HP HP-15C

188 Appendix: Accuracy of Numerical Calculations
i would generallythen in consequence the output y+Ay = f(
be contaminated by noise Ay = f(x + Ax) — f(x).
Ax
= f(x)
No Noise
Noisy Input
Some transformations / are stable in the presence of input noise;
they keep Ay relatively small as long as A x is relatively small.
Other transformations / may be unstable in the presence of noise
because certain relatively small input noises Ax cause relatively
huge perturbations Ay in the output. In general, the input noise Art
will be colored in some way by the intended transformation f on the
way from input to output noise Ay, and no diminution in Ay is
possible without either diminishing Ax or changing /. Having
accepted / as a specification for performance or as a goal for
design, we must acquiesce to the way / colors noise at its input.
The real system F differs from the intended / because of noise or
other discrepancies inside F. Before we can appraise the
consequences of that internal noise we must find a way to
represent it, a notation. The simplest way is to write
where the perturbation 8f represents the internal noise in F.
One Small Output Perturbation (Level 1)
We hope the noise term 6/is negligible compared with /. When that
hope is fulfilled, we classify F in Level 1 for the purposes of