Reality is more difficult. Big trouble can arise from just a modest clustering of poles at a moderate distance from the unit circle. This is shown in Figure 16, where the result is completely wrong.

Figure 16

The spectrum should look like the spectrum in Figure 8 multiplied by itself about six or seven times, once for each pole. The effect of such repetitive multiplication is to make the small spectral values become very small. When I added the last pole to Figure 16, however, the spectrum suddenly became rough. The time response now looks almost divergent. Moving poles slightly creates very different plots. I once had a computer that crashed whenever I included one too many poles.

To understand this,
notice that the peak spectral values in Figure 16
come from the
*minimum*
values of the denominator.
The denominator will not go to a properly small value
if the **precision** of its terms is not adequate
to allow them to extinguish one another.
Repetitive multiplication has caused
the dynamic range
(the range between the largest and smallest amplitudes
as a function of frequency)
of single-precision arithmetic,
about 10^{6}.

When single-word precision becomes a noticeable problem,
the obvious path is to choose double precision.
But considering that most geophysical data has a precision of
less than one part in a hundred, and only rarely do we
see precision of one part in a thousand, we can conclude that
the failure of single-word precision arithmetic,
about one part in 10^{-6},
is more a sign of conceptual failure
than of numerical precision inadequacy.

If an application arises for which you really need an operator that
raises a polynomial to a high degree,
you may be able to accomplish your goal by applying
the operator in stages.
Say, for example, you need the all-pass filter
(.2-*Z*)^{100}/(1-.2*Z*)^{100}.
You should be able to apply this filter in a hundred stages of
(.2-*Z*)/(1-.2*Z*),
or maybe in ten stages of (.2-*Z*)^{10}/(1-.2*Z*)^{10}.

Other ways around this precision problem are suggested by reflection-coefficient modeling in a layered earth, described in FGDP.

10/21/1998