HP HP-15C Advanced Functions Handbook

202 Appendix: Accuracy of Numerical Calculations
As A + AA runs through matrices with || AA|| at least about as large
as roundoff in ||A||, its inverse (A + AA)"1 must roam at least about
as far from A"1 as the distance from A"1 to the computed 11/x|(A).
All these excursions are very small unless A is too near a singular
matrix, in which case the matrix should be preconditioned away
from near singularity. (Refer to section 4.)
If among those neighboring matrices A + AA lurk some that are
singular, then many (A + AA)"1 and 11/x|(A) may differ utterly
from A"1. However, the residual norm will always be relatively
small:
-I|| I|AA|| _ „
This last inequality remains true when 11/xj(A) replaces
(A + AA)'1.
If A is far enough from singularity that all
'I|AA||,
then also
HA"1 - (A + AA)'1! . I|AA||||(A
||(A + AA)-1!! 1 - II AA|| ||(A + AA)-
This inequality also remains true when 11/x|(A) replaces
(A + AA)"1, and then everything on the right-hand side can be
calculated, so the error in 11/x|(A) cannot exceed a knowable
amount. In other words, the radius of the dashed ball in the figure
above can be calculated.
The estimates above tend to be pessimistic. However, to show why
nothing much better is true in general, consider the matrix
X =
0.00002 -50,000 50,000.03 -45
0 50,000 -50,000.03 45
0 0 0.00002 -50,000.03
0 00 52,000

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