HP HP-15C Advanced Functions Handbook

192 Appendix: Accuracy of Numerical Calculations
For matrix operations, the magnitudes \dx\, x\ \df\, and |/| must be
replaced by matrix norms ||<5x||, ||x||, ||<5f||, and ||f|| respectively,
which are explained in section 4 and evaluated using | MATRIX]7 or
| MATRIX] 8. Then all matrix functions not in Level 1 fall into Level 2
with roughly
17 s? 10~12tt and e < 10~9 for matrix operations (other than
determinant I MATRIX!9, 0, and|l/x|)
77 sg 10~9rc and e < 1(T9 for determinant | MATRIX]9, M/x|.
and 0 with a matrix divisor
where n is the largest dimension of any matrix involved in the
operation.
The implications of successful backward error analysis look simple
only when the input data x comes contaminated by unavoidable
and uncorrelated noise Ax, as is often the case. Then when we wish
to calculate f(x), the best we could hope to get is f(x + Ax), but we
actually get F(x + Ax) = (f+8f)(x + Ax + dx), where \6f\ e\f\d
\8x\^r,\x\.
What we get is scarcely worse than the best we could hope for
provided the tolerances e and r? are small enough, particularly if
| Ax\s likely to be at least roughly as big as rj\x\. Of course, the best
we could hope for may be very bad, especially if f possesses a
singularity closer to x than the tolerances upon x's perturbations
Ax and 8x.
Backward Error Analysis Versus Singularities
The word "singularity" refers to both a special value of the
argument x and to the way f(x) misbehaves as x approaches that
special value. Most commonly, f(x) or its first derivative f'(x) may
become infinite or violently oscillatory as x approaches the
singularity. Sometimes the singularities of ln|/| are called
singularities of /, thereby including the zeros of / among its
singularities; this makes sense when the relative accuracy of a
computation of / is at issue, as we shall see. For our purposes the
meaning of "singularity" can be left a little vague.
What we usually want to do with singularities is avoid or
neutralize them. For instance, the function
!
(l-cosx)/x2 ifx^O
1/2 otherwise

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