exponential growth

[David Hough]

> Is it not the case that the
> naive algorithm if implemented in real arithmetic would experience the
> same kind of exponential error growth? But, without interval arithmetic,
> the naive user might not be alerted to the fact that errors have grown
> exponentially. Without this warning, he may not be motivated to search
> for a more elegant solution.

As I recall, this example did not feature exponential growth
of error.  It featured an exponential growth of error BOUND if
the bound were computed by a naive application of interval arithmetic.

As a simplified example consider a particle circling the origin under gravity.
Small uncertainty in its initial position and velocity might manifest
itself as a solution set consisting of a skinny arc of a portion of
a circle.  Expressing the uncertainty in terms of uncertainty in the
radius and argument, the error bounds would grow slowly, compared to
expressing the uncertainty in the x and y coordinates centered on the
origin, which must necessarily grow quickly.  But even error bounds
on radius and argument aren't good enough; for the growth of the error
bound to match the growth of the error, an elliptical representation
is required.

So a naive application of interval arithmetic would only serve to
unduly alarm the user.  The art in expert mathematical software
for interval analysis is that required to keep the computed error bounds
growing not much faster than the actual error.