HP 15c User Manual

Page 158

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158

Appendix: Accuracy of Numerical Calculations

158

Some transformations f are stable in the presence of input noise; they keep Δy relatively
small as long as Δx is relatively small. Other transformations f may be unstable in the
presence of noise because certain relatively small input noises Δx cause relatively huge
perturbations Δy in the output. In general, the input noise Δx will be colored in some way by
the intended transformation (on the way from input to output noise Δy, and no diminution in
Δy is possible without either diminishing Δx or changing f. Having accepted f as a
specification for performance or as a goal for design, we must acquiesce to the way f colors
noise at its input.

The real system F differs from the intended f 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

F(x) = (f + δf)(x)

where the perturbation δf represents the internal noise in F.

We hope the noise term δf is negligible compared with f. When that hope is fulfilled, we
classify F in Level 1 for the purposes of exposition; this means that the noise internal to F
can be explained as one small addition of δf to the intended output f.

For example F(x) = N(x) is classified in Level 1 because the dozens of small errors
committed by the HP-15C during its calculation of F(x) = (f + δf)(x) amounts to a
perturbation of δf(x) smaller than 0.6 in the last (10th) significant digit of the desired output
f(x)
= ln(x). But F(x) = [(x) is not in Level 1 for radian x because F(x) can differ too
much from f(x) = sin(x); for instance F(10

14

$) = 0.799... is opposite in sign from

f(10

14

$)= −0.784…, so the equation F(x) = (f + δf)(x) can be true only if δf is sometimes

rather bigger than f, which looks bad.

Real systems more often resemble [than N. Noise in most real systems can
accumulate occasionally to swamp the desired output. at least for some inputs. and yet such
systems do not necessarily deserve condemnation. Many a real system F operates reliably
because its internal noise, though sometimes large, never causes appreciably more harm than
might be caused by some tolerably small perturbation δx to the input signal x. Such systems
can be represented as

F(x) = (f + δf) (x + δx)

where δf is always small compared with f and δx is always smaller than or comparable with
the noise Δx expected to contaminate x. The two noise terms δf and δx are hypothetical noises

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