HP 15c - Manual (Page 159)

Appendix: Accuracy of Numerical Calculations 159
introduced to explain diverse noise sources actually distributed throughout F. Some of the
noise appears as a tolerably small perturbation δx to the input
— hence the term "backward
error analysis." Such a system F, whose noise can be accounted for by two tolerably small
perturbations, is therefore classified into Level 2 for purposes of exposition.
No difference will be perceived at first between Level 1 and Level 2 by readers accustomed
to linear systems and small signals because such systems' errors can be referred
indiscriminately to output or input. However, other more general systems that are digital or
nonlinear do not admit arbitrary reattribution of output noise to input noise nor vice-versa.
For example, can all the error in \ be attributed, merely by writing
\ (x) = cos(x + δx), to an input perturbation δx small compared with the input x? Not
when x is very small. For instance, when x approaches 10
-5
radians, then cos(x) falls very
near 0.99999999995 and must then round to either 1 = cos(0) or 0.9999999999
= cos(1.414 ... × 10
-5
). Therefore \ (x) = cos(x + δx) is true only if δx is allowed to be
relatively large, nearly as large as x when x is very small. If we wish to explain the error in
\ by using only relatively small perturbations, we need at least two of them: one a
perturbation δx = (−6.58 ... × 10
-14
) x smaller than roundoff in the input; and another in the
output comparable with roundoff there, so that \ (x) = (cos + δcos)(x + δx) for some
unknown |δcos| ≤ (6×10
-10
)|cos|.
Like \ every system F in Level 2 is characterized by just two small tolerances—call
themεand η—that sum up all you have to know about that system's internal noise. The
tolerance ε constrains a hypothetical output noise, |δf| ≤ε|f|, and η constrains a hypothetical
input noise, |δx| ≤ η|x|, that might appear in a simple formula like
F(x) = (f + δf) (x + δx) for |δf| ≤ε|f| and |δx| ≤ η|x|.
The goal of backward error analysis is to ascertain that all the internal noise of F really can
be encompassed by so simple a formula with satisfactorily small tolerancesεand η. At its
best, backward error analysis confirms that the realized value F(x) scarcely differs from the
ideal value f(x + δx) that would have been produced by an input x + δx scarcely different
from the actual input x, and gives the word "scarcely" a quantitative meaning (εand η). But,
backward error analysis succeeds only for systems F designed very carefully to ensure that
every internal noise source is equivalent at worst to a tolerably small input or output
perturbation. First attempts at system design, especially programs to perform numerical
computations, often suffer from internal noise in a more complicated and disagreeable way
illustrated by the following example.