectrum, for
which the flux is assumed to be  constant. The difference
spectrum is represented by the simple analytical function (usually
by a 2nd order polynomial). Then a {chi}^2 of this correspondence
is minimized by a grid search method by successive variations of 3
input parameters: a flux scaling factor, a wavelength shift, and a
difference in resolution of the spectrum ({Delta} FWHM). For the
latter, a convolution with Gaussian function is selected. For data
with the S/N ratio ~10, the error of the flux scaling factor
is <5%. However, the method of van Groningen & Wanders
(1992) is unstable to the selection of initial parameters (the zero 
approximation is done manually). In  order to circumvent
this problem, we have modified the method. The difference between
the individual spectra (obj) and the ``reference'' one (ref) is
represented by a 3rd degree polynomial, and for the minimization
of the differences, a downhill simplex method by Nelder and Mead
(1965) is used.  The latter is more stable than the grid
search method used by van Groningen & Wanders (1992). As a
zero approximation, the flux  in the spectrum lines was determined
automatically after subtraction of a linear continuum determined
by the beginning and the end of a  given spectral interval. The
scaling procedure is then carried out with the program means in
IDL, the program is fast and stable. The program output is similar
to that of van Groningen & Wanders (1992): the flux scaling
factor, relative wavelength shift and  Gaussian width which is
used for convolution with one of the spectra for spectral
resolution correction, values of {chi}^2 and  {sigma} for the
power approximation to the spectrum difference, the scaled and
difference spectra (obj-ref). The latter are obtained after
reducing the spectra to the same spectral  resolution. In order to
check the correctness of the scaling method several tests have
been carried out.

   1. We have tested the accuracy in the determination of the flux scaling
factor by means of a model spectrum of different S/N ratios.  As
model data, we adopted the result of a multi Gaussian approximation to
the Jan 21,1998  NGC 5548 spectrum, for the wavelength interval
(4700-5400){AA} including the [OIII]{lambda}{lambda} 4959, 5007 and 
H{beta} emission lines.

   In testing, the oxygen line intensities remained constant while the
H{beta} intensity was varied between 10 and 500 percent (H{beta}=100%
corresponded to the observed spectrum of NGC 5548 in Jan 21
1998). A continuum level was added to the lines and the spectrum
was convolved with grey noise, to a chosen value of the S/N ratio.
200 experiments were carried out for each of the selected S/N
ratios. From the simulated spectra with S/N=20, the average values of
the flux scaling factor show systematic decrease from 1.015 to
0.995 for changes of the H{beta} intensity from 10% to 500%.
The dispersion of the flux scaling factor being about 2.5% 
As the S/N ratio is increased to 40 (typical value for
spectra obtained in our monitoring campaign), systematic errors
and dispersion values went down to 0.5% and 1% respectively.

   2. With the  model spectra we tested the method accuracy for the
determination of the flux scaling factor in the case in which
spectra of  different resolution are compared. In this case the
model spectrum adopted is the same as in case 1., with H{beta}
intensity 100%, but before introducing noise, a convolution with
a Gaussian of the proper width is done. The value of the Gaussian
dispersion ({sigma}=FWHM/2.35) varied from 0.01 to 5.59 pixels
with steps of 0.01 pixel (dispersion being 2{AA}/px). We found
that for spectra with S/N ratio ~20, the value of {sigma}>=0.51 
and the flux scaling factor varies by less than 1%.
We also checked  for changes of the flux scaling factor in the
observed blue spectra of NGC 5548. For this purpose, we first
scaled the spectra adopting a spectrum with a resolution (~8{AA}) 
as a referenced one. Then, the scaled spectra were
processed by the program adopting a spectrum of lower resolution
(~15{AA}) as a the reference one. The values for the flux
scaling factors so derived, coincided within an accuracy of 1%.

   3. The wavelength shift ({Delta}{lambda}) is always recovered with
high accuracy by the method.

   Thus  by scaling our AGN spectra by means of the modified method
of van Groningena & Wanders (1992) as described above, one
can  obtain correct values of the scale parameters, their errors
being dependent only on the quality of the spectra.

References:

Van Groningen, E. & Wanders, I., 1992PASP..104..700V

Nelder, J.A. & Mead, R., 1965, Computer Journal, 7, 308